Rings of differential operators and étale
homomorphisms
by
Gı́sli Másson
B.S., University of Iceland (1985)
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1991
c Massachusetts Institute of Technology 1991
Signature of Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Mathematics
May 2, 1991
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michael Artin
Professor of Mathematics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sigurdur Helgason
Chairman, Departmental Graduate Committee
Department of Mathematics
Rings of differential operators and étale homomorphisms
by
Gı́sli Másson
Submitted to the Department of Mathematics
on May 2, 1991, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Rings of differential operators over rings of krull dimension 1 were studied by Musson,
Smith and Stafford, and Muhasky. In particular it was proved that if k is a field of
characteristic zero, and A is a finitely generated reduced k-algebra, then
(i) D(A), the ring of differential operators on A, has a minimal essential twosided
ideal J ,
(ii) A contains a minimal essential left D(A)-submodule, J, and
(iii) the algebras C(A) := A/J and H(A) := D(A)/J are finite dimensional vector
spaces over k.
In this case the algebras C(A) and H(A) were studied by Brown and Smith.
We use étale homomorphisms to obtain similar results in a somewhat more general
setting. While an example of Bernstein, Gelfand and Gelfand shows that the statements
above do not hold for all reduced finitely generated k-algebras of higher krull dimension,
we present sufficient conditions on commutative, noetherian and reduced rings so that
analogues of the ideal J and module J can be constructed, and for which statements
similar to (i) and (ii) hold. Besides being applicable to rings of higher krull dimension,
our method also reveals that the earlier results are a reflection of the (rather natural)
fact that differential operators preserve supports of regular functions.
We then use our results to prove the existence of minimal reflexive modules for rings
of differential operators over domains which satisfy the condition S2 . This is related to
results of Chamarie and Stafford. We also compute the dimension of C(A) (in terms
of intersection multiplicities) when A is a completion of an affine plane curve over an
algebraically closed field of characteristic zero, and we give a description of H(A) for
some algebras A of higher krull dimension, an analogue of a theorem of Brown.
Finally we point out that we give a generalization of the formula for applying differential operators to inverses, extending a result of Nakai and Hart.
Thesis Supervisor: Michael Artin
Title: Professor of Mathematics
2
Acknowledgements
First of all I would like to thank my advisor Michael Artin, who always had the right
ideas, and who set the record straight on countless matters. I would also like to thank
him for the financial support I received as a research assistant the last two years of my
studies.
The mathematics department at MIT I thank for its support and virtually unlimited
resources, and I would in particular like to thank Phyllis Ruby, our graduate administrator, for all the letters and other help which made real life considerably simpler.
I would like to thank all the friends I made during my time at MIT, especially all
my officemates for providing me with good surroundings and nice time on and off work.
Special thanks go to Amnon Yekutieli for all the mathematical discussions and for the
good time in Texas, to Wolfram Gerdes for the beer and the intercontinental cultural
excursions, to Tom Roby for being my primary reference on the english language, and
to Jan Pedersen for the coffee and Danish. You all made my time in the Boston area an
unforgettable experience.
I thank my teachers at the University of Iceland for providing me with a solid undergraduate education, and I also thank them for their hospitality during my stay in Iceland
in the spring and summer of 1990.
All my friends in Iceland I thank for making it a place where I want to return to, and
Hans Jakob Beck and Snorri Bergmann get special thanks for keeping me in the illusion
that every place has the same luxurious computer resources as I got spoiled by at MIT.
My parents, Sigrún Gı́sladóttir and Jóhann Már Marı́usson, deserve very special
thanks for always being supportive of my decisions, and father, this is probably what our
early games with a pencil and graph paper finally lead to.
Finally my very best thanks go to my wife, my companion and my very best friend,
Freyja Hreinsdóttir, for taking care of me and our son, Nökkvi, while I was out working.
I thank her for the mathematics, and for pointing out all the mistakes and the places
where there was room for improvement in my thesis. Last but not least I do thank her
for all the patience and willingness to go along even if it meant that she had to sacrifice
the most. Freyja, I hope that now I can start giving something in return for all your
help.
3
Contents
Introduction
6
1 Definition and first properties of rings of differential operators
9
1.1
1.2
Basic properties of differential operators . . . . . . . . . . . . . . . . . .
9
1.1.1
Endomorphisms, derivations and the derivation ring . . . . . . . .
9
1.1.2
Definition of rings of differential operators . . . . . . . . . . . . .
11
Elementary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.2.1
A collection of basic lemmas . . . . . . . . . . . . . . . . . . . . .
14
1.2.2
Modules of differentials . . . . . . . . . . . . . . . . . . . . . . . .
17
2 Étale homomorphisms and differential operators
2.1
2.2
2.3
20
Basic properties of étale homomorphisms . . . . . . . . . . . . . . . . . .
21
2.1.1
Formally étale homomorphisms . . . . . . . . . . . . . . . . . . .
21
2.1.2
Étale homomorphisms, essentially of finite type . . . . . . . . . .
24
Behavior of differential operators under étale homomorphisms . . . . . .
26
2.2.1
Differentials and formally étale homomorphisms . . . . . . . . . .
26
2.2.2
Differential operators and formally étale morphisms . . . . . . . .
30
2.2.3
Finitely presented modules of differentials . . . . . . . . . . . . .
36
2.2.4
Étale descent for differential operators . . . . . . . . . . . . . . .
43
Differential operators and algebraic relations . . . . . . . . . . . . . . . .
47
2.3.1
47
Elements satisfying polynomial relations . . . . . . . . . . . . . .
4
2.3.2
Proof of theorem 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . .
52
2.3.3
Right ideals and étale homomorphisms . . . . . . . . . . . . . . .
63
3 Minimal twosided ideals in rings of differential operators
3.1
3.2
Supports in reduced commutative rings . . . . . . . . . . . . . . . . . . .
69
3.1.1
Associated prime ideals . . . . . . . . . . . . . . . . . . . . . . . .
69
3.1.2
Geometrical unibranches . . . . . . . . . . . . . . . . . . . . . . .
76
Minimal twosided ideals and minimal left submodules . . . . . . . . . . .
84
3.2.1
Differential operators and supports . . . . . . . . . . . . . . . . .
84
3.2.2
Essential submodules . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.2.3
Minimal nonzero ideals in rings of differential operators . . . . . .
91
3.2.4
Applications of theorem 3.2.11 . . . . . . . . . . . . . . . . . . . .
96
4 Reflexive modules and rings of differential operators
4.1
4.2
4.1.1
Rings of quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.2
Reflexive modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Rings satisfying the condition S2
. . . . . . . . . . . . . . . . . . . . . . 111
Minimal reflexive modules . . . . . . . . . . . . . . . . . . . . . . 111
5 Algebras related to rings of differential operators
117
The algebra C(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.1.1
5.2
101
Some noncommutative ring theory . . . . . . . . . . . . . . . . . . . . . . 102
4.2.1
5.1
68
A dimension formula related to affine plane curves . . . . . . . . . 118
The algebra H(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.1
A description when the irreducible components of A are regular . 123
Bibliography
130
5
Introduction
In [25] it was proved that if k is an algebraically closed field of characteristic zero, and
A is a finitely generated k-algebra, and a domain of krull dimension 1, then
(i) D(A), the ring of differential operators on A, has a minimal essential twosided
ideal J ,
(ii) A contains a minimal essential left D(A)-submodule, J, and
(iii) the algebras C(A) := A/J and H(A) := D(A)/J are finite dimensional vector
spaces over k.
A similar result was independently obtained in [17], in the slightly more general case that
k is (any) field of characteristic zero and A is a finitely generated reduced k-algebra of
krull dimension 1. In [25] it was moreover shown that the result above also holds if A is
replaced by its localization at a maximal ideal.
In both papers, one of the main techniques was to use the fact that if A is as above,
and the map Spec Ae → Spec A is set theoretically injective (where Ae denotes the nore are Morita equivalent (a result motivated
malization of A), then the rings D(A) and D(A)
by an example of [18]). Since Ae is regular in this case, its ring of differential operators is
well known, and using the Morita equivalence it is hence possible to derive many of the
e
properties of D(A) from the same properties of D(A).
There are two reasons why this method fails for rings of higher krull dimension: First
e are no longer Morita equivalent if A is not of
of all it may happen that D(A) and D(A)
krull dimension 1; and secondly, rings of differential operators over normal domains may
be rather unpleasant. A well known example is the ring of differential operators on the
6
cubic cone, originally given in [1].
The goal of this thesis is to generalize the results stated at the beginning of the introduction to some rings of higher krull dimension, and to show how étale homomorphisms
may be used to do that. In particular we show that if A is a commutative, noetherian
and reduced k-algebra such that A has an étale covering B with irreducible components
that are geometrical unibranches, then an analogue of the module J, as above, may be
constructed. We call this analogue χA . If furthermore the modules of differentials of A
are finitely presented, the ideal D(A, χA ) corresponds to J . We then prove that if the
rings of differential operators on the irreducible components of B are simple, results as
(i), (ii) from above are true for A.
The key to our method is the fact that if f : A → B is a formally étale homomorphism
of k-algebras, then it extends in a natural way to a homomorphism f : D(A) → D(B) of
filtered rings. This result is implicit in [26] albeit not stated explicitly there, so we give
a complete proof. If in addition the modules of differentials of A are finitely presented,
we show that there is a natural isomorphism B ⊗A D(A) ∼
= D(B); b⊗D 7→ bf (D), i.e.,
differential operators on B are given by simply extending the coefficients of differential
operators on A.
We prove an étale descent theorem for left modules over rings of differential operators,
and in order to apply it to twosided ideals, we prove the following formula: If B is a kalgebra and b ∈ B is a root of a polynomial P ∈ B[T ] such that P 0 (b) is an invertible
element of B, then
Db = bD +
∞
X
νd (D)
D ∈ D(B)
d=1
where the νd s are B-bilinear endomorphisms on D(B), computed recursively from b and
the coefficients of the polynomial P . This is a generalization of the formula for applying
differential operators to inverses, proved in [20, 8].
As mentioned earlier, the example in [1] shows that D(A) can be quite complicated,
even when A is a finitely generated normal domain over an algebraically closed field of
characteristic zero. However, it was proved in [4] that if k is a field of characteristic zero,
7
and A is a geometrical unibranch domain, which is essentially of finite type over k and
satisfies the condition S2 , then D(A) has some nice properties. For example it has no
nontrivial reflexive twosided ideals. As an application of our results, we prove a related
theorem. Namely if A is a domain, essentially of finite type over a field of characteristic
zero, and it satisfies the condition S2 , then the set
{J|J is a reflexive nonzero ideal of A and a D(A)-submodule of A },
ordered by inclusion, has a unique minimal element.
We also apply our results to the algebras C(A) and H(A), mentioned at the beginning
of this introduction. They have been studied in [3, 17, 23, 24, 25]. We give a formula for
the dimension of C(A), in terms of intersection multiplicities, when A is the completion
at a maximal ideal of an affine reduced plane curve over an algebraically closed field of
characteristic zero, thus giving a partial answer to a question asked in [24]. The algebra
H(A) embeds naturally in the ring D(A/χA ), and our last application is to identify the
image of this embedding when A is a reduced algebra, essentially of finite type over a
field of characteristic zero, and with irreducible components that are regular. This is an
analog of [3, theorems 4.5, 4.6], and the proof is similar.
8
Chapter 1
Definition and first properties of
rings of differential operators
This chapter contains most of the definitions and elementary lemmas needed in this
thesis. All results stated here are well known, and most of them may be found e.g. in [7],
[15] or [25].
1.1
1.1.1
Basic properties of differential operators
Endomorphisms, derivations and the derivation ring
Let A be a commutative k-algebra, and let M and N be A-modules. Let Homk (M, N)
denote the linear space of all k-linear mappings M → N. This space obviously contains
HomA (M, N) as a subset. We write Endk M for Homk (M, M), and we call the elements
of Endk M (k-linear) endomorphisms on M.
If φ ∈ Homk (M, N) and m ∈ M, we let φ∗m ∈ N denote the image of m under φ. For
a ∈ A and φ ∈ Homk (M, N), we let aφ and φa in Homk (M, N) be the elements defined
by (aφ)∗m := a(φ∗m) and (φa)∗m := φ∗(am) for all m ∈ M. In this way it is clear that
Homk (M, N) has the structure of an A-bimodule, and that HomA (M, N) is a twosided
A-submodule of Homk (M, N). If N = M, then we can also define multiplication on
9
Endk M , and it is clear that it then becomes a k-algebra. We define one more structure:
Definition Let A, M and N be as above. For φ ∈ Homk (M, N) and a ∈ A, we let
[φ, a] denote the element φa − aφ ∈ Homk (M, N). We call it the commutator of φ and
a. Similarly if φ, ψ ∈ Endk M , we let [φ, ψ] := φψ − ψφ ∈ Endk M and we call it the
commutator of φ and ψ.
With this, the following lemma is obvious:
Lemma 1.1.1 Let A be a commutative k-algebra, and let M and N be A-modules. With
the definitions as above, we have that
HomA (M, N) = {ψ ∈ Homk (M, N)|[ψ, a] = 0 for all a ∈ A}
Let A be a commutative k-algebra and let M be an A-module. A k-derivation on
M, or derivation on M for short, is an homomorphism δ ∈ Homk (A, M) such that
δ ∗ (ab) = a(δ ∗ b) + (δ ∗ a)b for all a, b ∈ A. We let Derk (A, M) denote the k-linear space
of all derivations on M. We write Derk A for Derk (A, A). Since A is commutative, it is
clear that Derk (A, M) is a left A-submodule of Homk (A, M). On the other hand it is in
general not a right A-submodule of HomA (A, M), even if M = A.
Let A be a commutative k-algebra. Since A is naturally isomorphic to HomA (A, A), it
is now clear by lemma 1.1.1 that we may regard A in a natural way as a k-subalgebra of
Endk A. We will do so without further comment.
Definition Let A be a commutative k-algebra. We let ∆(A) denote the k-subalgebra of
Endk A generated by the left A-submodules A and Derk A, and we call it the derivation
ring on A. For all d ≥ 0 we let ∆d (A) be the space generated by all multiples of at most
d derivations on A.
It is not hard to see that if D ∈ ∆d (A), then [D, a] ∈ ∆d−1 (A) for all a ∈ A. By
definition we have that ∆0 (A) = A and it is also easy to see that ∆1 (D) = A ⊕ Derk A
(the definitions readily imply that A ∩ Derk A = 0). The structure of ∆(A) is fairly
10
well known, especially in the case when A is regular and of finite type over a field of
characteristic zero. In that case it turns out that ∆(A) is a simple noetherian ring and
it is generated by finitely many elements over A. See for example [15, chapter 15] for an
account of the derivation ring.
1.1.2
Definition of rings of differential operators
We can now define the central objects of this work. Let A be a commutative k-algebra,
and let M and N be A-modules. We define subsets of Homk (M, N) as follows: Let
D 0 (M, N ) := {D ∈ Homk (M, N)|[D, a] = 0 for all a ∈ A},
and then for all integers d > 0 define subsets inductively by
D d (M, N) := {D ∈ Homk (M, N)|[D, a] ∈ D d−1 (M, N) for all a ∈ A}.
From these definitions it is clear that all the sets D d (M, N) are in fact twosided Asubmodules of Homk (M, N) and that we have a chain of inclusions
D 0 (M, N) ⊂ D 1 (M, N) ⊂ . . . .
We define
D(M, N ) :=
∞
[
D d (M, N ).
d=0
0
0
If L is a third A-module, we have by induction that D d (N, L)D d (M, N) ⊂ D d+d (M, L)
for all integers d, d0 ≥ 0. Hence it follows that D(M, M) is a k-subalgebra of Endk M , and
it is filtered (in the sense of [15, 1.6.1]) by the subgroups D d (M, M ), d ≥ 0. Moreover
D(M, N) is a D(N)-D(M)-bimodule, and it is also filtered (on both sides) by the subgroups D d (M, N ), d ≥ 0. We write D(A) and D d (A) instead of D(A, A) and D d (A, A)
respectively. We make:
11
Definition Let A be a commutative k-algebra. We call the filtered k-algebra D(A),
defined as above, the ring of (k-linear) differential operators on A. We call the elements of
D(A) (k-linear) differential operators on A, and if D ∈ D d (A) we say that the differential
operator D has order ≤ d.
Remarks 1. If M and N are A-modules, we call elements of D(M, N ) (k-linear) differential operators from M into N, and we define order of differential operators in the
same way.
2. We note that lemma 1.1.1 implies that we may identify D 0 (M, N ) and HomA (M, N)
for all A-modules M and N. It follows especially that D 0 (A) = A for all commutative
k-algebras A.
The first result on rings of differential operators is the following well known lemma:
Lemma 1.1.2 Let A be a commutative k-algebra. Then
D 1 (A) = A ⊕ Derk A = ∆1 (A)
Proof:
First let D ∈ D 1 (A) be such that D ∗ 1 = 0. Then for a and b in A we have that
0 = [[D, a], b] ∗ 1 = D ∗ (ab) − aD ∗ b − bD ∗ a − baD ∗ 1
and since by assumption D ∗ 1 = 0, we thus have that
D ∗ (ab) = aD ∗ b + bD ∗ a.
This shows that D ∈ Derk A. More generally if D is any element of D 1 (A), the element
D0 = D − D ∗ 1 is in D 1 (A) and D 0 ∗ 1 = 0, so D ∈ A ⊕ Derk A. The converse inclusion
2
is obvious from the definitions.
From this lemma it may be tempting to think that ∆d (A) = D d (A) for all d ≥ 0, and
12
hence that ∆(A) = D(A). If k is a field of characteristic zero and A is a regular finitely
generated algebra over k, this is in fact the case (cf. e.g. [15, corollary 15.5.6]), and hence
the study of rings of differential operators reduces to the study of the derivation ring.
In particular it follows in this case that D(A) is a simple noetherian ring. However, it
is not always true that D(A) = ∆(A). Regularity of A seems in fact to be essential for
this equality to hold, and it has been conjectured by Nakai (cf. [15, section 15.6.5]) that
if A is a domain and an algebra of finite type over a field of characteristic zero, then
D(A) = ∆(A) if and only if A is regular.
We end this section by some remarks on notation. If f : A → B is a homomorphism of
commutative k-algebras, and M and N are B-modules, the notation D(M, N) is clearly
ambiguous. Hence we will write DA/k (M, N) if we must clarify all the objects which
go into the definition. Usually, however, we can omit the underlying ring k from this
notation, and we write DA (M, N) instead. The same considerations apply for the spaces
D d (M, N ). Aside from this, we will reserve the notation D(A) for the object DA (A, A).
At this point it is also appropriate to justify the notation D ∗ a for the action of
D ∈ D(A) on a ∈ A, rather than the more conventional notations D(a) or even Da. At
several points throughout this work, we will be regarding D(A) as a right A-module, while
we regard A as a left D(A)-module. Hence there is a danger that the traditional notation
D(a) leads to confusion, as is exemplified by the expression D(a+b). For the same reason
would juxtaposition be ambiguous. Therefore we will follow the example of [25] and write
D ∗ a for the operation of D ∈ D(A) on a ∈ A and retain Da for the right operation of
a ∈ A on D ∈ D(A). We will also use this notation for elements of A-modules whenever
there is a danger of confusion. We note of course that Da ∗ b = Dab ∗ 1 = D ∗ ab for all
D ∈ D(A) and all a, b ∈ A.
13
1.2
1.2.1
Elementary results
A collection of basic lemmas
We will now state some basic results on differential operators, starting with:
Lemma 1.2.1 Let A be a commutative k-algebra, and let M be an A-module. Let N be
an A-submodule of M. Then there is a canonical isomorphism
D(M, N) ∼
= {D ∈ D(M)|D ∗ M ⊂ N} ⊂ D(M)
Proof:
2
Trivial.
This lemma applies especially if we take M = A and N = I, an ideal of A. The
lemma shows that we can identify D(A, I) with the subset {D ∈ D(A)|D ∗ A ⊂ I}, a
right ideal of D(A). In the sequel we will do so without further comment. From this
description it follows immediately that I ⊂ D(A, I) is a left (and hence a twosided) ideal
of D(A) if and only if I is a left D(A)-submodule of A. We have in fact a lemma:
Lemma 1.2.2 Let A be a commutative k-algebra and assume that I ⊂ A is a left D(A)submodule of A (and hence an ideal of A). Let π : A → A/I denote the canonical
projection. Then there is a canonical homomorphism π : D(A) → D(A/I) of filtered
rings, and for all D ∈ D(A), π(D) is the unique element of D(A/I) such that π(D) ∗
π(a) = π(D ∗ a) for all a ∈ A. Moreover we have an exact sequence
π
0 −→ D(A, I) −→ D(A) −→ D(A/I)
14
Proof:
We get the canonical ring homomorphism π : D(A) → D(A/I) by chasing in the
diagram
0
0
- I
pp
pp
pp
pp D|I
pp
p?
- I
- A
π
D
?
- A
π
- A/I
pp
pp
pp
pp π(D)
pp
p?
- A/I
- 0
- 0
The diagram shows that π(D), defined by the formula π(D) ∗ π(a) = π(D ∗ a) for all
a ∈ A, is a well defined differential operator on A/I, of order less than or equal to the
order of D. Uniqueness follows immediately and subsequently we have that π is a ring
homomorphism. The formula implies directly that D(A, I) is the kernel of π.
2
Important Warning Later we will show that the homomorphism π : D(A) → D(A/I)
need not be surjective.
As mentioned above, D(A, I) is a twosided ideal of D(A) whenever I is a left D(A)submodule of A. Moreover, since I = D 0 (A, I) it is clear that D(A, I) is nontrivial
whenever I is. On the other hand it may happen that D(A) has nontrivial twosided ideals
while A is a simple left D(A)-module. The next lemma helps in applying commutative
theory to the study of twosided ideals in rings of differential operators:
Lemma 1.2.3 Let A be a commutative k-algebra and let I ⊂ D(A) be a twosided ideal.
Then I 6= 0 if and only if I contains a nonzero element of A (i.e., a nonzero differential
operator of order zero).
Proof:
The implication ⇐ is clear. For the other direction assume that D ∈ I \ A. For
all a ∈ A we have that ord([D, a]) < ord(D), and [D, a] = Da − aD ∈ I since I is a
twosided ideal. Since D 6∈ A, it follows that [D, a] 6= 0 for some a ∈ A. An induction
then implies that I ∩ A 6= ∅.
2
15
Recall that a ring is said to be semiprime if it has no nonzero nilpotent twosided
ideals (cf. [15, section 0.2.7]). The lemma above immediately implies another:
Lemma 1.2.4 Let A be a commutative and reduced k-algebra. Then D(A) is a semiprime ring.
Proof:
Assume that I is a nonzero nilpotent twosided ideal of D(A). By lemma 1.2.3,
there is a nonzero element a ∈ A ∩ I, and by assumption a must be nilpotent.
2
We will on several occasions need to study rings of differential operators on direct
products of commutative k-algebras. The structure of rings of differential operators on
those is closely related to the rings of differential operators on the factors, as the next
lemma shows:
Lemma 1.2.5 Let A := A1 × . . . × An where n is an integer ≥ 1 and A1 , . . . , An are
commutative k-algebras. Then there is a natural isomorphism:
D(A) ∼
= D(A1 ) × . . . × D(An )
given by the formula D 7→ (π1 (D), . . . , πn (D)), where for i = 1, . . . , n, πi (D) ∈ D(Ai )
is defined by the formula πi (D) ∗ πi (a) = πi (D ∗ a). (πi denotes the natural projection
A → Ai .)
Proof:
2
[17, proposition 1.12]
This implies that we may identify the rings D(A) and D(A1 ) × . . . × D(An ). We will
do so without further comment.
We end this section with a triviality that will be needed later
16
Lemma 1.2.6 Let φ : A → B be a homomorphism of commutative k-algebras, and let
M and N be B-modules. If φ is surjective, then DB (M, N ) = DA (M, N)
Proof:
1.2.2
2
Trivial.
Modules of differentials
In [10, p. 214] and [7, IV (16.3)] the modules of n-differentials, for n ≥ 0, are defined.
We recall some basic properties of them.
Let A be a commutative k-algebra. Then A ⊗k A carries a natural k-algebra structure
by componentwise multiplication. We have two natural morphisms A → A ⊗k A of kalgebras, a 7→ a⊗1 and a 7→ 1⊗a. We fix these morphisms, and from now on we will
always regard A ⊗k A as a (left) A-module via the former morphism and we denote the
latter morphism by dA .
Let mA : A ⊗k A → A denote the natural multiplication map a⊗a0 7→ aa0 . It is a
morphism of k-algebras, and we observe that it is A-linear by the A-module structure we
just defined on A ⊗k A. This morphism is of course surjective, so if we let IA denote the
kernel of mA , we have an exact sequence
mA
0 −→ IA −→ A ⊗k A −→ A −→ 0.
of A-modules, and we note that IA is an ideal of A ⊗k A. For all integers n ≥ 0 we let
PnA denote the k-algebras A ⊗k A/IAn+1 . These algebras are left A-modules and for all
n ≥ 0 we call PnA the module of n-differentials on A. For n ≥ 0, we let dnA : A → PnA
denote the composite of dA with the natural projection A ⊗k A → PnA . These maps
are homomorphisms of k-algebras, and for n ≥ 0 we call dnA : A → PnA the universal
n-derivation on A.
17
Fix an integer n ≥ 0. The morphism mA : A ⊗k A → A factors naturally to an
A-linear k-algebra map PnA → A with kernel IA /IAn+1. By abuse of language we will also
denote this map by mA and we denote its kernel also by IA . Having set that notation,
we have an exact sequence
mA
0 −→ IA −→ PnA −→ A −→ 0.
of left A-modules, and IA is an ideal of PnA such that IAn+1 = 0. The definitions of
dnA : A → PnA and mA : PnA → A now imply that mA is a left inverse for dnA . That is, we
have a commutative diagram
A
1
- A
@
@
@
dn
A
@
R
@
6
mA
PnA
This leads to
Lemma 1.2.7 Let A be a commutative k-algebra and let n ≥ 0 be an integer. Then
dnA ∈ D n (A, PnA ), and it has the following universal property: For all A-modules M and
all D ∈ D n (A, M) there is a unique φ ∈ HomA (PnA , M) completing the diagram
A
dn
A
- Pn
A
pp
p
@
pp
@
pp φ
pp
D@
p
@
R p?
@
M
Proof:
2
[10, theorem 2.2.6]
This lemma implies that if A is a commutative k-algebra and M is an A-module, then
18
for all integers n ≥ 0 there are natural isomorphisms D n (A, M ) ∼
= HomA (PnA , M).
Again fix an integer n ≥ 0. We note that by definition of PnA it is in fact isomorphic
n+1
to Pn+1
A /IA , and hence we have a natural surjective A-linear k-algebra homomorphism
π : Pn+1
→ PnA making the diagram
A
Pn+1
A
dn+1
A
A
dn
A
π
@
@
?
- Pn
A
mA
@
@
R
@
- A
mA
commutative. Therefore for all A-modules M, we get a natural injective A-linear homomorphism HomA (PnA , M) → HomA (Pn+1
A , M). It is then an easy task to verify that
this map in fact corresponds to the inclusion D n (A, M ) ⊂ D n+1 (A, M ). That is, for all
integers n ≥ 0 we have a commutative diagram
D n+1 (A, M )
∼
=
6
D n (A, M )
HomA (Pn+1
A , M)
6
∼
=
19
HomA (PnA , M)
Chapter 2
Étale homomorphisms and
differential operators
In this chapter, the theory of étale homomorphisms is applied to rings of differential
operators.
The first result is that if f : A → B is a formally étale homomorphism of k-algebras,
then it extends in a natural way to a homomorphism f : D(A) → D(B) of filtered rings.
This result is implicit in [26] and the arguments in the proof are essentially the same,
but the proof is carried out here for the sake of completion and for the convenience of
the reader.
If f : A → B is as above, and in addition the modules of differentials of A are finitely
presented, we prove that the morphism f induces in a natural way an isomorphism
D(B) ∼
= B ⊗A D(A) of B-D(A)-bimodules. This is then used to prove a descent theorem
for left modules over rings of differential operators, a technique which will be useful in
chapter 3.
Let B be a commutative k-algebra, and let b ∈ B be a root of a polynomial P ∈ B[T ]
such that P 0 (b) is an invertible element of B (here P 0 denotes the formal derivative of
20
the polynomial P ). We give a formula
Db = bD +
∞
X
νd (D)
D ∈ D(B)
d=1
where the νd s are B-bilinear endomorphisms on D(B), computed recursively from b and
the coefficients of the polynomial P . A model for this result is the formula
∂a−1 = a−1 ∂ − a−2 ∂ ∗ a,
where ∂ is a derivation on an algebra and a is an invertible element in the algebra. We
show also how an earlier result of [20, 8] is a special case of our formula.
Finally use this last result to show that if f : A → B is an étale homomorphism,
and the modules of differentials of A are finitely presented, then B ⊗A J is naturally
isomorphic to a twosided ideal of D(B) whenever J is a twosided ideal of D(A).
2.1
2.1.1
Basic properties of étale homomorphisms
Formally étale homomorphisms
As in [11, Chapter II.1] we make the following definitions:
Definition Let A denote a ring. A covariant functor
F : A − algebras −→ Sets
is said to be formally étale if for all pairs (C, I) where C is an A-algebra and I is an ideal
in C with I 2 = 0, the map
F (C) −→ F (C/I)
21
is bijective.
Definition A ring homomorphism f : A → B of commutative rings is said to be formally
étale if HomA (B, −) is a formally étale functor from A − algebras to Sets.
Remark We define formally smooth and formally unramified functors and ring homomorphisms in a similar way, by replacing the word “bijective” with the weaker conditions
“surjective” or “injective” respectively. It follows that a ring homomorphism is formally
étale if and only if it is both formally smooth and formally unramified.
Let f : A → B be a homomorphism of commutative rings. Then the definition implies
that f is formally étale if and only the following condition holds: For all commutative
diagrams
B
u
- C/I
6
6
f
A
v
- C
where C is a commutative ring, I ⊂ C is an ideal such that I 2 = 0 and u, v are ring
homomorphisms, there is a unique ring homomorphism u0 : B → C completing the
diagram
- C/I
pp
6 pp
6
p p u0
f
pp
pp
Rp
- C
A
B
u
v
The ring homomorphism f is formally smooth if and only if there is at least one such u0 ,
and it is formally unramified if and only if there is at most one such u0 . In this way are
0-étale, 0-smooth and 0-unramified homomorphisms defined in [14].
Remarks 1. An induction shows that this last condition is equivalent to the same
condition with “I 2 = 0” replaced by “I nilpotent”.
22
2. If in the diagrams above, A, B and C are algebras over some commutative ring
k, and the morphisms f , u and v are k-algebra homomorphisms, it follows immediately
that u0 is also a k-algebra homomorphism whenever it exists.
We list some well known properties of formally étale homomorphisms:
Lemma 2.1.1 The following properties hold:
(i) The composite of two formally étale homomorphisms is formally étale.
(ii) If f : A → B is formally étale, and A0 is any commutative A-algebra, then
f ⊗ 1 : A0 → B ⊗A A0 is formally étale.
(iii) Let f : A → B be a formally unramified morphism and let g : B → C be any
homomorphism of commutative rings. If g ◦ f is formally étale, then g is formally
étale.
(iv) For any commutative ring A and any multiplicatively closed subset S ⊂ A, the
localization map A → S −1 A is formally étale.
(v) If f : A → B is formally étale and S and T are multiplicatively closed subsets of
A and B such that f (S) ⊂ T , then the corresponding morphism S −1 A → T −1 B
is also formally étale.
Remark Properties (i), (ii), (iv) and (v) of the lemma hold also with “étale” replaced
by either “smooth” or “unramified”.
Proposition 2.1.2 Formally smooth morphisms are flat.
Proof:
By part (v) of lemma 2.1.1 and the remark following it, we have that localizations
of formally smooth morphisms are formally smooth, and since flatness is a local property
(cf. [14, Theorem 7.1]) it suffices to prove the proposition for local homomorphisms.
This case, however, follows directly from [7, (0III , 19.7.1)] once we observe that formal
23
smoothness in the sense of Grothendieck implies formal smoothness in our sense (cf. [14,
§28]).
2.1.2
2
Étale homomorphisms, essentially of finite type
Definition We say that a homomorphism f : A → B of noetherian commutative rings
is of finite type if B is generated (as an algebra over A) by finitely many elements. We
say that f is essentially of finite type if it can be factored as A → C → B where A → C
is of finite type and B ∼
= S −1 C for some multiplicatively closed subset S of C.
We follow [11, Chapter III] for the definition of étale homomorphisms (note however
that since we are assuming our rings to be noetherian, the notions “essentially of finite
type” and “essentially of finite presentation” are the same).
Definition A homomorphism of commutative noetherian rings is said to be étale if it
is formally étale and essentially of finite type.
Remark We define smooth and unramified morphisms in a similar manner, and it follows
that a morphism is étale if and only if it is both smooth and unramified.
We list some properties of étale homomorphisms:
Lemma 2.1.3 Lemma 2.1.1 holds with the word “formally” deleted. Furthermore we
have:
(vi) Let f : A → B be essentially of finite type. Then f is étale if and only if
Af −1 (P ) → BP is étale for all P ∈ Spec B if and only if Af −1 (M ) → BM is étale
for all maximal ideals M of B.
(vii) Let k be a field. A ring homomorphism k → A is étale if and only if A is a finite
direct product of finite separable field extensions of k.
24
We have a local structure theorem for étale morphisms:
Theorem 2.1.4 Let A and B be local noetherian rings, and let f : A → B be an étale
local morphism. Then B ∼
= (A[T ]/P )M , where:
(i) P ∈ A[T ] is a monic polynomial,
(ii) M is a prime ideal in A[T ] such that P ∈ M and A ∩ M is the maximal ideal of
A, and
(iii) P 0 6∈ M, where P 0 denotes the formal derivative of P .
Conversely for any P and M satisfying these three condition, the natural map A →
(A[T ]/P )M is an étale local morphism.
Corollary 2.1.5 An unramified morphism is étale if and only it is flat.
We will be working with étale algebras over reduced rings, so it is useful to note the
following:
Lemma 2.1.6 If A is a reduced noetherian ring, and A → B is étale, then B is reduced
too.
Proof:
The ring A is reduced if and only if Ap is reduced for all p ∈ Spec A. Thus
by lemma 2.1.3 we may assume that f : A → B is an étale local homomorphism.
Since étale homomorphisms are flat, the lemma now follows from [14, Theorem 23.9] and
2
lemma 2.1.3.
25
2.2
Behavior of differential operators under étale
homomorphisms
2.2.1
Differentials and formally étale homomorphisms
Let f : A → B be a homomorphism of commutative k-algebras and fix an integer n ≥ 0.
From section 1.2.2 we recall the exact sequence
mA
0 −→ IA −→ PnA −→ A −→ 0
of A-modules. By tensoring it on the left by B, we get an exact sequence of B-modules,
B⊗A mA
0 −→ I˜B −→ B ⊗A PnA −→ B −→ 0
where I˜B is the image of the map B ⊗A IA → B ⊗A PnA .
We recall that the (left) A-module PnA is a k-algebra and that mA is a homomorphism
of k-algebras. Therefore we can define a k-algebra structure on the (left) B-module
B ⊗A PnA by componentwise multiplication. Then B ⊗A mA becomes a homomorphism
of k-algebras, and its kernel, I˜B , is an ideal of B ⊗A PnA . Moreover, I˜B is generated by
elements of the form b ⊗ x where b ∈ B and x ∈ IA , and since IAn+1 = 0 it follows that
I˜Bn+1 = 0.
Let 1⊗dnA : A → B ⊗A PnA denote the map a 7→ 1⊗dnA (a). It is obviously a morphism
of k-algebras, and for all a ∈ A we have the formula
(B ⊗A mA ) ◦ (1⊗dnA )(a) = (B ⊗A mA )(1 ⊗ dnA (a)) = f (a).
26
Therefore we have a commutative diagram
1
B
6
- B
6
f
B⊗A mA
A
- B ⊗A Pn
A
1⊗dn
A
of k-algebras and k-algebra homomorphisms, and I˜B = ker (B ⊗A mA ) is such that I˜Bn+1 =
0. From these considerations we now get:
Proposition 2.2.1 Let f : A → B be a formally étale morphism of k-algebras, and let
n ≥ 0 be an integer. Then there is a unique k-algebra morphism d˜nB : B → B ⊗A PnA
making the diagram
1
- B
pp
6 pp
6
p p d˜n
B
f
pp
B⊗A mA
pp
Rp
- B ⊗A Pn
A
B
1⊗dn
A
A
commutative. This map is a differential operator of order ≤ n
Proof:
As remarked earlier, I˜B = ker B ⊗A mA is nilpotent (I˜Bn+1 = 0). Thus the
existence and uniqueness of d˜nB : B → B ⊗A PnA follows directly from the definition of
étale morphisms. It is a morphism of k-algebras by a remark in section 2.1.1. From
the diagram we now have clearly that bd˜nB (1) − d˜nB (b) ∈ ker B ⊗A mA for all b ∈ B, and
since (ker B ⊗A mA )n+1 = 0 it then follows that d˜nB is a differential operator of order ≤ n
(recall that d˜nB is a ring homomorphism).
2
By this proposition and the universal property of dnB : B → PnB , it follows that if
f : A → B is a formally étale homomorphism of k-algebras, then for all n ≥ 0, there is
27
a unique B-linear map ψB : PnB → B ⊗A PnA such that d˜nB = ψB ◦ dnB . The next theorem
shows that this homomorphism is in fact an isomorphism, and thus we have a very close
relation of the k-algebras PnA and PnB for all integers n ≥ 0 if f : A → B is a formally
étale morphism of k-algebras.
Theorem 2.2.2 Let f : A → B be a formally étale homomorphism of k-algebras, and
∼
=
let n ≥ 0 be an integer. Then there is a unique B-module isomorphism B ⊗A PnA →
PnB
making the diagram
B
d˜n
B-
B ⊗A PnA
@
@
@
dn
B
@
R
@
pp
pp
pp
pp
pp
p?
PnB
commutative. Hence d˜nB : B → B ⊗A PnA solves the same universal problem as dnB : B →
PnB .
Proof:
Once we have shown the existence of this isomorphism, uniqueness follows from
the universal property of dnB : B → PnB . We will show existence by constructing an
inverse to the map ψB : PnB → B ⊗A PnA mentioned above.
The map dnB : B → PnB is a differential operator of B-modules of order ≤ n. Since f
is A-linear we have that dnA ◦ f : A → PnB is a differential operator of A-modules of order
≤ n. Therefore by the universal property of dnA : A → PnA there is a unique A-linear
morphism φA : PnA → PnB such that
φA ◦ dnA = dnB ◦ f.
By adjointness of Hom and ⊗ we have a natural isomorphism of groups
HomA (PnA , PnB ) ∼
= HomB (B ⊗A PnA , PnB ),
28
so there is a unique B-linear homomorphism φB : B ⊗A PnA → PnB such that φB ◦(1⊗id) =
φA . Then by composing this with dnA and using the formula above, we get that
φB ◦ (1⊗dnA ) = dnB ◦ f
By the definitions of mB : PnB → B and B ⊗A mA : B ⊗A PnA → B we now have for all
a ∈ A that
(mB ◦ φB )(1⊗dnA (a)) = mB (dnB (f (a))) = f (a) = (B ⊗A mA )(1⊗dnA (a)).
By the construction of PnA it follows that the B-module B ⊗A PnA is generated by the
elements 1⊗dnA (a), a ∈ A and therefore we have that
mB ◦ φB = B ⊗A mA
The two formulas we have now proved, along with the definition of d˜nB , now imply
that
mB ◦ (φB ◦ d˜nB ) = B ⊗A mA ◦ d˜nB = idB
and
(φB ◦ d˜nB ) ◦ f = φB ◦ (1 ⊗ dnA ) = dnB ◦ f
I.e, we have a commutative diagram
B
1
- B
6@
6
@ φB ◦d˜nB
mB
f
@
@
R
@
Pn
A
dn
B ◦f
B
and it is trivial to show that φB ◦ d˜nB is a ring homomorphism. Since f : A → B is
formally étale and the kernel of mB is nilpotent, the diagonal is unique. However, it can
29
be replaced by the map dnB : B → PnB , so we must have dnB = φB ◦ d˜nB .
Now we can finish the proof. By combining the formulas we have for φB and ψB , we
get that
dnB = φB ◦ d˜nB = (φB ◦ ψB ) ◦ dnB
and so the universal property of dnB : B → PnB implies that φB ◦ ψB = id. Conversely we
recall that ψB ◦ dnB = d˜nB , and then we can read from our diagram that
ψB ◦ φB ◦ (1⊗dnA ) = ψB ◦ dnB ◦ f = d˜nB ◦ f = 1⊗dnA .
Since B ⊗A PnA is generated as a (left) B-module by the elements 1⊗dnA (a), a ∈ A, this
implies that ψB ◦ φB = id.
2.2.2
2
Differential operators and formally étale morphisms
In this section we will study the relationship between differential operators on A-modules
and differential operators on B-modules if f : A → B is a formally étale homomorphism
of k-algebras.
We start with a general observation. Let f : A → B be a homomorphism of kalgebras and let n ≥ 0 be an integer. If M is a B-module, the groups HomA (PnA , M)
n
n
(A, M) have natural (left) B-module structures: For b ∈ B, D ∈ DA
(A, M)
and DA
and a ∈ A, we let (bD) ∗ a = b(D ∗ a). Moreover it is clear that the isomorphism
n
(A, M) is an isomorphism of B-modules. This implies that the
HomA (PnA , M) ∼
= DA
n
functors HomA (PnA , −) and DA
(A, −) are naturally equivalent.
sequel without further comment. We start with a lemma:
30
We will use this in the
Lemma 2.2.3 Let f : A → B be a formally étale homomorphism of k-algebras and let
n ≥ 0 be an integer. Then
(i) there is a natural equivalence
n
n
φ n : DA
(A, −) −→ DB
(B, −)
of functors from B − Mod into B − Mod;
n
(ii) if M is a B-module and D ∈ DA
(A, M), then φnM (D) is the unique element of
n
DB
(B, M) completing the commutative diagram
f
A
D
- B
pp
pp
pp
p
p
n
p p φM (D)
p
?
M
Proof:
By theorem 2.2.2, d˜nB : B → B ⊗A PnA has the same universal property as
dnB : B → PnB . Hence by using the adjointness of Hom and ⊗ we have a chain of
natural equivalences
n
n
DB
(B, −) ∼
(A, −)
= HomB (B ⊗A PnA , −) ∼
= HomA (PnA , −) ∼
= DA
of functors from B − Mod into B − Mod. This proves the first part.
For the second part we look at the following diagram:
d˜n
B
B
pp
6 p p(3)
Rp
M
f
D
A
dn
A
31
- B ⊗A Pn
A
p
p
p
6
p p (2)
ι
p
Ip p
(1) p p
- Pn
A
We are just repeating the argument from the first part of the proof. We fill in the first
arrow by the universal property of dnA : A → PnA . The adjointness of Hom and ⊗ gives
us the second arrow, and finally the third arrow comes from the universal property of
d˜nB : B → B ⊗A PnA . It is clear that this last arrow is equal to φnM (D). All these arrows
n
(B, M)
are unique, and we note that if we start with any differential operator D 0 ∈ DB
such that D = D0 ◦ f , we can successively fill in the diagram in reverse order. Hence we
conclude that D 0 = φnM (D).
2
Proposition 2.2.4 Let f : A → B be a formally étale homomorphism of k-algebras.
n
n
The natural equivalences φn : DA
(A, −) → DB
(B, −) commute with the inclusions
n+1
n+1
n
n
DA
(A, −) ⊂ DA
(A, −) and DB
(B, −) ⊂ DB
(B, −) for all n ≥ 0, and thus they
define a natural equivalence
φ : DA (A, −) −→ DB (B, −)
of functors from B − Mod into B − Mod. Moreover if M is a B-module and D ∈
DA (A, M) then ord(φM (D)) ≤ ord(D) and φM (D) is the unique element of DB (B, M)
which completes the diagram
A
D
f
p
pp
p
pp
? p
- B
pp
pp
M
Proof:
n
Fix an integer n ≥ 0, let M be a B-module and assume that D ∈ DA
(A, M) ⊂
n+1
n+1
(A, M). We must show that φnM (D) = φn+1
DA
M (D) ∈ DB (B, M). But this is clear
since both these element fill in the diagram in the second part of lemma 2.2.3 (with
n replaced by n + 1). Hence the natural equivalences commute with inclusions, and
32
we conclude that they define a natural equivalence DA (A, −) → DB (B, −) of functors
B − Mod into B − Mod.
The definition of the map φM implies that it preserves order of differential operators.
Finally for the last part let D ∈ DA (A, M). If D 0 ∈ DB (B, M) fills in the diagram, we
n
n
(A, M) and D 0 ∈ DB
(B, M). Then uniqueness follows
pick n large enough so D ∈ DA
2
from lemma 2.2.3.
Theorem 2.2.5 Let f : A → B be a formally étale homomorphism of k-algebras. It
induces a homomorphism D(A) → D(B) of filtered k-algebras, which we also denote
by f . The filtered homomorphism f : D(A) → D(B) is characterized by the following
property: For D ∈ D(A), f (D) is the unique element of D(B) such that
f (D) ∗ f (a) = f (D ∗ a)
for all a ∈ A.
Proof:
We apply proposition 2.2.4 with M = B, and thus get a morphism φB :
DA (A, B) → D(B) of filtered groups. The map f : A → B is a morphism of A-modules,
so it clearly induces a morphism ψ : D(A) → DA (A, B) of filtered groups, given by the
formula ψ(D) := f ◦ D. We let f := φB ◦ ψ. With f : D(A) → D(B) so defined, it is
now clear that f is k-linear and a morphism of filtered groups.
Let D be an element of D(A). Then f (D) = φB (f ◦D), and hence by proposition 2.2.4
it follows that f (D) is the unique element of D(B) which completes the diagram
A
f ◦D
f
p
pp
p
pp
? p
B
33
- B
pp
p
p
This is precisely what we want.
It remains to show that f : D(A) → D(B) is a homomorphism of k-algebras. We
have already seen that f is k-linear. We must show that f preserves multiplication. Let
D, D 0 ∈ D(A). Then for all a ∈ A we have that
(f (D)f (D 0)) ∗ a = f (D ∗ (D 0 ∗ a)) = f (DD 0 ∗ a) = f (DD 0) ∗ a
so by the uniqueness already proved it follows that f (D)f (D 0) = f (DD 0). The uniqueness
2
also shows that f (1) = 1.
We have an immediate corollary to this:
Corollary 2.2.6 Let f : A → B be an injective formally étale homomorphism of kalgebras. Then f : D(A) → D(B) is also injective, and its image is equal to the filtered
ring {D ∈ D(B)|D ∗ f (A) ⊂ f (A)}.
Proof:
Let D ∈ D(A) be such that f (D) = 0. Then for all a ∈ A, 0 = f (D) ∗ f (a) =
f (D ∗ a), and since f : A → B is injective, this implies that D = 0. For the second
claim we first observe that f (D) ∗ f (A) ⊂ f (A) for all D ∈ D(A). Conversely let
D ∈ D(B) be such that D ∗ f (A) ⊂ f (A). Since f is injective, D induces a unique
element D 0 of Endk A with the property that f (D 0 ∗ a) = D ∗ f (a) for all a ∈ A. From
this it follows that D 0 ∈ D(A) is of order ≤ ord(D). Then we have for all a ∈ A that
f (D 0 ) ∗ f (a) = f (D 0 ∗ a) = D ∗ f (a), so by uniqueness we must have that D = f (D 0).
2
Remarks 1. Using this corollary, we will from now on identify the rings D(A) and
{D ∈ D(B)|D ∗ f (A) ⊂ f (A)} whenever f : A → B is an injective formally étale homomorphism of k-algebras, and we find it convenient.
34
2. The corollary applies especially to the case where A is a commutative reduced
k-algebra and K is its total ring of fractions. We conclude that
D(A) = {D ∈ D(K)|D ∗ A ⊂ A}
3. Let f : A → B be a faithfully flat and formally étale homomorphism of k-algebras.
Then there is an exact sequence
f1
f
0 −→ A −→ B −→
B ⊗A B,
−→
f
2
and from the corollary it is more or less immediate that this exact sequence induces an
exact sequence
f1
f
0 −→ D(A) −→ D(B) −→
D(B ⊗A B)
−→
f
2
of filtered rings.
4. Note that the proof of the corollary did not use the fact that f : A → B was formally
étale, it only used the existence and uniqueness of the morphism f : D(A) → D(B) with
the given properties, and injectivity of f . Later we will in fact encounter situations where
this is useful to know.
For later usage we now return to the map
φB : DA (A, B) −→ D(B)
defined in proposition 2.2.4 when f : A → B is a formally étale homomorphism of
k-algebras. We had already seen that it is an isomorphism of left B-modules.
It is clear that DA (A, B) has a natural right D(A)-module structure via composite of
differential operators. Moreover by using last theorem, we can define a right D(A)-module
structure on D(B). We fix these structures and then we have:
35
Proposition 2.2.7 Let f : A → B be a formally étale k-algebras homomorphism. The
morphism φB : DA (A, B) → D(B), defined in proposition 2.2.4, is an isomorphism of
B-D(A)-bimodules.
Proof:
It only remains to show that φB is right D(A)-linear. For that let D ∈ DA (A, B)
and D0 ∈ D(A). By the last part of theorem 2.2.5 we have that f (D 0 ) ◦ f = f ◦ D 0 . It
follows that
φB (D) ◦ f (D 0 ) ◦ f = φB (D) ◦ f ◦ D 0 = DD 0
= φB (DD 0) ◦ f
Since φB (D) ◦ f (D 0) ∈ D(B), last part of proposition 2.2.4 thus implies that φB (D) ◦
f (D0 ) = φB (DD 0 ). This is precisely to say that φB is right D(A)-linear.
2.2.3
2
Finitely presented modules of differentials
Theorem 2.2.5 implies that if f : A → B is a formally étale homomorphism of k-algebras,
then we can define the homomorphism B ⊗A D(A) → D(B) of B-D(A)-bimodules by the
formula b ⊗ D 7→ bf (D). We will now introduce some conditions which ensure that this
homomorphism is in fact an isomorphism.
We start by recalling some homological algebra: Let f : A → B be a morphism of
commutative rings. We can form the following two bifunctors
B ⊗A HomA (−1 , −2 )
and
HomA (−1 , B ⊗A −2 )
36
from A − Mod×A − Mod into B − Mod. Both these bifunctors are contravariant in the
first variable and covariant in the second variable. We can define a natural transformation
τ between these two functors as follows: For A-modules M and N we define a B-linear
homomorphism
τN,M : B ⊗A HomA (N, M) −→ HomA (N, B ⊗A M)
by τN,M (b⊗g) = gb where gb (n) := b⊗g(n). This transformation is functorial in both
variables, that is, τN,− : B ⊗A HomA (N, −) → HomA (N, B ⊗A −) is a natural transformation of functors for all N and so is τ−,M : B ⊗A HomA (−, M) → HomA (−, B ⊗A M )
for all M.
Definition An A-module N is said to be finitely presented if there are integers r and s
and an exact sequence of A-modules
Ar −→ As −→ N −→ 0
With this in hand we can state the following:
Lemma 2.2.8 If f : A → B is flat homomorphism of commutative rings, and N is a
finitely presented A-module, then τN,− : B ⊗A HomA (N, −) → HomA (N, B ⊗A −) is a
natural equivalence of functors.
Proof:
This follows directly from the observations above and [22, lemma 3.83].
2
We can apply this result to differential operators:
Proposition 2.2.9 Let f : A → B be a flat homomorphism of commutative k-algebras,
and let n ≥ 0 be an integer. If PnA is a finitely presented A-module, then there is a
37
canonical isomorphism of B-modules,
n
τBn : B ⊗A D n (A) −→ DA
(A, B),
defined by by the formula τBn (b⊗D) := Db , where Db ∗ a = bf (D ∗ a).
If the A-modules PnA are finitely presented for all integers n ≥ 0, the isomorphisms τBn
commute with inclusions, and hence they induce an isomorphism of B-D(A)-bimodules
τB : B ⊗A D(A) −→ DA (A, B)
defined by the same formula as above.
Proof:
The first statement follows directly by applying lemma 2.2.8 to N = PnA and the
natural equivalence of functors
n
DA
(A, −) ∼
= HomA (PnA , −).
It is trivial to show that the maps τBn commute with inclusions, and by passing to direct
limits we get the B-linear isomorphism
τB : B ⊗A D(A) −→ DA (A, B)
It remains to show that τB is right D(A)-linear. Let D 0 ∈ D(A). Then (b⊗D)D 0 =
b⊗(DD0 ) by definition, and hence τB ((b⊗D)D 0 ) = (DD 0)b . On the other hand we have
τB (b⊗D)D 0 = Db D 0 . But for all a ∈ A we have that
Db D 0 ∗ a = Db ∗ (D ∗ a) = bf (D ∗ (D 0 ∗ a)) = bf (DD 0 ∗ a) = (DD 0 )b ∗ a
and therefore is Db D 0 = (DD 0 )b . This implies that τB is right D(A)-linear.
38
2
We can now prove:
Theorem 2.2.10 If f : A → B is a formally étale homomorphism of k-algebras and PnA
is finitely presented for all integers n ≥ 0, then f induces an isomorphism of B-D(A)bimodules
B ⊗A D(A) −→ D(B),
given by (b⊗D) 7→ bf (D).
Proof:
From proposition 2.1.2 it follows that f : A → B is flat, and hence this iso-
morphism is just the composite of the B-D(A)-isomorphisms from propositions 2.2.7
and 2.2.9. It remains only to establish the formula stated in the theorem. For that let
b ∈ B and D ∈ D(A). The isomorphism from proposition 2.2.9 maps b ⊗ D to the
differential operator Db ∈ DA (A, B), and then proposition 2.2.4 implies that φB (Db ) is
the unique differential operator in D(B) such that
φB (Db ) ∗ f (a) = Db ∗ a = bf (D ∗ a)
for all a ∈ A. It is clear, however, that bf (D) is a differential operator in D(B), and
theorem 2.2.5 implies that bf (D) ∗ a = bf (D ∗ a) for all a ∈ A. Therefore we have that
2
φB (Db ) = bf (D).
Remark A similar statement holds for modules. In fact if A and B are as in theorem 2.2.10 and M is an A-module, then we can proceed as in the proof of proposition 2.2.9 to get a canonical isomorphism B ⊗A DA (A, M) ∼
= DA (A, B ⊗A M ). Furthermore proposition 2.2.4 implies that there is a canonical isomorphism DA (A, B ⊗A M ) ∼
=
DB (B, B ⊗A M ), so by combining these we have a canonical isomorphism
B ⊗A DA (A, M) ∼
= DB (B, B ⊗A M )
39
Having theorem 2.2.10, it becomes natural to ask when PnA is finitely presented for
all integers n ≥ 0. The following two results show that this is often the case:
Proposition 2.2.11 If f : A → B is a formally étale homomorphism of k-algebras, and
PnA is finitely presented for some integer n ≥ 0, then PnB is also finitely presented.
Proof:
By assumption we have an exact sequence
Ar −→ As −→ PnA −→ 0
of A-modules, and by applying the right exact functor B ⊗A − to this sequence, we get
an exact sequence of B-modules
B r −→ B s −→ B ⊗A PnA −→ 0
Finally, since f : A → B is formally étale, theorem 2.2.2 tells us that B ⊗A PnA is
2
isomorphic to PnB as a B-module.
Proposition 2.2.12 If A is a finitely generated k-algebra, then PnA is finitely generated
for all integers n ≥ 0.
Proof:
This follows directly from [7, IV (16.4.22)]. However, for the sake of completion
we supply a proof here:
As before, we let IA denote the kernel of the multiplication map mA : A ⊗k A → A.
We will first prove that IA /IA 2 is a finitely generated (left) A-module. It is well known
(and an easy exercise to see) that IA is generated by the elements {a ⊗ 1 − 1 ⊗ a|a ∈ A}.
40
Hence we have for all a, b ∈ A that
1⊗ab = a ⊗ b + b ⊗ a − ab ⊗ 1
(mod IA 2 ),
and from this it follows that
ab ⊗ 1 − 1 ⊗ ab ≡ ab ⊗ 1 − a ⊗ b − b ⊗ a + ab ⊗ 1 (mod IA 2 )
= a(b ⊗ 1 − 1 ⊗ b) + b(a ⊗ 1 − 1 ⊗ a)
for all a, b ∈ A. Hence if the finitely many elements {ai } generate A as a k-algebra,
then the left A-module IA /IA 2 is generated by the canonical images of the elements
ai ⊗ 1 − 1 ⊗ ai .
Now let r > 0 be an integer and assume that IA r−1 /IA r is finitely generated. We want
to show that IA r /IA r+1 is finitely generated. Let x ∈ IA r and write x =
P
j
xj yj where
xj ∈ IA and yj ∈ IA r−1 for all j. We then use the assumption and the first part of the
proof to write the xi s and the yi s in terms of generators (modulo IA n and IA 2 ) and the
result follows.
Finally to prove that the modules PnA are finitely generated for all n ≥ 0 we use the
exact sequences
0 −→ IA n /IA n+1 −→ PnA −→ Pn−1
−→ 0.
A
We have already proved that the terms on the left end are finitely generated and since
P0A ∼
= A as a left A-module, we get immediately by induction that PnA is finitely generated
for all n (we bear in mind the elementary fact that if the two ends in a short exact sequence
are finitely generated modules, then the middle term is a finitely generated module as
2
well.)
Remark We recall of course that all finitely generated modules over noetherian rings
are automatically finitely presented.
41
Earlier it was pointed out that localizations of commutative rings are always formally
étale. Therefore these two propositions give a large class of algebras to which theorem 2.2.10 applies, e.g. all algebras which are essentially of finite type over a noetherian
ring.
Next we point out a lemma which follows directly from theorem 2.2.10:
Lemma 2.2.13 Let f : A → B be a formally étale homomorphism of k-algebras, and
assume that PnA is finitely presented for all integers n ≥ 0. Then there is a natural
equivalence
B ⊗A − −→ D(B) ⊗D(A) −
of functors from D(A) − Mod into B − Mod.
Proof:
By theorem 2.2.10, D(B) ∼
= B ⊗A D(A) as a B-D(A)-bimodule. Then cancella2
tion gives the desired result.
This lemma implies that when studying the behavior of the functor D(B) ⊗D(A) −
on exact sequences of left D(A)-modules, we may just as well look at the better known
functor B ⊗A −. For example we have immediately that D(B) ⊗D(A) − is exact. Hence
in the sequel we will sometimes use B ⊗A M in place of D(B) ⊗D(A) M if M is a left
D(A)-module (and the assumptions of the lemma hold).
We end this section by discussing localizations. Localizations are formally étale and
hence flat. Therefore if A is a commutative k-algebra such that PnA is finitely presented for
all integers n ≥ 0, and S ⊂ A is any multiplicatively closed subset of A, by theorem 2.2.10
we have a natural isomorphism D(S −1 A) ∼
= S −1 D(A) (where we regard D(A) as a left
A-module). We will thus from now on treat these two objects as being equal. If J is a left
42
A-submodule of D(A), by the above remarks and flatness we may regard S −1 J as a left
S −1 A-submodule of D(S −1A). Furthermore, if J is a left ideal of D(A) then lemma 2.2.13
applies, and we have that S −1 J = D(S −1 A) ⊗D(A) J is a left ideal of D(S −1A).
With the assumptions and notation as above and for p ∈ Spec A, we will from now
on write Jp for Ap ⊗A J, and we will regard it as a left Ap -submodule of D(Ap ). It is a
left ideal if J is.
2.2.4
Étale descent for differential operators
If f : R → S is a ring homomorphism, we will sometimes write f∗ − for the functor S⊗R −
from the category of left R-modules into the category of left S-modules. This notation
lacks the reference to the rings, but has some obvious advantages, for example if we need
to study tensor products with respect to two or more ring homomorphisms R → S.
For now we note that if f : A → B is a homomorphism of commutative k-algebras such
that the conditions of lemma 2.2.13 hold, the equivalence of the functors D(B) ⊗D(A) −
and B ⊗A − implies that there is no danger of confusion when we write f∗ M where M
is a left D(A)-module.
For f : A → B, a homomorphism of commutative rings, let f1 , f2 : B → B ⊗A B be the
two natural base extensions of f ,defined by f1 (b) = 1⊗b, f2 (b) = b⊗1. For a B-module
M, it follows immediately that we have natural B ⊗A B-isomorphisms, f1 ∗ M ∼
= B ⊗A M
and f2 M ∼
= M ⊗A B. As in [12, p. 34], we let θi , i = 1, 2, 3 denote the three B⊗A B⊗A B∗
homomorphisms induced by a B ⊗A B-homomorphism θ : B ⊗A M → M ⊗A B. We will
fix this notation in what follows.
We will now define descent data. In light of the notational convention we have established and [12, proposition II.3.1], it follows that our definition is equivalent to the one
given in [12]:
Definition Let f : A → B be a homomorphism of commutative rings and let M be
43
a B-module. An isomorphism θ : B ⊗A M → M ⊗A B of B ⊗A B-modules such that
θ2 = θ3 θ1 is called a descent data for M over A.
Now assume that A is a commutative k-algebra, and that PnA is finitely presented for
all integers n ≥ 0. Furthermore assume that f : A → B is a formally étale homomorphism
of k-algebras. Then by proposition 2.2.11, we have that PnB is also finitely presented for
all n ≥ 0, and by lemma 2.1.1 it follows that the morphisms fi : B → B ⊗A B, i = 1, 2
are formally étale. Hence if M is a left D(B)-module, we can form the left D(B ⊗A B)modules f1 ∗ M and f2 ∗ M. It was pointed out earlier that these modules are naturally
isomorphic as B ⊗A B-modules to the modules B ⊗A M and M ⊗A B respectively, and
therefore we can define a left D(B ⊗A B)-module structure on those as before. If θ :
B ⊗A M → M ⊗A B is a D(B ⊗A B)-homomorphism, restriction of scalars implies that
it is especially a B ⊗A B-homomorphism, and hence we can define θi , i = 1, 2, 3 as before.
Definition Let A be a commutative k-algebra such that PnA is finitely presented for all
n ≥ 0, and let f : A → B be a formally étale homomorphism of k-algebras. Let M be a
left D(B)-module. An isomorphism θ : B ⊗A M → M ⊗A B of left D(B ⊗A B)-modules
such that θ2 = θ3 θ1 is called a descent data for M over D(A).
We get immediately:
Theorem 2.2.14 Let A be a commutative k-algebra such that PnA is finitely presented
for all n ≥ 0, and let f : A → B be a faithfully flat and formally étale extension of
k-algebras. Let M be a left D(B)-module and let θ : B ⊗A M → M ⊗A B be a descent
data for M over D(A). Then
(i) there is a left D(A)-module N and an isomorphism
η : B ⊗A N −→ M
44
of left D(B)-modules such that the diagram
B ⊗A B ⊗A N
1⊗η -
∼
=
B ⊗A M
θ
?
B ⊗A N ⊗A B
η⊗1
?
- M⊗ B
A
commutes;
(ii) the pair (N, η) is determined up to a unique isomorphism;
(iii) we may in fact take N = {m ∈ M|θ(1⊗m) = m⊗1}, and the isomorphism η :
B ⊗A N → M to be the multiplication map b⊗n 7→ bn.
Proof:
It is clear from the definition that θ : f1 ∗ M → f2 ∗ M is also a descent data
for the B-module M over A. Hence the existence and uniqueness of the pair (N, η)
(as a B-module) is a direct consequence of the theorem of faithfully flat descent (cf. [12,
Théorème II.3.2]). The only fact remaining to be checked is that N, defined as in part (iii),
is actually a D(A)-submodule of M. It then follows directly from the definition of the
left D(B)-module structure on B ⊗A N that η : B ⊗A N → M is D(B)-linear.
We must show that θ(1⊗(f (D)∗m)) = (f (D)∗m)⊗1 if D ∈ D(A) and m ∈ M is such
that θ(1⊗m) = m⊗1. We recall that B ⊗A M is naturally isomorphic to f1 ∗ M . Under
that isomorphism, the element 1⊗(f (D) ∗ m) corresponds to (1⊗1)⊗f1 (f (D) ∗ m) =
f1 (f (D))⊗f1 m = f2 (f (D))⊗f1 m, since by definition f1 ◦ f = f2 ◦ f . By assumption on
the element m we have that θ(1⊗m) = m⊗1 which in turn corresponds to the element
(1⊗1)⊗f2 m ∈ f2 ∗ M . Since θ is D(B ⊗A B)-linear, this implies that θ(1⊗(f (D) ∗ m))
corresponds to the element f2 (f (D))⊗f2 m = (1⊗1)⊗f2 (f (D) ∗ m) ∈ f2 ∗ M . Finally, as
before, we have that the element (1⊗1)⊗f2 (f (D) ∗ m) ∈ f2 ∗ M corresponds to (f (D) ∗
m)⊗1 ∈ M ⊗A B, and thus we have that θ(1⊗(f (D) ∗ m)) = (f (D) ∗ m)⊗1.
2
Let A be a commutative k-algebra such that PnA is finitely presented for all n ≥ 0
45
and let f : A → B be a faithfully flat formally étale homomorphism of k-algebras. We
can immediately apply theorem 2.2.14 to two cases: Taking M = D(B), we have natural
isomorphisms f1 ∗ D(B) ∼
= D(B ⊗A B) ∼
= f2 ∗ D(B). Let θ : B ⊗A D(B) → D(B) ⊗A B 1
be the D(B ⊗A B)-linear homomorphism that corresponds to the composite of these
natural isomorphisms. Since D(B ⊗A B) ∼
= (B ⊗A B)⊗A D(A), it follows that θ is in
fact a descent data for D(B) over D(A). Hence the theorem implies that there is a left
D(A)-module N, and an isomorphism η : B ⊗A N → D(B), making the diagram there
commutative. Of course D(A) and the multiplication map B ⊗A D(A) → D(B); b⊗D 7→
bf (D) is such a pair, and thus we have that D(A) ∼
= {D ∈ D(B)|f1 (D) = f2 (D)}.
Another natural case to consider is M = B. In that case we have that f1 ∗ B ∼
=
B ⊗A B ∼
= f2 B, and this D(B ⊗A B)-isomorphism simply corresponds to the identity
∗
map. Hence it is obviously a descent data for B over D(A), and the theorem implies
that there is a left D(A)-module N, and an D(B)-isomorphism η : B ⊗A N → B. We
get of course that N ∼
= A. This leads however to a little less trivial statement that will
be useful later, namely:
Proposition 2.2.15 Let A be a commutative k-algebra such that PnA is finitely presented
for all n ≥ 0, and let f : A → B be a faithfully flat and formally étale extension
of k-algebras. Let I be a left D(B)-submodule of B such that B ⊗A I = I ⊗A B as
subsets of B ⊗A B. Then there is a unique D(A)-submodule J of A such that B ⊗A J is
isomorphic to I via the ordinary multiplication map b⊗x 7→ bf (x). The module J is in
fact isomorphic to {x ∈ I|f1 (x) = f2 (x)}.
Proof:
Uniqueness of J follows from faithful flatness. The identity map on B ⊗A B
restricts to a D(B ⊗A B)-isomorphism θ : B ⊗A I → I ⊗A B, and it follows immediately
that all the maps θi , i = 1, 2, 3 are also the identity. Therefore is θ a descent data for I
over D(A). Then the existence and the description of J follows from theorem 2.2.14.
1
This is rather unfortunate notation. We are considering D(B) as a left B-module here
46
2
2.3
2.3.1
Differential operators and algebraic relations
Elements satisfying polynomial relations
Let B be a commutative k-algebra. Let b ∈ B and P =
Pn
r=0 ar T
r
∈ B[T ] be such that
P (b) = 0 and P 0 (b) is an invertible element of B (P 0 denotes the formal derivative of the
polynomial P ). For such a pair and all integers i ≥ 0 we let γ(P,b),i and α(P,b),i be the
B-B-linear endomorphisms on D(B) defined on D ∈ D(B) by the formulas:
0
γ(P,b),i (D) = −P (b)
−1
n X
r
i
br−i ar D
r=0
and
α(P,b),i (D) = −P 0 (b)−1
n X
r
i
br−i [D, ar ]
r=0
We will drop the subscript (P, b) from this notation when the context makes it clear.
Sometimes we will write γi(D) = bi D and αi (D) =
Pn
r=0 cr,i [D, ar ]
for D ∈ D(B) if the
precise formulas for the coefficients bi and cr,i are unimportant.
Fix a b ∈ B and P ∈ B[T ] as above. From the definitions, and the assumption that
B is commutative, it follows that all the endomorphisms γi and αi commute. We let
ZP,b be the commutative k-subalgebra of EndB D(B)B generated by them, and we let
JP,b be the ideal of ZP,b generated by the endomorphisms αi , i ≥ 0. We observe that
γ1 = −id = −1, γ0 = 0 and γi = αi = 0 whenever i > n, so the algebra ZP,b is in fact
finitely generated over k, 1 ∈ ZP,b and JP,b is a finitely generated ideal of ZP,b .
The usefulness of these definitions lies in the following: By definition every element
of ZP,b is a polynomial in the variables γi , αi , and for all D ∈ D(B) we may write
γi (D) = bi D and αi (D) =
Pn
r=0 cr,i [D, ar ].
Therefore it follows that for every ν ∈ ZP,b
47
there are b0j ∈ B and a0j,1 , . . . , a0j,mj ∈ {a0 , . . . , an } such that
ν(D) =
X
b0j [[. . . [[D, a0j,1], a0j,2 ], . . .], a0j,mj ]
j
for all D ∈ D(B) (if mj = 0 we define the iterated commutator to be equal to D). With
this we can now prove the following useful lemma:
Lemma 2.3.1 Let B be a commutative k-algebra, and let b ∈ B, P ∈ B[T ] be such that
P (b) = 0 and P 0 (b) is an invertible element of B. Let ZP,b be defined as above and let
ν ∈ ZP,b . Assume that A is a subring of B which contains all the coefficients of the
polynomial P . Then there are finitely many elements bi ∈ B, ai ∈ A such that
ν(D) =
X
bi Dai
i
for all D ∈ D(B).
Proof:
This follows simply by expanding the commutators in the expression preceding
2
the lemma.
We can now state the following theorem, which in certain sense tells us how move an
element b, as above, to the left of every differential operator on B.
Theorem 2.3.2 Let B be a commutative k-algebra, and let b ∈ B and P ∈ B[T ] be such
that P (b) = 0 and P 0 (b) is an invertible element of B. Then there are endomorphisms
d
νd ∈ JP,b
⊂ ZP,b for d = 1, 2,. . . such that
Db = bD +
∞
X
d=1
for all D ∈ D(B).
48
νd (D)
Moreover, a family of such endomorphisms νd may be computed using the following
formulas:
νd =


 σ1
1
if d = 1

 σ d − σ d−1
1
1
if d > 1
where for integers d ≥ i ≥ 1, the elements σid are defined by the recursive formulas
σ11 = α0
σ1d+1 = α0 +
d+1
σi+1
=
d
X
d−(i−1)
σid (γi+1σ1
i=1
d d+1
σi σ1
+ αi )
if d ≥ 1
if d ≥ 1 and i = 1, . . . , d
The proof of this theorem is elementary, but it uses some rather involved inductions,
and is highly unilluminating in general. Hence it will be deferred until the next section.
Here we only point out the following fact:
Remark From the definition of the γis and αi s it is obvious that γi (D d (B)) ⊂ D d (B)
and αi (D d (B)) ⊂ D d−1 (B) for all i = 0, . . . , n and all integers d ≥ 0 (we let D −1 (B) := 0).
Since JP,b is generated by the elements αi , it thus follows that ν(D d (B)) ⊂ D d−1 (B) for
all ν ∈ JP,b and all integers d ≥ 0. This implies that if D ∈ D d (B) for some d ≥ 0, then
0
d
ν(D) = 0 whenever ν ∈ JP,b
and d0 > d. Therefore the sum
P∞
d=1
νd (D) is well defined
for every D ∈ D(B).
The above also implies that if we are given any D ∈ D(B), say of order ≤ d, then
we may write Db = bD + ν(D) where ν = ν1 + . . . + νd ∈ ZP,b . From this we get the
following consequence of theorem 2.3.2:
Lemma 2.3.3 Let B be a commutative k-algebra, and let b ∈ B and P ∈ B[T ] be such
that P (b) = 0 and P 0(b) is an invertible element of B. Let A be a subring of B which
contains all the coefficients of the polynomial P . Then for all D ∈ D(B) and all integers
49
j ≥ 0 there are finitely many elements bi ∈ B, ai ∈ A such that
Dbj =
X
bi Dai
i
Proof:
Fix an element D ∈ D(B), say of order ≤ d, and an integer j ≥ 0. By theo-
rem 2.3.2 we have that Db = bD + ν(D) where ν = ν1 + . . . + νd . We recall that ν is
B-bilinear and hence by using the formula for Db twice, we get that
Db2 = bDb + ν(Db) = b(bD + ν(D)) + ν(bD + ν(D))
= b2 D + 2bν(D) + ν 2 (D)
An induction then shows that
j
Db =
j X
j
l
bj−l ν l (D).
l=0
and the result follows by applying lemma 2.3.1 to all the endomorphisms ν l .
2
We end this section by showing how it is possible to use theorem 2.3.2 to prove an
older result on invertible elements (cf. [8, 20]): Let B be a commutative k-algebra and let
s be an invertible element of B. Then s−1 is a root of the polynomial P = sT − 1 ∈ B[T ]
and P 0(s−1 ) = s is invertible. Thus theorem 2.3.2 applies.
First we compute αi for i ≥ 0. We have that deg P = 1, so by our definitions we have
for all D ∈ D(B) that
αi (D) = −P 0 (s−1 )
= −s−1
1
i
−1
0
i
(s−1 )
si−1 [D, s]
50
0−i
[D, −1] +
1
i
(s−1 )
1−i
[D, s])




−s−2 [D, s]



=
−s−1 [D, s]






0
if i = 0
if i = 1
if i ≥ 2
Next we compute the σid s. Since γi = 0 = αi if i ≥ 2, the recursive formulas become
σ11 = α0
σ1d+1 = α0 + σ1d α1
if d ≥ 1
d+1
σi+1
= σid σ1d+1
if i ≥ 1 and d ≥ 1
and then an easy induction on the first two formulas shows that
σ1d =
d−1
X
α0 (α1 )i
i=0
for all d ≥ 1. Thus from the definition of the νd s it follows directly that
νd = α0 (α1 )d−1
for all d ≥ 1. That is, for all d ≥ 1 and all D ∈ D(B) we have that
νd (D) = (−1)d s−d−1 [D, s](d)
where [D, s](d) := [. . . [[D, s], s], . . . , s] is the commutator repeated d times. Having this,
theorem 2.3.2 immediately implies the following
Proposition 2.3.4 If B is a commutative k-algebra and s is an invertible element of B,
then for all D ∈ D(B) we have the formula
Ds−1 = s−1 D +
∞
X
(−1)d s−d−1 [D, s](d)
d=1
51
where for all d ≥ 1, [D, s](d) := [. . . [[D, s], s], . . . , s] is the commutator repeated d times.
Especially, if we apply both sides of this equation to 1 ∈ B, we get that
D ∗ s−1 = s−1 D ∗ 1 +
∞
X
(−1)d s−d−1 [D, s](d) ∗ 1
d=1
As before we note that [D, s](d) = 0 if d > ord(D), so the infinite sums in the
proposition are well defined. This proposition has also been proved in [8, 20].
2.3.2
Proof of theorem 2.3.2
Let k be a commutative ring and let B be a commutative k-algebra. It is clear by
definition that D(B), the ring of (k-linear) differential operators on B, is a B-B-bimodule
in a natural way. We let EndB D(B)B denote the space of B-B-linear endomorphism on
D(B). It is clear that EndB D(B)B is a k-algebra in a natural way.
Since B is commutative, the formulas
ρ(b)(D) := Db
λ(b)(D) := bD
for b ∈ B and D ∈ D(B) define two maps ρ, λ : B → EndB D(B)B . It is trivial to check
that ρ and λ are in fact injective morphisms of k-algebras. Therefore the morphisms ρ
and λ are in fact two canonical ways of embedding B into EndB D(B)B . We want to
study the interplay of these two embeddings and hence we make the following definition:
Definition We let A(B) denote the smallest k-subalgebra of EndB D(B)B containing
the images of ρ and λ. That is, A(B) is the k-subalgebra of EndB D(B)B generated by
the set {ρ(b), λ(b)|b ∈ B}.
We get immediately:
Lemma 2.3.5 Let B be a commutative k-algebra. Then
52
(i) A(B) is commutative,
(ii) D(B) is a (left) A(B)-module via the natural action, and
(iii) for every d ≥ 0, D d (B) is a A(B)-submodule of D(B).
Remark In light of the lemma we will sometimes write σD instead of σ(D) for elements
σ ∈ A(B) and D ∈ D(B). Moreover, if there is no confusion about the underlying
k-algebra B, we will from now on write A, D and D d instead of A(B), D(B) and D d (B).
In general, elements of D do not commute with elements of B and hence are ρ(b) and
λ(b) usually different elements of A. We want to study the difference of them and thus
we make
Definition Let B be a commutative k-algebra. We let
ad := ρ − λ
and we let I be the ideal of A generated by the set {ad(b)|b ∈ B}.
We observe that for all b ∈ B, all d ≥ 0 and all D ∈ D d , we have that
ad(b)(D) = ρ(b)(D) − λ(b)(D) = Db − bD = [D, b] ∈ D d−1
(we recall that D −1 = 0 by definition). It follows immediately that ID d ⊂ D d−1 for all
d ≥ 0, and a trivial induction then implies that
I n D n+d ⊂ D d
for all integers n ≥ 0 and d ≥ −1.
We will now establish some technical lemmas concerned with computations in A, starting with:
53
Lemma 2.3.6 Let B be a commutative k-algebra, and let b ∈ B.
(i) For all integers r ≥ 0 we have a formula
r−1
X
ρ(br ) = λ(br ) +
λ(bi )ad(b)ρ(br−1−i )
i=0
(ii) Assume that b is a root of the polynomial P =
0=
n r−1
X
X
i
r−1−i
λ(ar b )ad(b)ρ(b
Pn
r=0
)+
r=0 i=0
Proof:
ar T r ∈ B[T ]. Then
n
X
ad(ar )ρ(br )
r=0
(i) We prove this by an induction on r. For r = 0 the formula reads ρ(1) = λ(1)
so there is nothing to prove. For higher powers we have
ρ(br+1 )
ρ(b)ρ(br )
=
ind. hyp.
=
r−1
X
ρ(b) λ(br ) +
!
λ(bi )ad(b)ρ(br−1−i )
i=0
r
=
(ad(b) + λ(b)) λ(b ) +
r−1
X
λ(bi )ad(b)ρ(br−i )
i=0
λ(br+1 ) +
=
r−1
X
λ(bi )ad(b)ρ(br−i ) + λ(br )ad(b)
i=0
λ(br+1 ) +
=
r
X
λ(bi )ad(b)ρ(br−i )
i=0
(ii) By assumption P (b) = 0 and hence is
0 = ρ(P (b))
=
=
=
n
X
ρ(ar br )
r=0
n X
λ(ar ) + ad(ar ) ρ(br )
r=0
n
X
r
λ(ar )ρ(b ) +
r=0
n
X
r=0
54
ad(ar )ρ(br )
By part (i) of the lemma we have that
n
X
n
X
λ(ar )ρ(br ) =
r=0
r=0
n
X
=
λ(ar ) λ(br ) +
r−1
X
!
λ(bi )ad(b)ρ(br−1−i )
i=0
λ(ar br ) +
r=0
n r−1
X
X
λ(ar bi )ad(b)ρ(br−1−i )
r=0 i=0
= λ(P (b)) +
n r−1
X
X
λ(ar bi )ad(b)ρ(br−1−i )
r=0 i=0
n r−1
X
X
=
λ(ar bi )ad(b)ρ(br−1−i )
r=0 i=0
and by combining these two formulas the proof is complete.
2
Before we can state our next lemma we need to introduce some additional notation.
d
For all integers d ≥ 0 we define equivalence relations ≡ on A as follows:
d
σ ≡ σ0
σ − σ0 ∈ I d
⇔
d
The careful reader notices of course that ≡ is just the same equivalence relation as
“congruence modulo I d ” but for our purposes this somewhat different notation is more
convenient. We observe immediately the following: Let d, d0 ≥ 0 be integers and let
d
0
d+d0
σ, σ 0 ∈ A be such that σ ≡ σ 0 . Then for all τ ∈ I d , στ ≡ σ 0 τ . With this notation in
hand we can now state:
Lemma 2.3.7 Let d ≥ 0 be an integer and let σ ∈ A, b ∈ B be such that
d
ρ(b) ≡ λ(b) + σ
Then for all integers r ≥ 0 we have that
d
ρ(br ) ≡ λ(br ) +
r
X
i=1
55
λ(br−i )σρ(bi−1 )
Proof:
d
For r = 0 the formula reads ρ(1) ≡ λ(1), a triviality and for r = 1 the implication
of the lemma becomes just our assumption. Then we proceed by induction on r:
ρ(br+1 )
=
d
≡
ρ(b)ρ(br )
r
ρ(b) λ(b ) +
r
X
!
r−i
λ(b
i−1
)σρ(b
)
i=1
=
r
X
r
ρ(b)λ(b ) +
λ(br−i )σρ(bi )
i=1
d
≡
=
(λ(b) + σ) λ(br ) +
r
X
λ(br−i )σρ(bi )
i=1
r
X
λ(br+1 ) + λ(br )σ +
λ(br−i )σρ(bi )
i=1
=
λ(br+1 ) +
r+1
X
λ(br+1−i )σρ(bi−1 )
i=1
2
where the last equality we get by a change of variables.
Let B be a commutative k-algebra, and let b ∈ B and P =
Pn
r=0 ar T
r
∈ B[T ] be such
that P (b) = 0 and P 0 (b) is an invertible element of B. As in section 2.3.1 we can then
define the endomorphisms γi and αi , the algebra ZP,b and the ideal JP,b of ZP,b. It is
clear from the definitions that ZP,b is a subalgebra of A and that JP,b ⊂ I.
It follows of course that
n
JP,b
D n+d (B) ⊂ D d (B)
for all integers n ≥ 0 and d ≥ −1.
We now recall the recursive definitions from the statement of theorem 2.3.2:
σ11 = α0
σ1d+1 = α0 +
d+1
σi+1
=
d
X
i=1
d d+1
σi σ1
d−(i−1)
σid (γi+1σ1
+ αi )
if d ≥ 1
if d ≥ 1 and i = 1, . . . , d
56
It is clear from these definitions that all the endomorphisms σid are elements of ZP,b . But
we have more, namely:
Proposition 2.3.8 Let b ∈ B be a root of the polynomial P =
that P 0(b) =
Pn
r−1
r=0 rar b
Pn
r=0 ar T
r
∈ B[T ] such
is an invertible element in B. Let the elements σid be defined
as above. Then:
i
(i) For all integers i, d such that 1 ≤ i ≤ d we have that σid ∈ JP,b
and
d+1
σid+1 − σid ∈ JP,b
;
(ii) For all integers s, d ≥ 0 we have a formula
d+1
ρ(b ) ≡ λ(b ) +
s
s
d X
s
i
λ(bs−i )σid ;
i=1
Proof:
i
(i) First we prove that σid ∈ JP,b
. We do this by a double induction, noting that
the case i = d = 1 follows right from the definitions. We assume that the statement holds
for all d0 ≤ d and all 1 ≤ i0 ≤ d0 . From that it follows directly that γj+1σ1
d−(j−1)
+αj ∈ JP,b
for all j such that 1 ≤ j ≤ d and then we have by the definition that
σ1d+1
= α0 +
d
X
d−(j−1)
σjd (γj+1σ1
+ αj ) ∈ JP,b
j=1
Then we use the induction hypothesis again to get that
d+1
i+1
i
σi+1
= σid σ1d+1 ∈ JP,b
JP,b = JP,b
for all 1 ≤ i ≤ d.
d+1
Next we prove that σid+1 − σid ∈ JP,b
. We also do that by a double induction.
57
First of all, for d = i = 1 we get from the recursive formulas that
σ12 − σ11 = α0 + σ11 (γ2 σ11 + α1 ) − α0
2
= α0 (γ2 α0 + α1 ) ∈ JP,b
since α0 and α1 are in JP,b.
We now assume that d > 1 and that we have proved the statement for all d0 < d and
all 1 ≤ i0 ≤ d0. Then by using the recursive formulas we get that
σ1d+1
−
σ1d
d
X
= α0 +
d−(j−1)
σjd (γj+1σ1
+ αj )
j=1
−α0 −
d−1
X
d−1−(j−1)
σjd−1 (γj+1σ1
+ αj )
j=1
= σdd (γd+1 σ11 + αd )
+
d−1
X
γj+1(σjd σ1d−j+1 − σjd−1 σ1d−j ) + (σjd − σjd−1 )αj
j=1
Here we pause to observe that we may write
σjd σ1d−j+1 − σjd−1 σ1d−j = σjd σ1d−j+1 − σjd σ1d−j + σjd σ1d−j − σjd−1 σ1d−j
= σjd (σ1d−j+1 − σ1d−j ) + (σjd − σjd−1 )σ1d−j
Thus, by putting this into the previous equation we get that
σ1d+1 − σ1d = σdd (γd+1 α0 + αd ) +
d−1
X
γj+1σjd (σ1d−j+1 − σ1d−j )
j=1
+
d−1
X
γj+1(σjd
−
σjd−1 )σ1d−j
j=1
+
d−1
X
(σjd − σjd−1 )αj
j=1
j
We have already shown that σjd ∈ JP,b
for all integers j such that 1 ≤ j ≤ d. We also
recall that αj ∈ JP,b for all j ≥ 0. We use this and our induction hypothesis to conclude
58
that
σdd (γd+1 α0 + αd ) ∈
d
JP,b
JP,b
d+1
= JP,b
j
d−j+1
d+1
γj+1σjd (σ1d−j+1 − σ1d−j ) ∈ JP,b
JP,b
= JP,b
γj+1(σjd − σjd−1 )σ1d−j ∈
d
JP,b
JP,b
d+1
= JP,b
(σjd − σjd−1 )αj ∈
d
JP,b
JP,b
d+1
= JP,b
for all j such that 1 ≤ j ≤ d − 1. Then it follows directly from the equation above that
d+1
σ1d+1 − σ1d ∈ JP,b
.
In order to finish the induction we must show the lemma holds for all i such that
1 < i ≤ d. But this is easy, since by the recursive formulas and the inductive hypothesis
we have for d > 1 and 1 < i ≤ d that
d−1 d
d
σid+1 − σid = σi−1
σ1d+1 − σi−1
σ1
d−1 d
d
d
d
= σi−1
σ1d+1 − σi−1
σ1d + σi−1
σ1d − σi−1
σ1
d−1
d
d
= σi−1
(σ1d+1 − σ1d ) + (σi−1
− σi−1
)σ1d
i−1 d+1
d
∈ JP,b
JP,b + JP,b
JP,b
d+1
= JP,b
This proves part (i) of the proposition.
(ii) We observe that the formula makes sense for all s, d ≥ 0. This is because σid is
defined for all 1 ≤ i ≤ d and
s
i
= 0 whenever s < i. Furthermore we observe that for
s = 0 the statement reduces to
d+1
ρ(1) ≡ λ(1),
for all d ≥ 0, a triviality since ρ(1) = λ(1) = 1. Also if d = 0, then the statement reads
1
ρ(bs ) ≡ λ(bs )
for all s ≥ 0, again a triviality which follows right from the definitions.
Having taken care of all the trivial cases, we will do the main body of the proof by
59
an induction on d.
We assume that the formula holds for all integers d0 ≤ d and we prove it for d + 1.
First we recall that ad(b) and ad(ar ) are all in I, so by lemma 2.3.6 and the induction
hypothesis (on d0 = d) we have that
0
n r−1
X
X
=
λ(ar bi )ad(b)ρ(br−1−i ) +
r=0 i=0
n r−1
X
X
d+2
≡
n
X
r=0

+
ad(ar ) λ(b ) +
r=0
d X
r
i
λ(br−1−i−j )σjd 
j
j=1
r

d X
r−1−i
λ(ar bi )ad(b) λ(br−1−i ) +
r=0 i=0
n
X
ad(ar )ρ(br )
r−i
λ(b
!
)σid
i=1
We rewrite this last expression, using the elementary fact that
r−1
X r−1−i
s
i=0
=
r−1
X
i
s
r
s+1
=
i=0
and the definitions of γi and αi , to obtain
0
d+2
≡
n
X
r−1
X
r=0
i=0
n
X
+
=
r=0
n
X
λ(
!
r−1
λ(ar b
) ad(b) +
r
λ(b )ad(ar ) +
d
X
+
r−1
rar b
)ad(b) +
r
λ(b )ad(ar ) +
r=0
=
r−1
X
r=0 j=1
i=0
n X
r
i
r−i
λ(b
r=0
d
X
n X
j=1
r=0
!
r−1−i
j
λ(ar br−1−j )ad(b)σjd
!
)ad(ar ) σid
i=1
r=0
n
X
n X
d
X
λ(
d
X
λ(P 0 (b))ad(b) +
n X
r
i
i=1
d
X
r
j+1
!
r−(j+1)
ar b
)σjd
ad(b)
!
r−i
λ(b
)ad(ar ) σid
r=0
λ(−P 0 (b))γj+1σjd ad(b)
j=1
0
0
−λ(P (b))α0 − λ(P (b))
d
X
αi σid
i=1
If we now multiply trough this equation by λ(P 0 (b)−1 ) and write ρ(b) − λ(b) for ad(b),
60
we get that
d+2
0 ≡ ρ(b) − λ(b) −
d
X
γj+1 σjd ad(b) − α0 −
d
X
j=1
or
d+2
ρ(b) ≡ λ(b) +
d
X
αi σid
i=1
γj+1ad(b)σjd + α0 +
j=1
d
X
αi σid
i=1
j
By part (i) of the proposition we have that σjd ∈ JP,b
for all j = 1, . . . , d. Then, since
ad(b) ∈ I, we can apply the induction hypothesis again (on d0 = d + 1 − j for j = 1, . . . , d
and s = 1) and we have
ad(b)σjd
=
ρ(b)σjd − λ(b)σjd
d+2
≡
λ(b)σjd +
=
σ1d+1−j σjd
d+1−j
X
1
i
λ(b1−i )σid+1−j σjd − λ(b)σjd
i=1
for all j = 1, . . . , d.
If we combine this with the formula we had before, we then have that
ρ(b)
d+2
≡
λ(b) +
d
X
γj+1σ1d+1−j σjd + α0 +
j=1
=
d
X
αi σid
i=1
λ(b) + α0 +
d
X
d−(i−1)
σid (γi+1 σ1
+ αi )
i=1
=
λ(b) + σ1d+1
where
σ1d+1
= α0 +
d
X
d−(i−1)
σid (γi+1 σ1
+ αi )
i=1
This proves the statement we want to prove in the special case when s = 1. For higher
powers on b we use lemma 2.3.7 to get the formula
d+2
ρ(bs ) ≡ λ(bs ) +
s
X
λ(bs−i )σ1d+1 ρ(bi−1 )
i=1
61
Then we recall that σ1d+1 ∈ JP,b and thus we can use the induction hypothesis still once
more (on d0 = d) to get the equations
d+2
σ1d+1 ρ(bi−1 ) ≡ λ(bi−1 )σ1d+1 +
d X
i−1
j
λ(bi−1−j )σjd σ1d+1
j=1
for i = 1, . . . , s. If we put this into the formula above, we finally get that
ρ(bs )
d+2
≡
λ(bs ) +
s
X

λ(bs−i ) λ(bi−1 )σ1d+1 +
i=1
=
=
s
λ(b ) +
λ(bs ) +
s
X
j
j=1
s−1
λ(b
)σ1d+1
+
s X
i−1
d
X
j
i=1
j=1
d X
s
1
d X
i−1
λ(bs−1 )σ1d+1 +

λ(bi−1−j )σjd σ1d+1 
!
λ(bs−1−j )σjd σ1d+1
i=1
s
j+1
λ(bs−(j+1) )σjd σ1d+1
j=1
=
λ(bs ) +
d+1
X s
λ(bs−j )σjd+1
j
j=1
2
d+1
where σj+1
= σjd σ1d+1 for j = 1, . . . , d
This proposition contains in fact everything we need in order to prove theorem 2.3.2:
Proof of theorem 2.3.2:
It suffices to demonstrate that the family {νd }∞
d=1 , defined by
the formulas
νd =


 σ1
1
if d = 1

 σ d − σ d−1
1
1
if d > 1
does satisfy the requirements of the theorem.
By part (i) of proposition 2.3.8 we have that ν1 = σ11 ∈ JP,b and that
d
νd = σ1d − σ1d−1 ∈ JP,b
for all d > 1.
62
0
d+1
Now fix an integer d0 ≥ 0, and an element D ∈ D d (B). Since JP,b
annihilates D d (B)
for all d ≥ 0, it follows that νd (D) = 0 for all d > d0 . Hence to finish the proof we must
show that
0
Db = bD +
d
X
νd (D).
d=1
But now by the definition of νd we have immediately that
0
d
X
0
νd = σ1d
d=1
so by applying part (ii) of proposition 2.3.8 to d = d0 and s = 1 we have that
ρ(b)
d0 +1
≡
0
λ(b) + σ1d
0
=
λ(b) +
d
X
νd
d=1
Pd0
d0 +1
By definition of the symbol ≡ this implies that ρ(b) − λ(b) −
0
0
d +1
annihilates the space D d (B), it follows that Db − bD −
JP,b
2.3.3
Pd0
0
d=1
d=1
d +1
νd ∈ JP,b
. Since
νd (D) = 0.
2
Right ideals and étale homomorphisms
Let A be a commutative k-algebra such that PnA is finitely presented for all integers n ≥ 0.
Let f : A → B be an étale homomorphism of k-algebras. Then theorem 2.2.10 gives an
isomorphism B ⊗A D(A) ∼
= D(B) of B-D(A)-bimodules. This implies immediately that
if I is a left ideal of D(A), then B ⊗A I is a left ideal of D(B). A similar argument is
obviously not available to us if I is a right ideal of D(A) (it need not even be a left Asubmodule). However, if a right ideal I of D(A) is at the same time a left A-submodule,
we will in this section show that B ⊗A I is indeed a right ideal of D(B). This applies
especially to twosided ideals of D(A). We start with a lemma and proposition which
proves this claim for localizations and local étale morphisms:
63
Lemma 2.3.9 Let A be a commutative k-algebra and let P ∈ A[T ] be a monic polynomial. Let B be a localization of the k-algebra A[T ]/P such that P 0 (T ) is invertible in B.
Let f : A → B be the composite A → A[T ] → A[T ]/P → B. Then f is étale, and for all
b ∈ B and D ∈ D(A) there are bi ∈ B and ai ∈ A such that
f (D)b =
X
bi f (Dai)
i
Proof:
By [16, Example 3.4] we have that A → (A[T ]/P )P 0(T ) =: B 0 is étale (and thus
formally étale). The algebra B is a localization of B 0 , and then the results of section 2.1.2
imply that f : A → B is étale. By theorem 2.2.5, f then extends to a homomorphism
f : D(A) → D(B) of filtered rings.
For convenience of notation, let us write C := A[T ]/P , let t ∈ C be the image of
T ∈ A[T ], and let λ : C → B be the localization map.
Let us now fix elements D ∈ D(A) and b ∈ B. Having set the notation as above,
we may write b = λ(c0 )λ(s)−1 with c0 , s ∈ C. Then by expanding the commutators in
proposition 2.3.4 it follows that there are bi ∈ B and integers ni ≥ 0 such that
f (D)b = (f (D)λ(c0))λ(s)−1 =
X
bi (f (D)λ(c0))λ(s)ni
i
=
X
bi f (D)λ(c0sni )
i
This implies that we may assume that b = λ(c) for some c ∈ C. Since A[T ] → C is
surjective, and t is the image of T under this map, it follows that we may write c = Q(t)
for some Q =
P
i
a0i T i ∈ A[T ]. Then
f (D)λ(c) =
X
f (Da0i )λ(ti )
i
and therefore we may assume that b = λ(c) = λ(ti ) for some integer i ≥ 0. Finally we
64
observe that by our assumptions P (λ(t)) = 0 and P 0 (λ(t)) is invertible in B, and all the
coefficients of the polynomial P are in f (A). Hence lemma 2.3.3 applies and we conclude
that
f (D)λ(ti) =
X
bj f (D)f (aj ) =
X
j
bj f (Daj )
j
for some bj ∈ B, aj ∈ A.
2
Remark Taking P = T ∈ B[T ] in this lemma shows especially that it holds when B
is any localization of A. Furthermore, theorem 2.1.4 implies that the lemma holds also
whenever f : A → B is a local étale homomorphism of local rings.
Proposition 2.3.10 Let A be a commutative k-algebra such that PnA is finitely presented
for all integers n ≥ 0, and assume that f : A → B is either a localization of k-algebras,
or a local étale homomorphism of local k-algebras. Let J be a left A-submodule of D(A).
If J is a right ideal of D(A), then B ⊗A J is a right ideal of D(B).
Proof:
By definition, the left B-submodule B ⊗A J of D(B) is generated by the elements
f (D), D ∈ J, and since f is étale, we have by theorem 2.2.10 that B ⊗A D(A) ∼
= D(B)
via b⊗D 7→ bf (D). Since we are assuming that J is a right ideal of D(A), it therefore
suffices to show that f (D)b ∈ B ⊗A J if D ∈ D(A) and b ∈ B. However by lemma 2.3.9
and the remark following it, we have that
f (D)b =
X
bi f (Dai)
i
for some bi ∈ B and ai ∈ A. Every Dai ∈ J by assumption and we are done.
2
Next we prove a lemma which shows that in order to determine if a left A-submodule
of D(A) is a right ideal, it is necessary and sufficient to do so locally:
65
Lemma 2.3.11 Let A be a commutative k-algebra such that PnA is finitely presented for
all integers n ≥ 0, and let J be a left A-submodule of D(A). Then J is a right ideal of
D(A) if and only if Jp (= Ap ⊗J) is a right ideal of D(Ap ) for all p ∈ Spec A.
Proof:
The direction “⇒” follows directly from proposition 2.3.10.
For the other direction assume that Jp is a right ideal of D(Ap ) for all p ∈ Spec A.
For p ∈ Spec A, let λp : D(A) → D(Ap ) denote the localization map. By definition we
have that λp (J) ⊂ Jp , and if D ∈ D(A) is such that λp (D) ∈ Jp for all p ∈ Spec A,
then [14, theorem 4.6] implies that D ∈ J (the localizations are just localizations as left
A-modules). But all the λp s are ring homomorphisms, so if Jp is a right ideal for all p,
we have for D ∈ J and D 0 ∈ D(A) that
λp (DD 0 ) = λp (D)λp (D 0) ∈ Jp D(Ap ) ⊂ Jp ,
so DD0 ∈ J.
2
We can now prove the theorem we have been aiming at in this section:
Theorem 2.3.12 Let A be a commutative k-algebra such that PnA is finitely presented
for all integers n ≥ 0, and let f : A → B be an étale homomorphism of k-algebras. Let
J be a left A-submodule of D(A). If J is a right ideal of D(A), then B ⊗A J is a right
ideal of D(B).
66
Proof:
Fix a P ∈ Spec B and let p = f −1 (P ) ∈ Spec A. Then f induces a local étale
morphism Ap → BP , and we get a commutative diagram
D(A)
?
D(Ap )
f
- D(B)
?
- D(BP )
It follows that (B ⊗A J)P = BP ⊗Ap Jp , and by applying proposition 2.3.10 twice, we
thus have that (B ⊗A J )P is a right ideal of D(BP ). This holds for all P ∈ Spec B and
then lemma 2.3.11 implies that B ⊗A J is a right ideal of D(B).
2
Left ideals of D(A) are automatically left A-submodules, and thus the theorem implies
the following result which we state as a corollary:
Corollary 2.3.13 Let A be a commutative k-algebra such that PnA is finitely presented
for all integers n ≥ 0, and let f : A → B be an étale homomorphism of k-algebras. If J
is a twosided ideal of D(A), then B ⊗A J is a twosided ideal of D(B).
67
Chapter 3
Minimal twosided ideals in rings of
differential operators
Let k be an algebraically closed field of characteristic zero, and let A be a finitely generated k-algebra and a domain of krull dimension 1. In this case it was proved in [25]
that A contains a minimal nonzero D(A)-module J, and that D(A) contains a minimal
nonzero twosided ideal J . The same result, in a slightly more general setting but with a
similar proof, was independently obtained in [17]. In [25] it was moreover shown that the
result above also holds if A is replaced by its localization at a singular maximal ideal.
The method employed in [17, 25] was to compare D(A) to the ring of differential
operators on the normalization of A and use the fact that in this case the normalization
is a regular ring.
In this chapter we prove some generalizations of the results described above, notably
to certain reduced rings of krull dimension more than 1. If the krull dimension of A
is more than 1, the method described above can no longer be used, partly because the
normalization need not be regular. Instead, for a commutative, noetherian and reduced
k-algebra A, we use supports of regular functions on Spec A to construct a canonical
D(A)-submodule of A, called ΓA . Then if the irreducible components of A have rings
of differential operators that are simple, we prove that statements analogous to those
68
of [17, 25] hold for the module ΓA and the ideal D(A, ΓA ). In many cases this method
gives no new information, e.g. when A is a domain. However, if A is as before and
it has an étale covering B which is a componentwise geometrical unibranch(we define
this in the chapter), we can construct the D(B)-module ΓB as above, and we use the
descent theory from chapter 2 to construct a canonical D(A)-submodule χA of A such
that ΓB ∼
= B ⊗A χA . If in addition A has finitely presented modules of differentials, and
the irreducible components of B have rings of differential operators that are simple, we
show that χA is the minimal essential left D(A)-submodule of A and that D(A, χA ) is
the minimal essential twosided ideal of D(A). Hence these are proper analogues of the
module J and the ideal J mentioned at the beginning.
Our method applies especially to local reduced algebras of krull dimension 1. Hence
apart from being generalizations to rings of higher krull dimension, the results of this
chapter show how the theorems of [17, 25] reflect the (rather natural) fact that differential
operators preserve supports of functions.
We should point out that from [1] it follows that the ring of differential operators on
the cubic cone has an infinite descending chain of twosided ideals (cf. [4]). Hence one
can not hope for a generalization of the main results of this chapter which holds for all
reduced finitely generated algebras over algebraically closed fields of characteristic zero.
Finally we note that rings of differential operators over commutative rings with zero
divisors have been studied in [19], but there the theory is approached in a somewhat
different way and results of different type are obtained.
3.1
3.1.1
Supports in reduced commutative rings
Associated prime ideals
The following definition appears in [14].
69
Definition Let A be a commutative ring and let p ⊂ A be a prime ideal.
(i) p is said to be associated if p = Ann(x) for some nonzero x ∈ A.
(ii) An associated prime of A is embedded if it strictly contains some other associated
prime of A. Otherwise it is said to be isolated.
We let Ass(A) denote the set of associated prime ideals of A. We list a few useful
elementary facts about associated primes in noetherian rings:
Lemma 3.1.1 Let A be a commutative, noetherian ring. Then:
(i) the set Ass(A) is finite;
(ii) the ideal
\
p
p∈Ass(A)
is equal to the set of all nilpotent elements of A;
(iii) the set
[
p
p∈Ass(A)
is equal to the set of all zero-divisors of A.
(iv) If A is also reduced, then it has no embedded primes.
Proof:
The first three parts are implied by [14, Theorems 1.2, 6.1, 6.5], and the last
2
past is an easy exercise.
For a commutative noetherian ring A, we let Min(A) denote the set of minimal prime
ideals of A, ordered by inclusion. It is not hard to see that Min(A) ⊂ Ass(A) whenever
A is a noetherian commutative ring (cf. [14, Theorem 6.5]). Thus it follows from part 4
of last lemma that Min(A) = Ass(A) for a reduced noetherian commutative ring. In the
sequel we will mostly be working with reduced rings and we will use this equality without
further comment.
We need another definition from [14]:
70
Definition If A is a commutative ring and I ⊂ A is an ideal, let
ΓI = {a ∈ A; I n a = 0 for some n > 0}
Let A be a commutative ring and let I be an ideal of A. From algebraic geometry we
recall the definition of the variety corresponding to I, V (I) := {p ∈ Spec A|I ⊂ p}. Let
a ∈ ΓI . If p 6∈ V (I), then there is an x ∈ I \ p such that xn a = 0 for some positive integer
n. However xp is a unit in Ap , and therefore it follows that ap = 0 in Ap . Conversely
assume that a ∈ A is such that ap = 0 in Ap for all p 6∈ V (I). Choose an x ∈ I. Then for
all p ∈ Spec A \ V (Ax), we have by assumption that ap = 0, and hence it follows that
a(x) = 0 in A(x) . This implies that xn a = 0 for some positive integer n. This shows that
if A is noetherian, then ΓI is the ideal of regular functions on Spec A that are supported
on V (I).
For a prime ideal p ∈ Spec A, it follows right from the definitions that Γp 6= 0 if and
only if p ∈ Ass(A). Using lemma 3.1.1 we can characterize Γp when p is an associated
prime ideal of a commutative noetherian reduced ring:
Lemma 3.1.2 Let A be a commutative, noetherian and reduced ring, and let p ∈ Ass(A).
Then
(i) Γp =
T
q∈Ass(A)
q6=p
q,
(ii) p = ΓΓp
Proof:
Let x ∈ Γp . Then pn x = 0 for some integer n. Since all the associated primes
are minimal, for all q ∈ Ass(A), with p 6= q, we can find an y ∈ p \ q. Then y n x = 0 ∈ q,
so x ∈ q. Conversely if x is contained in all associated primes of A except p, then px is
contained in all the associated primes of A. Since A is reduced, it follows that px = 0.
Hence x ∈ Γp . This proves part (i).
71
Since the intersection of all the associated primes of a reduced ring is zero, part (i)
of the lemma implies immediately that p ⊂ ΓΓp . Conversely assume that x ∈ ΓΓp . Then
for each q ∈ Ass(A), q 6= p choose an yq ∈ q \ p (possible since all the associated primes
are minimal), and let y be the multiple of these elements. Then by lemma 3.1.2 we have
that y ∈ Γp , and hence it follows that y n x = 0 ∈ p for some integer n ≥ 0. Since y 6∈ p
and p is a prime ideal, this implies that x ∈ p.
2
The sum of all the Γp for p ∈ Ass(A), will play an important role in the sequel. Thus
we give it a name:
Definition Let A be a commutative, noetherian and reduced ring. We let
X
ΓA :=
Γp
p∈Ass(A)
By using the basic properties of varieties and some elementary set theory, it follows
easily that
V (ΓA ) =
\
[
V (q)
p∈Ass(A) q∈Ass(A)
q6=p
=
[
V (p) ∩ V (q)
p,q∈Ass(A)
q6=p
That is, the variety corresponding to ΓA is the closed subset of Spec A consisting of all
points that lie in at least two different irreducible components of Spec A. We will now
give some descriptions of ΓA . First we have:
Lemma 3.1.3 Let A be a commutative, noetherian and reduced ring. Then ΓA =
L
p∈Ass(A)
Γp and it contains a regular element of A.
72
Proof:
Assume that xp ∈ Γp are elements such that
Then xp = −
P
q6=p
P
xp = 0. Choose a p ∈ Ass(A).
xq , and by lemma 3.1.2 we have that the right hand side of this
equation lies in p. Therefore we have that xp ∈ p ∩ Γp = 0. This shows that ΓA is equal
to the direct sum of all the Γp for p ∈ Ass(A).
It is clear from the definition that Γp 6= 0 for all p ∈ Ass(A). Thus we may choose
0 6= xp ∈ Γp for all p ∈ Ass(A). Let x =
P
xp . From lemma 3.1.2 it then follows that x
is not contained in any associated prime ideal of A, and then lemma 3.1.1 implies that x
2
is regular.
Let A be a commutative, noetherian and reduced ring and write Ass(A) = {p1 , . . . , pn }
(we recall that this is also the set of minimal prime ideals of A). For i = 1, . . . , n we
let Ai := A/pi , πi : A → Ai the natural projections, Ā :=
Qn
i=1
Ai and λi : Ai → Ā the
natural injections. Finally we let ι : A → Ā be the map a 7→ (πi (a)), and for all i we
let ei := λi (1). It is clear that each ei is idempotent, ei ej = 0 whenever i 6= j and that
Pn
i=1 ei
= 1.
Since A is reduced, we have that
Tn
i=1
pi = 0 and hence it follows that ι is injective.
We may thus view A as a subring of Ā via this map. Then K, the total ring of fractions
of A, is also the total ring of fractions of Ā and it is equal to the direct product
Qn
i=1
Ki
where Ki is the field of fractions of Ai , i = 1, . . . , n.
Now fix an integer 1 ≤ i ≤ n. From the definitions and identifications above, and
lemma 3.1.2 it then follows that Γpi = (0, . . . , 0, πi(Γpi ), 0, . . . , 0), which is clearly an
ideal of Ā. Hence under the identification A ⊂ Ā we may identify ΓA with the ideal
(π1 (Γp1 ), . . . , πn (Γpn )). We have more, namely:
Proposition 3.1.4 Let A be a commutative, noetherian and reduced ring, and let Ā be
as above. Then ΓA is equal to the conductor of Ā into A.
73
Proof:
Since ΓA is an ideal of Ā and contained in A, it follows immediately that it is
contained in the conductor. Conversely let a ∈ A be such that aĀ ⊂ A. Fix an integer
1 ≤ i ≤ n. We have by assumption that aei = (0, . . . , 0, πi (a), 0, . . . , 0) ∈ A, and it
follows that πj (aei ) = 0 for all j 6= i. This, and lemma 3.1.2 then implies that aei ∈ Γpi .
This holds for all i and then we have by definition that a = ae1 + . . . + aen ∈ ΓA .
2
We end this section by giving a certain combinatorial description of ΓA , which we
need when we discuss étale homomorphisms.
Definition Let U be a finite set. A partition of U is a finite family S = {Si } of subsets
of U such that
S
i
Si = U and Si ∩ Sj = ∅ whenever i 6= j.
We will study partitions of the set of associated primes in a reduced noetherian ring.
Hence we make:
Definition Let A be a commutative, noetherian and reduced ring. A partition S = {Si }
of the finite set Ass(A) is said to be fine if p + p0 = A whenever p and p0 lie in the same
Si and p 6= p0 .
Remark First of all note that we allow some of members of a partition to be the empty
set. Also we observe that by this definition, any partition of Ass(A) into singletons and
empty sets is automatically fine.
Lemma 3.1.5 Let {I1 , . . . , Ir }, r ≥ 2 be a set of pairwise comaximal ideals of a commutative ring A. Then
A=
r
X
(
ν 0 =1
Proof:
r
\
Iν )
ν=1
ν6=ν 0
We have that
V(
Pr
ν 0 =1
(
Tr
ν=1
ν6=ν 0
r
r
\
[
Iν )) =
ν 0 =1 ν=1
ν6=ν 0
74
V (Iν )
[
=
(V (Iν ) ∩ V (Iν 0 ))
1≤ν<ν 0 ≤r
= ∅
since by our assumption, V (Iν ) ∩ V (Iν 0 ) = ∅ whenever ν 6= ν 0 . This proves the lemma.
2
Proposition 3.1.6 Let A be a commutative, noetherian and reduced ring, and let S =
{Si }ni=1 be a fine partition of Ass(A). Then
ΓA = IS :=
n
X
\
(
p)
i=1 p∈Ass(A)\Si
Proof:
Let T = {Tj }m
j=1 be a partition of Ass(A) into singletons. By definition of ΓA
and lemma 3.1.2 we have that ΓA = IT . Moreover, since S is a partition of Ass(A) it
follows that for every j there is an i such that Tj ⊂ Si . From this we get that IT ⊂ IS .
We prove the other inclusion. By symmetry it clearly suffices to show that
\
p ⊂ IT
p∈Ass(A)\S1
If S1 = ∅, the statement holds trivially since the intersection of all the associated primes
of a reduced ring is equal to zero. Let us then assume that S1 contains only one element,
say S1 = {q}. Since T is a partition of Ass(A), there is an j such that q ∈ Tj . It clearly
follows that
\
p∈Ass(A)\{q}
\
p⊂
p∈Ass(A)\Tj
75
p ⊂ IT .
If S1 contains more than one element, we use the assumption that S is a fine partition
of Ass(A) and lemma 3.1.5 to write
A=
X
(
\
q)
q 0 ∈S1 q∈S1
q6=q 0
and then we can write
\
p =
X
(
\
q)(
q 0 ∈S1 q∈S1
q6=q 0
p∈Ass(A)\S1
⊂
X
(
\
p)
p∈Ass(A)\S1
\
p)
q 0 ∈S1 p∈Ass(A)\{q 0 }
Finally we can use the case when S1 contains only one element to show that each term
2
of the last sum is contained in IT .
3.1.2
Geometrical unibranches
Recall our definition of a partition from section 3.1.1. We start this section by proving a
lemma on the behavior of minimal prime ideals under étale homomorphisms:
Lemma 3.1.7 Let A be a commutative noetherian ring and let A → B be an étale
homomorphism. For all p ∈ Min(A) let Sp := {P ∈ Spec B|P ∩ A = p}. Then
(i) the family {Sp }p∈Min(A) is a partition of Min(B);
(ii) for all p ∈ Min(A) we have that
pB =
\
P
P ∈Sp
(where the right hand side is defined to be equal to B if Sp = ∅).
76
Proof:
(i). Flat local homomorphisms are faithfully flat. Hence if P ∈ Spec B and
p = P ∩ A, then the morphism Spec BP → Spec Ap is surjective. If P ∈ Min(B), this
implies that Ap has only one prime ideal, and thus p must be a minimal prime of A.
Conversely let p be a minimal prime of A. If Sp = ∅ there is nothing to prove (recall
that we allow members of a partition to be the empty set). Hence we may assume that
p = P ∩ A for some P ∈ Spec B. Then P contains a minimal prime P 0 of B, and
by the minimality of p it follows that p = P 0 ∩ A also. By [9, Ex. II.3.10] we have
that the fiber over p is homeomorphic to Spec B ⊗A κ(p), and since A → B is étale it
follows from lemma 2.1.3 that B ⊗A κ(p) is a finite direct product of fields, and hence
of krull dimension 0. This implies that P 0 = P and thus P ∈ Min(B). This shows that
Sp ⊂ Min(B).
(ii). By base extension, the ring B/pB is an étale A/p-algebra. Hence it is reduced
by lemma 2.1.6, so the intersection of all the minimal (= associated) primes of B/pB is
equal to zero. This means precisely that the intersection of all the minimal elements of
the set S := {P ∈ Spec B|p ⊂ P } is equal to pB. Thus it only remains to show that the
minimal elements of S are actually minimal primes of B. Assume that p ⊂ P 0 for some
P 0 ∈ Spec B. Then p ⊂ p0 = P 0 ∩ A and since the going-down theorem holds for flat
homomorphisms, we have that there is a P ∈ Spec B such that P ⊂ P 0 and p = P ∩ A.
Thus P ∈ S and part (i) of the lemma implies that P ∈ Min(B).
2
Let A be a commutative noetherian local ring and let ι : κ(A) → Ω be a separable
closure of κ(A). As in [7, (IV,18.8)], all diagrams of the type
- Ω
B
6
6
ι
A
can.
77
- κ(A)
where A → B is a local étale homomorphism and B → Ω is a local homomorphism, form
a direct system (observe that our definition of “local étale homomorphism” coincides
with the definition of “essentiellement étale” in [7, (IV, 18.6.1)]).
Definition Let A be a local noetherian ring. The direct limit of the direct system
defined above is called the strict henselization of A, and it is denoted by Ash .
If A is a local noetherian ring, then Ash is also a local noetherian ring and A → Ash
is a local homomorphism (cf. [7, (IV, 18.8.8)]). This morphism is faithfully flat since
directs limit of flat modules are flat and flat local homomorphisms are faithfully flat.
This implies that A → Ash is injective and Spec Ash → Spec A is surjective. Finally [7,
(IV, 18.8.13)] implies that Ash is reduced if and only A is reduced.
Remark We should note that the definition of the strict henselization of the local ring
A depends on the choice of the separable closure ι : κ(A) → Ω. Hence if A0 is the
strict henselization of A with respect to some other separable closure ι0 : κ(A) → Ω0 of
κ(A), then Ash and A0 are not canonically isomorphic. Nevertheless as A-algebras, these
strict henselizations are isomorphic (cf. [7, (IV, 18.8.8)]). For our purposes we may fix a
separable closure once and for all.
Lemma 3.1.8 Let A be a noetherian and reduced local ring. Every local étale A-algebra
is isomorphic to an A-subalgebra of Ash , and every element of Ash is contained in a local
étale A-subalgebra of Ash .
Proof:
Fix a separable closure κ(A) → Ω. Let A → B be a local étale homomorphism.
Then by lemma 2.1.3, κ(B) is a finite separable field extension of κ(A), and thus there is
a κ(A)-linear morphism α : κ(B) → Ω. By [7, (IV,18.8.4)], this morphism corresponds
78
to a unique local homomorphism B → Ω making the diagram
- Ω
B
6
A
commutative. From the definition of Ash we thus have a local A-algebra homomorphism
B → Ash , and Ash is also the strict henselization of B. Injectivity follows directly.
The second claim follows directly from the definitions of Ash and basic properties of
2
direct limits.
Lemma 3.1.9 Let A be a local noetherian ring.
(i) A prime ideal P ∈ Spec Ash is in Min(Ash ) if and only if P ∩ A is in Min(A).
(ii) Let p ∈ Min(A). Then
pAsh =
\
P
P ∈Min(Ash )
P ∩A=p
Proof:
(i). By [7, (IV, 18.8.12)] we have for all p ∈ Spec A that Ash ⊗A κ(p) is a finite
direct product of fields. Since A → Ash is flat, we can thus proceed precisely as in the
proof of part (i) of lemma 3.1.7.
(ii). We already pointed out that Ash is reduced if and only A is. Then by using
part (i), we can proceed as in the proof of part (ii) of lemma 3.1.7.
2
Remark By [7, IV, corollaire (18.8.14)] we also have that a prime ideal P ∈ Spec Ash
is in Ass(Ash ) if and only if P ∩ A is in Ass(A). If A is reduced, this fact implies the
lemma above.
79
The lemma, combined with the earlier fact that Spec Ash → Spec A is surjective,
implies that we have a surjective morphism Min(Ash ) → Min(A); P 7→ P ∩ A. We use
this in the next definition:
Definition We say that a noetherian ring A is a componentwise geometrical unibranch
at p ∈ Spec A if the the map Min(Ash
p ) → Min(Ap ) is injective (and hence bijective). We
say that A is a componentwise geometrical unibranch if A is a componentwise geometrical
unibranch at all p ∈ Spec A.
Let A be a noetherian local ring and assume that A is irreducible (i.e., that it has
only one minimal prime ideal). Then the definition above clearly implies that A is a
componentwise geometrical unibranch at its maximal ideal if and only if Ash has only
one minimal prime ideal. This shows that for such a ring, the definition of a geometrical
unibranch in [7, 21] is similar to our definition of a componentwise geometrical unibranch
at the maximal ideal. However:
Important Warning We must note that a local noetherian ring may be a componentwise geometrical unibranch at its maximal ideal without being a componentwise
geometrical unibranch (at all points).
We should also point out that the definition above is not quite in line with the usual
definition of geometrical unibranches (cf. e.g. [21, définition IX.2] or [7, (0IV , 23.2.1) and
(IV, 18.8.15)]), since usually they are irreducible by definition. The next lemma shows
though, that being a componentwise geometrical unibranch is in fact a condition on the
irreducible components:
Lemma 3.1.10 Let A be a noetherian ring. It is a componentwise geometrical unibranch
if and only if all its irreducible components are geometrical unibranches.
Proof:
Assume that A is a componentwise geometrical unibranch. Let p be a minimal
prime ideal of A and let q ∈ Spec A be such that p ⊂ q. By assumption there is a unique
80
minimal prime ideal of Ash
q , say P , such that p = P ∩ Aq . Then lemma 3.1.9 implies that
sh ∼
sh
sh
sh
P = pAsh
q , and thus we have by [7, (IV, 18.8.10)] that (A/p)q = Aq /pAq = Aq /P .
This shows that (A/p)sh
q is a domain and hence A/p is a geometrical unibranch at q.
This holds for all minimal prime ideals p of A and all prime ideals q containing p, and
therefore by definition all the irreducible components are geometrical unibranches.
For the converse let q be a prime ideal of A and assume that P, P 0 ∈ Min(Ash
q ) are
such that p := P ∩ Aq = P 0 ∩ Aq . By lemma 3.1.9, p is necessarily a minimal prime ideal
0
of Aq (which we may take to be a minimal prime ideal of A). Then pAsh
q ⊂ P ∩ P , and
sh
so P and P 0 are both minimal prime ideals of Ash
q /pAq which in turn is isomorphic to
(A/p)sh
q , again by [7, (IV, 18.8.10)]. Since we are assuming that A/p is a geometrical
sh
0
unibranch, this implies that Ash
q /pAq is a domain, and hence P = P .
2
Next lemma (cf. [21, corollaire IX.1, définition IX.2]) gives a different description of
geometrical unibranches:
Lemma 3.1.11 Let A be a noetherian local integral domain. The following two conditions on A are equivalent:
(i) Ash , the strict henselization of A, has a unique minimal prime.
e the normalization of A, is a local ring and the corresponding extension of
(ii) A,
residue fields has separable degree 1.
For the next result we recall the definition of fine partitions from section 3.1.1:
Proposition 3.1.12 Let A be a componentwise geometrical unibranch, let A → B be an
étale homomorphism and for p ∈ Min(A) let Sp := {P ∈ Min(B)|P ∩ A = p}. Then the
family {Sp }p∈Min(A) is a fine partition of Min(B).
Proof:
Let M be a maximal ideal of B and let m := M ∩ A. Then the local morphism
Am → BM is also étale by the results in section 2.1.2, so by lemma 3.1.8 we have that BM
81
is isomorphic to a subalgebra of Ash
m . Since A is a componentwise geometrical unibranch,
this implies that the morphism Min(BM ) → Min(Am ) is injective (in fact bijective).
Now let P, P 0 ∈ Min(B) such that P ∩ A = p = P 0 ∩ A where p ∈ Min(A) and P 6= P 0 .
If P + P 0 6= B, then there is a maximal ideal M ∈ Spec B containing both the ideals,
and hence P and P 0 are distinct minimal prime ideals of BM lying over the same minimal
prime ideal of Am , where m = M ∩ A. This is a contradiction to the first paragraph of
2
the proof.
For a reduced noetherian ring A we recall the definition of ΓA from section 3.1.1.
Having the result above we can now prove:
Theorem 3.1.13 Let A be a reduced componentwise geometrical unibranch and let A →
B be an étale homomorphism. Then there is a canonical isomorphism B ⊗A ΓA ∼
= ΓB .
Proof:
Since A is reduced we have that Ass(A) = Min(A). The same is true for B
by lemma 2.1.6. For p ∈ Min(A) we let Sp := {P ∈ Min(B)|P ∩ A = p} and we let S
denote the family {Sp }p∈Min(A) . This family is a fine partition of Min(B) = Ass(B) by
proposition 3.1.12.
For p ∈ Ass(A) we have by lemma 3.1.2 that
Γp =
\
q
q∈Ass(A)
q6=p
Thus, since B is a flat A-module, it follows from [14, Theorem 7.4] that
B ⊗A Γp =
\
q∈Ass(A)
q6=p
82
B ⊗A q
By lemma 3.1.7 we have for all q ∈ Ass(A) a canonical isomorphism
B ⊗A q ∼
=
\
Q
Q∈Sq
and thus it follows that
X
B ⊗A ΓA =
B ⊗A Γp
p∈Ass(A)
X
=
\
B ⊗A q
p∈Ass(A) q∈Ass(A)
q6=p
X
∼
=
\
\
Q
p∈Ass(A) q∈Ass(A) Q∈Sq
q6=p
Here we pause for a moment to observe the elementary set theoretic fact that since
S = {Sq }q∈Ass(A) is a partition of Ass(B) , we have
\
\
\
Q=
q∈Ass(A) Q∈Sq
q6=p
Q.
Q∈Ass(B)\Sp
This, along with the previous formula, implies that
B ⊗A ΓA ∼
=
X
\
Q = IS
p∈Ass(A) Q∈Ass(B)\Sp
where IS is defined as in proposition 3.1.6. Since S is a fine partition of Ass(B) , the
result now follows directly from proposition 3.1.6.
83
2
3.2
Minimal twosided ideals and minimal left submodules
3.2.1
Differential operators and supports
We will now relate supports of regular functions on spectra of commutative rings to their
rings of differential operators. For that we will need the following easy lemma from the
folklore. We include a proof here for the convenience of the reader.
Lemma 3.2.1 Let A be a commutative k-algebra and let I be an ideal of A. The ideal
ΓI is a left D(A)-submodule of A. That is, D(A) ∗ ΓI ⊂ ΓI .
Proof:
Let D d denote the set of differential operators on A of order ≤ d. We claim that
I n+d D d ⊂ D d I n for all integers d ≥ 0 and n ≥ 0.
The lemma follows immediately: Let a ∈ ΓI . Then there is an integer n ≥ 0 such that
I n a = 0. Hence if D ∈ D(A), say of order ≤ d, the claim implies
I n+d D ∗ a ⊂ D d I n ∗ a = D d ∗ (I n a) = 0
so by definition D ∗ a ∈ ΓI .
We now prove the claim. First we note that for n = 0 or d = 0 the statement is trivial.
We then proceed by a double induction. We prove the claim for a pair (n, d) where both
n and d are greater than zero. By induction we may assume that the statement is true
for all pairs (n0 , d0 ) where either n0 < n or d0 < d.
84
Let x ∈ I d+n−1 , y ∈ I and D ∈ D d . Our task is to show that xyD ∈ D d I n . However,
by our induction hypothesis we have that xD ∈ D d I n−1 and x[D, y] ∈ D d−1 I n (since
[D, y] ∈ D d−1 ), and then
xyD = xDy − x[D, y] ∈ D d I n−1 I + D d−1 I n = D d I n
2
We already pointed out that ΓI is the ideal of regular functions on Spec A whose
support is contained in the closed subvariety V (I). Hence in terms of geometry, this
lemma says that differential operators preserve supports of regular functions.
Now let A be a commutative, noetherian and reduced k-algebra. By the definition
of ΓA in section 3.1.1, the lemma above implies immediately that it is always a left
D(A)-submodule of A. Using this we can sometimes construct less obvious left D(A)submodules of A. But to simplify our notation we first make a definition:
Definition If A → B is a faithfully flat étale homomorphism, we say that B is an étale
covering of A.
This definition is justified by the fact that if A → B is a faithfully flat homomorphism of commutative rings, then the corresponding map Spec B → Spec A is surjective
(cf. [14, theorem 7.3]). We have:
Theorem 3.2.2 Let A be a commutative, noetherian and reduced k-algebra, and assume that A has an étale covering B which is a componentwise geometrical unibranch.
Then
(i) there is a unique ideal χA of A such that B ⊗A χA ∼
= ΓB via the multiplication
map;
(ii) this ideal is an invariant of A, i.e., χA is independent of the choice of a componentwise geometrical unibranch étale covering;
85
(iii) if A is a k-algebra, then χA is a left D(A)-submodule of A.
Proof:
Fix an étale covering f : A → B where B is a componentwise geometrical
unibranch. Let fi : B → B ⊗A B, i = 1, 2 denote the base extensions of f , b 7→ 1 ⊗ b
and b 7→ b ⊗ 1 respectively. These morphisms are étale and hence, by our assumption
on B, theorem 3.1.13 applies. We recall the notation of section 2.2.4, and we have by
theorem 3.1.13 canonical isomorphisms
B ⊗A ΓB = f1 ∗ ΓB ∼
= ΓB⊗A B ∼
= f2 ∗ ΓB = ΓB ⊗A B
which correspond to the restriction of the identity on B ⊗A B. The existence and uniqueness of the ideal χA follows now directly from the theorem on faithfully flat descent
(cf. [12])
Now assume that A → B 0 is another étale covering of A which is a componentwise
geometrical unibranch. Let χA 0 be the ideal defined by that covering. By base extension,
we get an étale covering B → B ⊗A B 0 , and then by theorem 3.1.13 there is a canonical
isomorphism
ΓB ⊗A B 0 ∼
= (B ⊗A B 0 ) ⊗B ΓB ∼
= ΓB⊗A B0 .
The same argument, with the roles of B and B 0 reversed, shows that
B ⊗A ΓB0 ∼
= ΓB⊗A B0
Hence by the definitions of χA and χA 0 we have that
(B ⊗A B 0 ) ⊗A χA ∼
= ΓB ⊗A B 0 ∼
= B ⊗A ΓB0 ∼
= (B ⊗A B 0 ) ⊗A χA 0
and since A → B ⊗A B 0 is faithfully flat, this implies that χA = χA 0 .
Finally assume that A is a commutative algebra over some ring k, and we have an
étale covering f : A → B as above. We define a k-algebra structure on B via f . By the
86
theorem on faithfully flat descent we have that
f (χA ) = {b ∈ ΓB |f1 (b) = f2 (b)}.
By a remark after corollary 2.2.6 we have an exact sequence
f1
f
0 −→ D(A) −→ D(B) −→
D(B ⊗A B).
−→
f
2
Let a ∈ χA and D ∈ D(A). Then f (a) ∈ ΓB , and since ΓB is a left D(B)-module, this
implies that f (D ∗ a) = f (D) ∗ f (a) ∈ ΓB . Furthermore
f1 (f (D ∗ a)) = f1 (f (D) ∗ f (a)) = (f1 ◦ f )(D) ∗ (f1 ◦ f )(a)
= (f2 ◦ f )(D) ∗ (f2 ◦ f )(a)
= f2 (f (D ∗ a))
and since f : A → B is injective, this and the description of f (χA ) above, implies that
D ∗ a ∈ χA . Therefore we have that χA is a left D(A)-submodule of A.
2
Remarks 1. We observe that if A is a k-algebra such that the modules PnA are finitely
presented for all n ≥ 0, and the conditions in the theorem hold, then the existence and
uniqueness of the left D(A)-module χA also follows from proposition 2.2.15.
2. If A, as in the theorem, is already a componentwise geometrical unibranch, then
obviously χA = ΓA . This is definitely no longer true if A is not a componentwise geometrical unibranch.
87
3.2.2
Essential submodules
Definition Let R be a ring and let M be an R-module. An R-submodule N ⊂ M is
said to be essential if for any R-submodule L ⊂ M, N ∩ L = 0 implies that L = 0. We
let Ess(M) denote the set of all essential R-submodules of M.
It is clear that Ess(M) is not empty (M ∈ Ess(M)). It also follows directly from the
definition that if N, N 0 ∈ Ess(M), then N ∩ N 0 ∈ Ess(M). Thus if we order Ess(M) by
inclusion, this implies that Ess(M) has at most one minimal element. For a general ring
R and an R-module M, the family Ess(M) need not have a minimal element. However if
Ess(M) has a minimal element N, we will say that N is the minimal essential submodule
of M.
Having this, we can identify essential twosided ideals of some rings of differential
operators:
Lemma 3.2.3 Let A be a reduced noetherian k-algebra and let I be a twosided ideal of
D(A). Then I is and essential ideal of D(A) if and only if I contains a regular element
of A.
Proof:
Assume that I is an essential ideal of D(A). For p ∈ Ass(A), we recall the
definition of Γp from section 3.1.1. We then have that D(A, Γp ) is a nonzero twosided
ideal of D(A), and hence is I ∩ D(A, Γp ) 6= 0. Lemma 1.2.3 now implies that I ∩ D(A, Γp )
contains a nonzero element of A, and subsequently any such element must be in Γp . This
holds for all p ∈ Ass(A) and therefore, as in the proof of lemma 3.1.3, we have that I
contains a regular element of A.
Conversely assume that I contains a regular element a ∈ A, and let 0 6= J be a
twosided ideal of D(A). By lemma 1.2.3 it contains a nonzero element b ∈ A, and hence
0 6= ab ∈ I ∩ J.
2
88
Let A be a commutative k-algebra. Besides essential twosided ideals of D(A), we
will also study essential left D(A)-submodules of A. They are also ideals of A, and a
priori it is not clear whether essential left D(A)-submodules of A are essential ideals of
A. Sometimes this is indeed the case as th following lemma shows:
Lemma 3.2.4 Let A be a reduced noetherian k-algebra and let I ⊂ A be an ideal of A.
The following two statements are equivalent:
(i) I is an essential ideal of A;
(ii) I contains a regular element of A.
If I is also a left D(A)-submodule of A, then the statements above are equivalent to the
following two statements:
(iii) I is an essential left D(A)-submodule of A;
(iv) D(A, I) is an essential twosided ideal of D(A).
Proof:
It is clear that (ii) implies (i). Conversely, a proof similar to that of lemma 3.2.3
shows that (i) implies (ii). Since the ideals Γp , p ∈ Ass(A) are left D(A)-submodules
of A, the same proof also shows that (iii) implies (ii). Conversely, since all left D(A)submodules of A are ideals, it is clear that (i) implies (iii). Finally, since I = D 0 (A, I) ⊂
D(A, I), lemma 3.2.3 and (ii) imply that (iii) is equivalent to (iv).
2
Remark From this and lemma 3.1.3 it follows that if A is a commutative, noetherian
and reduced k-algebra, then ΓA is an essential D(A)-submodule of A, and D(A, ΓA ) is an
essential twosided ideal of D(A). The lemma shows also that we can talk about essential
D(A)-submodules of A without a danger of confusion.
Next we will show how essential ideals behave under faithfully étale homomorphisms.
Before stating the result, we recall some properties of étale homomorphisms and differential operators. Let A be a commutative, noetherian and reduced k-algebra such that
89
the modules PnA are finitely presented for all n ≥ 0. Let f : A → B be an étale homomorphism of k-algebras. Then we showed in section 2.2.3 that D(B) = B ⊗A D(A)
as B-D(A)-bimodules, and hence it followed that B ⊗A M is a left D(B)-module in a
canonical way whenever M is a left D(A)-module. Furthermore, under the same assumptions we showed in section 2.3.3 that B ⊗A I is a twosided ideal of D(B) if I is a twosided
ideal of D(A). We use these facts in the statement and proof of the next proposition.
But first we need a lemma:
Lemma 3.2.5 Let f : A → B be an étale homomorphism of commutative, noetherian
and reduced rings. If a ∈ A is a regular element of A, then f (a) is a regular element of
B.
Proof:
We recall that for a reduced noetherian ring A we have Min(A) = Ass(A). By
lemma 3.1.1 we have that the set of zero-divisors in a noetherian ring is precisely the
union of all the associated primes. Thus if f (a) is a zero-divisor of B , then f (a) ∈ P
for some P ∈ Ass(B). Hence a ∈ f −1 (P ). Finally by lemma 3.1.7 we have f −1 (P ) ∈
Min(A) = Ass(A) and thus a is a zero-divisor of A.
2
Proposition 3.2.6 Let A be a commutative noetherian reduced ring, and let f : A → B
be a faithfully étale homomorphism. Let I be an ideal of A. Then
(i) I is an essential ideal of A if and only if B ⊗A I is an essential ideal of B.
Assume furthermore that f is a morphism of k-algebras, the modules PnA are finitely
presented for all n ≥ 0, and I is a left D(A)-submodule of A. We have:
(ii) If B ⊗A I is the minimal essential D(B)-submodule of B, then I is the minimal
essential D(A)-submodule of A.
We have similar statements for ideals of D(A). Namely, if I is a twosided ideal of D(A)
and all the conditions above hold, then we have:
90
(iii) I is an essential twosided ideal of D(A) if and only if B ⊗A I is an essential
twosided ideal of D(B);
(iv) if B ⊗A I is the minimal essential twosided ideal of D(B), then I is the minimal
essential twosided ideal of D(A).
Proof:
If I is an essential ideal of A, it contains a regular element a ∈ A, by lemma 3.2.4.
Then f (a) is regular by lemma 3.2.5, and it is contained in B ⊗A I. Hence B ⊗A I is
essential, again by lemma 3.2.4.
For the converse let 0 6= J be an ideal of A. Since B is faithfully flat over A, we have
that B ⊗A J 6= 0 and B ⊗A (J ∩ I) = (B ⊗A J) ∩ (B ⊗A I) by [14, Theorems 7.2, 7.4].
If B ⊗A I is an essential ideal of B, we thus have that (B ⊗A J) ∩ (B ⊗A I) 6= 0 which
in turn implies that J ∩ I 6= 0. This shows that I is an essential ideal of A and we have
proved (i).
For (ii), let J be an essential D(A)-submodule of A such that J ⊂ I. Then by the
first part of the proposition and lemma 3.2.4, B ⊗A J is an essential D(B)-submodule of
B, contained in B ⊗A I. Hence if B ⊗A I is minimal, we have that B ⊗A J = B ⊗A I,
and since B is faithfully flat over A, this implies that J = I.
The proofs of (iii) and (iv) are completely analogous, once we recall that under our
assumptions we have that D(B) = B ⊗A D(A), and that B ⊗A I is a twosided ideal of
D(B) if I is a twosided ideal of D(A).
3.2.3
2
Minimal nonzero ideals in rings of differential operators
Let A be a commutative, noetherian and reduced k-algebra.
of Ass(A).
Let p be an element
We recall that by lemmas 3.1.2 and 3.2.1 we have indeed that p is a
D(A)-submodule of A. This implies that D(A, p) (which we may identify with the set
{D ∈ D(A)|D ∗ A ⊂ p}) is an ideal of D(A). We have:
91
Lemma 3.2.7 Let A be a commutative, noetherian and reduced k-algebra, and let p be an
associated prime ideal of A. Let π : A → A/p be the natural projection. The image π(Γp )
is an ideal of A/p and π restricted to Γp is an isomorphism of A-modules Γp → π(Γp ).
Proof:
Since π is surjective it is clear that the image of Γp under π is an ideal. The
second statement follows directly from the fact that p ∩ Γp = 0.
2
It was pointed out in section 1.2.1 that π : A → A/p extends to a homomorphism
π : D(A) → D(A/p) of filtered rings, and it is characterized by the property that π(D) ∗
π(a) = π(D ∗ a) for all D ∈ D(A), a ∈ A. We use these observations in the statement of
next result:
Proposition 3.2.8 Let A be a commutative, noetherian and reduced k-algebra, and let p
be an associated prime ideal of A. Let π : A → A/p be the natural projection, and denote
the corresponding map D(A) → D(A/p) also by π. The space π(D(A, Γp )) ⊂ D(A/p) is
equal to the right ideal D(A/p, π(Γp)). Moreover π restricted to D(A, Γp ) is injective.
Proof:
If D ∈ D(A, Γp ) and π(D) = 0 then for all a ∈ A
0 = π(D) ∗ π(a) = π(D ∗ a).
Therefore D ∗ A ⊂ p ∩ Γp = 0. This proves the second statement.
For the first statement let D ∈ D(A, Γp ). Then for all a ∈ A
π(D) ∗ π(a) = π(D ∗ a) ∈ π(Γp )
and thus π(D) ∗ (A/p) ⊂ π(Γp ). That is, π(D) ∈ D(A/p, π(Γp)).
92
Conversely let D ∈ D(A/p, π(Γp)). We recall that by lemmas 1.2.1 and 1.2.6 we have
the following natural isomorphism
D(A/p, π(Γp )) ∼
= DA/p (A/p, π(Γp )) = DA (A/p, π(Γp ))
and so D is a differential operator between A-modules. By abuse of language we let
π −1 : π(Γp ) → Γp denote the inverse to the natural A-linear isomorphism in lemma 3.2.7.
Then the map π −1 Dπ : A → Γp is the composite of differential operators (two of which
are A-linear) between A-modules. Therefore π −1 Dπ ∈ DA (A, Γp ) ∼
= D(A, Γp ), again by
lemma 1.2.1. Finally we have for all a ∈ A that
π(π −1 Dπ) ∗ π(a) = π(π −1 (D ∗ π(a))) = D ∗ π(a)
since D ∗ π(a) ∈ π(Γp ). This shows that D = π(π −1 Dπ) ∈ π(D(A, Γp )).
2
We can now prove:
Theorem 3.2.9 Let A be a commutative noetherian and reduced k-algebra. If p ∈
Ass(A) is such that D(A/p) is a simple ring, then D(A, Γp ) is a minimal nonzero twosided
ideal of D(A)(i.e. it does not strictly contain any nontrivial twosided ideal of D(A)).
Moreover, if D(A/p) is a simple ring for all p ∈ Ass(A), then
(i) the set {D(A, Γp )|p ∈ Ass(A)} is precisely the set of all minimal nonzero twosided
ideals of D(A),
(ii) we have an equality
D(A, ΓA ) =
M
D(A, Γp ),
p∈Ass(A)
and
(iii) the ideal D(A, ΓA ) is the minimal essential twosided ideal of D(A).
93
Proof:
Let I ⊂ D(A, Γp ) be a nonzero twosided ideal of D(A). Then clearly I ⊃
D(A, Γp )ID(A, Γp ) and hence
π(I) ⊃ π(D(A, Γp ))π(I)π(D(A, Γp)).
Since I is a nonzero twosided ideal, by lemma 1.2.3 it must contain a nonzero element of A,
and since I ⊂ D(A, Γp ), such an element must lie in Γp . It follows that π(I)π(D(A, Γp )) 6=
0. Last proposition implies that π(D(A, Γp )) is a right ideal of D(A/p), and since we
assume that D(A/p) is a simple ring we then have that
π(I) ⊃ π(D(A, Γp ))π(I)π(D(A, Γp))
= (π(D(A, Γp ))D(A/p))π(I)(π(D(A, Γp))D(A/p))
= π(D(A, Γp )).
Since π is injective on D(A, Γp ) it follows that D(A, Γp ) ⊂ I. This proves the first part
of the theorem.
For the remainder write Ass(A) = {p1 , . . . , pn } and for the moment let J be the ideal
Pn
i=1
D(A, Γpi ). We assume that D(A/p) is simple for all p ∈ Ass(A). For i = 1, . . . , n
let Di ∈ D(A, Γpi ) and assume that
Pn
i=1
Di = 0. Then D1 =
Pn
i=2
Di , so for every a ∈ A
we have that D1 ∗ a ∈ p ∩ Γp = 0. This shows that the sum of the D(A, Γpi )s is in fact
direct.
Let D be any nonzero differential operator of D(A)and let a ∈ A be such that D ∗ a 6=
0. Then by taking x = x1 + . . . + xn with 0 6= xi ∈ Γpi , we have that x is a regular
element of A, and therefore xD ∗ a 6= 0. This implies that xD = x1 D + . . . + xn D 6= 0,
and since xi D ∈ D(A, Γpi ) for all i, we have that xi D 6= 0 for some i. This implies that
if I is a nonzero twosided ideal of D(A), then there is an i such that D(A, Γpi ) ∩ I 6=
0. The first part of the theorem then implies that D(A, Γpi ) ⊂ I. This proves that
{D(A, Γp )|p ∈ Ass(A)} is the set of all nonzero minimal twosided ideals of D(A). This
also shows that J is the minimal essential twosided ideal of D(A).
94
By definition it is clear that J, defined as above, is contained in D(A, ΓA ). For the
other inclusion, let πi : ΓA → Γpi denote the natural projection. It is A-linear, and
thus it induces a natural morphism D(A, ΓA ) → D(A, Γpi ), D 7→ πi ◦ D. It is clear that
π1 +. . .+πn is the identity on ΓA , and therefore it follows immediately that D(A, ΓA ) ⊂ J.
2
From this theorem, a similar statement for left D(A)-submodules of A follows easily:
Corollary 3.2.10 Let A be a commutative, noetherian and reduced k-algebra. If p ∈
Ass(A) is such that D(A/p) is simple, then Γp is a minimal nonzero left D(A)-submodule
of A. Moreover, if the rings D(A/p) are simple for all p ∈ Ass(A), then
(i) the set {Γp |p ∈ Ass(A)} is precisely the set of all minimal nonzero left D(A)submodules of A, and
(ii) the ideal ΓA is the minimal essential left D(A)-submodule of A.
Proof:
If 0 6= I is a left D(A)-submodule of A, then D(A, I) is a nonzero twosided ideal
of D(A). From this observation, the corollary follows.
2
We observe that the theorem and its corollary say nothing about integral domains
since the assumptions become the implications. In order to deal with domains, we hence
must prove a more general version of these results. For that we will use étale coverings
(defined in section 3.2.1). Let A be a commutative, noetherian and reduced k-algebra.
Assume that there is a faithfully étale homomorphism f : A → B of k-algebras (i.e., an
étale covering) where B is a componentwise geometrical unibranch. We recall that B is
reduced, by lemma 2.1.6. Then we can define the left D(B)-submodule ΓB of B, and
theorem 3.2.2 implies the existence and uniqueness of χA , a left D(A)-submodule of A
such that ΓB = B ⊗A χA . Finally we have of course that D(A, χA ) is a twosided ideal
and if PnA is finitely presented for all n ≥ 0, then by the remark following theorem 2.2.10
it follows that D(B, ΓB ) = B ⊗A D(A, χA ).
95
We can now state and prove the generalization we want of theorem 3.2.9 and its
corollary:
Theorem 3.2.11 Let A be a commutative, noetherian and reduced k-algebra such that
PnA is finitely presented for all n ≥ 0. Assume that A has an étale covering f : A → B
where B is a componentwise geometrical unibranch and the irreducible components of B
have rings of differential operators that are simple.
(i) Then A has a minimal essential D(A)-submodule χA , and B ⊗A χA ∼
= ΓB via the
natural multiplication map.
(ii) Moreover, the ideal D(A, χA ) is the minimal essential twosided ideal of D(A),
and B ⊗A D(A, χA ) = D(B, ΓB ).
Proof:
We have already constructed χA in theorem 3.2.2. This ideal and D(A, χA ) are
essential by proposition 3.2.6. Thus it remains only to show that these objects are in fact
minimal. For that let I be an essential twosided ideal of D(A). Then by corollary 2.3.13
and a remark following proposition 3.2.6, we have that B ⊗A I is an essential twosided
ideal of D(B). Hence it follows from theorem 3.2.9 that B ⊗A D(A, χA ) = D(B, ΓB ) ⊂
B ⊗A I, and then faithful flatness implies that D(A, χA ) ⊂ I. We get the minimality of
2
χA in a similar way from corollary 3.2.10.
3.2.4
Applications of theorem 3.2.11
Theorems 3.2.2 and 3.2.11 raise two questions: 1. Do all commutative noetherian reduced
rings have an étale covering which is a componentwise geometrical unibranch?
2. When is it true for a commutative, noetherian algebra A that D(A) is a simple
ring?
In this section we will reflect on these two questions.
96
The answer to the first question is “no”, even with the additional assumption that the
ring is local and essentially of finite type over an algebraically closed field of characteristic
zero. The problem lies in the fact that even if a local ring is a componentwise geometrical
unibranch at its maximal ideal, it may not be so at other prime ideals. However if we
dispose of that problem, we have the following:
Proposition 3.2.12 Let A be a commutative noetherian local ring such that A is a componentwise geometrical unibranch at all prime ideals except possibly the maximal ideal.
Then A has a local étale covering B which is a componentwise geometrical unibranch (at
all prime ideals).
Proof:
In section 3.1.2 we pointed out that Ash , the strict henselization of A, is noethe-
rian. Therefore it has finitely many minimal prime ideals by lemma 3.1.1. Assume that
P, P 0 ∈ Min(Ash ) such that P 6= P 0 . Then there is an x ∈ P \ P 0 , so by lemma 3.1.8 there
is a local étale A-subalgebra A0 of Ash such that x ∈ A0 . Then obviously P ∩ A0 6= P 0 ∩ A0 .
We can repeat this procedure for all pairs of minimal prime ideals of Ash , and thus there
exists a local étale A-subalgebra B of Ash such that P ∩ B 6= P 0 ∩ B whenever P and
P 0 are two distinct minimal prime ideals of Ash . This implies that B is a componentwise
geometrical unibranch at its maximal ideal.
Finally if q is a prime ideal of B distinct from the maximal ideal, then p := q ∩ A is
distinct from the maximal ideal of A (fibers of étale homomorphisms have krull dimension
0) and Ap → Bq is a local étale homomorphism. Then, by assumption we have that the
map Min(Ash
p ) → Min(Ap ) is injective, and since Bq is a local étale extension of Ap ,
the same is obviously true for that ring. This shows that B is also a componentwise
geometrical unibranch at all prime ideals distinct from the maximal ideal.
2
All fields are automatically geometrical unibranches, and hence by using lemma 3.1.10
we can apply the proposition to rings of krull dimension 1:
97
Corollary 3.2.13 Let A be a commutative noetherian local ring of krull dimension 1.
Then A has an étale covering B which is a componentwise geometrical unibranch.
Proof:
If p ∈ Spec A is not the maximal ideal, then p is a minimal prime ideal and
2
Ap /p is a field.
Proposition 3.2.12 implies also the following:
Proposition 3.2.14 Let A be a commutative noetherian reduced ring, and a k-algebra.
If D(A) is a simple ring, then A is a domain and a geometrical unibranch.
Proof:
If A is not a domain and p is one of the minimal prime ideals of A, it follows
directly from the definition that Γp is neither A nor 0. Hence D(A, Γp ) is a nontrivial
twosided ideal of D(A).
We may now assume that A is a domain. Assume that A is not a geometrical unibranch, and let p be a minimal element of the set
{q ∈ Spec A|A is not a geometrical unibranch at q}.
Then Ap is a geometrical unibranch at all prime ideals not equal to p, and proposition 3.2.12 implies that Ap has a local étale covering B which is a componentwise geometrical unibranch. Since Ap is not a geometrical unibranch at p it follows that B has
more than one minimal prime ideal, and therefore is ΓB a nontrivial left D(B)-submodule
of B. By theorem 3.2.2 it then follows that Ap has a nontrivial left D(Ap )-submodule,
and by intersecting this module with A, we get a nontrivial left D(A)-submodule of A.
This shows that the ring D(A) is not simple.
98
2
Remark This proposition shows that the condition on geometrical unibranchedness
in theorem 3.2.11 is in fact redundant. We should also note that for reduced algebras,
essentially of finite type over a field of characteristic zero, proposition 3.2.14 is implied
by the results of [4].
We now turn to the second question stated at the beginning of this section. Proposition 3.2.14 gives a necessary condition for D(A) to be simple if A is a commutative
noetherian and reduced k-algebra. That is, A must be a domain and a geometrical unibranch. Furthermore it is not hard to see that A must also be equal to the intersection
of all its height one localizations (i.e., A must satisfy the condition S2 . See chapter 4 for
the definition). This follows also from [4]. On the other hand, these conditions are not
sufficient. The well known example of [1] shows that D(A) need not be simple even if
A is a normal domain, finitely generated over the complex numbers. However, if A is a
regular domain, essentially of finite type over a field of characteristic zero, it was pointed
out in the beginning that D(A) = ∆(A), and in that case it follows, for example from [15,
Chapter 15], that D(A) is simple. Furthermore, if krull dimension of A is equal to 1, we
have:
Theorem 3.2.15 Let A be a local domain of krull dimension 1, a localization of a finitely
generated k-algebra, where k is a field of characteristic zero. If A is a geometrical unibranch, then A has no proper nonzero D(A)-submodules, and D(A) is a simple ring.
Proof:
2
This is [4, corollary 3.6].
From this and theorem 3.2.11 we get an immediate corollary:
Corollary 3.2.16 Let k be a field of characteristic zero and let A be a local, reduced
k-algebra of krull dimension 1 and essentially of finite type over k. Then χA , as defined
99
in theorem 3.2.2, is the minimal essential left D(A)-submodule of A, and D(A, χA ) is the
minimal essential twosided ideal of D(A).
Remark We must note that the existence of such minimal modules was already known,
at least if A is the localization of an affine reduced curve over a field of characteristic
zero (cf. [17, 25]).
There are other examples of algebras A for which D(A) is simple (cf. [13]), but in
general the problem is quite open. In next chapter, however, we will show that if we
restrict the class of twosided ideals, then it is possible to prove a minimality condition
which holds quite generally.
100
Chapter 4
Reflexive modules and rings of
differential operators
In [4] it was proved that if A is a domain, essentially of finite type over a field of characteristic zero, then A satisfies the condition S2 and is a geometrical unibranch if and only
if D(A) is a krull domain with no proper reflexive ideals.
The goal of this chapter is to prove a related result. Namely, if A is a domain,
essentially of finite type over a field of characteristic zero, and satisfies the condition S2 ,
then the set
{J|J is a reflexive nonzero ideal of A and a D(A)-submodule of A },
ordered by inclusion, has a unique minimal element. In fact if we let X denote the set of
all height one prime ideals of A, it follows from section 3.2.4 that the ideals χAp can be
constructed for all p ∈ X . We show that I :=
T
p∈X
χAp is the unique minimal element
we are seeking. Moreover, in that case the ideal D(A, I) of D(A) is reflexive on the right
and it contains all nonzero twosided ideals of D(A) which are reflexive on either side.
101
4.1
Some noncommutative ring theory
4.1.1
Rings of quotients
We refer to [15, Section 2.1] for the general definition of rings of fractions of noncommutative rings (also called “rings of quotients”). We will only need the following special
case:
Definition Let R be a ring. The total right (resp. left) ring of fractions of R is a ring
Q containing R such that the following conditions hold:
(i) Every regular element of R has an inverse in Q;
(ii) Every element of Q may be written as rs−1 (resp. s−1 r) where r, s ∈ R and s is
a regular element.
If R is commutative, this is the classical definition, and we know that the ring Q exists.
However if R is noncommutative, the existence of its total (left or right) ring of fractions
is by no means guaranteed. However if it exists, it is unique up to an isomorphism by [15,
Corollary 2.1.4], and that result also shows that if R has both a left and a right total
rings of fractions, they must be isomorphic. Therefore if R has both left and right ring
of fractions, we will call it the ring of fractions of R (without the qualifiers “left” and
“right”).
In [15, Section 3.1] we have also the following definitions, related to that above:
Definition A ring Q is called a quotient ring if every regular element of Q is invertible.
Given a quotient ring Q, a subring R is called a right (left) order in Q if every q ∈ Q has
the form rs−1 (s−1 r) for some r, s ∈ R. A left and a right order is called an order.
Note that if R has a total ring of fractions Q, then Q is a quotient ring and R is an
order in Q.
102
In this section we will show that if k is a field of characteristic zero, and A is a reduced
k-algebra, essentially of finite type over k, then D(A) has a total ring of fractions. We
carry out some of the proofs for the sake of completeness, but essentially the same result
and proof may be found in [17].
Theorem 4.1.1 Let k be a field of characteristic zero, and let k → K be a field extension
of finite transcendence degree. Then D(K) is a simple noetherian integral domain.
Proof:
This is [15, Proposition 15.3.2(ii), Theorem 15.5.5(ii)]
2
The following is essentially proved in [17]:
Theorem 4.1.2 Let k be a field of characteristic zero, and let A be a commutative kalgebra, reduced and essentially of finite type over k. Let K be the total ring of fractions
of A. Then D(K) has a total ring of fractions QD(K) , and we have a chain of inclusions
D(A) ⊂ D(K) ⊂ QD(K)
Moreover, QD(K) is also the total ring of fractions of D(A).
Proof:
Since A is reduced and essentially of finite type over k, it follows that K ∼
=
K1 × . . . Kn where each Ki is a field of finite transcendence degree over k. This implies
especially that K is reduced, so D(K) is semiprime by lemma 1.2.4. Lemma 1.2.5 implies
that D(K) = D(Ki) × . . . D(Kn ), and therefore we have by theorem 4.1.1 that D(K) is
noetherian (a direct product of finitely many noetherian rings is noetherian).
Noetherian rings are Goldie (cf. [15, Section 2.3.1]), and hence [15, Theorem 2.3.6]
implies that D(K) has a total ring of fractions.
103
By a remark following theorem 2.2.5 we have that D(A) ⊂ D(K). Moreover, since A is
essentially of finite type over k, propositions 2.2.11 and 2.2.12 imply that theorem 2.2.10
applies for the extension A ⊂ K. Therefore every element of D(K) may be written as
s−1 D where s is a regular element of A and D ∈ D(A). The same is true for the other
side: We may write s−1 D = Ds−1 − [D, s−1 ], and by induction on order of differential
operators, we may assume that [D, s−1 ] = D 0 t−1 for some D 0 ∈ D(A) and t ∈ A, a regular
element. Therefore we have that s−1 D = Ds−1 − D 0 t−1 = (Dt − D 0 s)(st)−1 (this proof
comes from [17, Proposition 1.8]).
From this last result it now follows that regular elements of D(A) are also regular
elements of D(K) and hence invertible in QD(K) . To show that QD(K) is the total ring of
fractions of D(A), we copy the proof of [17, Lemma 3.2(c)]. Let DD 0−1 be an element of
QD(K) with D, D 0 ∈ D(K) and D 0 regular. We may write D = D 00 s−1 and D 0 s = D 000 t−1
with D 00 , D000 ∈ D(A) and s, t regular elements in A. Then
DD 0−1 = D 00 (D 0 s)−1 = D 00 (D 000 t−1 )−1 = (D 00 t)D 000−1 .
It follows in a similar way that DD 0−1 = D 000 −1 D 00 for some D 00 , D 000 ∈ D(A) where D 000
2
is regular.
Corollary 4.1.3 Let k and A be as in the theorem. Then D(A) is a semiprime (right
and left) Goldie ring.
Proof:
This follows directly from the theorem and [15, Theorem 2.3.6].
104
2
4.1.2
Reflexive modules
In this section we will introduce reflexive modules, and show how they apply to rings of
differential operators. We refer to [15] for most of the general theory, but some of the
commutative theory may also be found in [2, chapter VII.4]. The applications to rings
of differential operators can be found in [4].
Let R be a ring and let M be a right (left) R-module. The dual of M is defined as
the left (right) R-module M ∗ := HomR (M, R). The doubledual is again a right (left)
R-module, and there is an obvious R-module homomorphism M → M ∗∗ , m 7→ φm , where
φm : M ∗ → R is the map α 7→ α(m). The module M is said to be torsionless if M → M ∗∗
is injective and in that case we identify M with its image in M ∗∗ . To say that a module
M is torsionless is equivalent to saying that M embeds in some direct product of copies
of R (cf. [15, sections 5.1.7, 3.4.2]).
Definition Let R be a ring and let M be a torsionless (left or right) R-module. We say
that M is reflexive if M = M ∗∗ .
We quickly observe that for any R-module M, its dual M ∗ is torsionless and in fact
reflexive [15, sections 3.4.2, 5.1.7].
The definition of reflexive modules we have just stated is rather hard to work with, and
we will restrict our attention to a certain class of modules where duals can be described
in somewhat more concrete terms. From section 4.1.1 we recall the definition of orders.
The following definition appears in [15, section 3.1.11].
Definition Let R be an order in a quotient ring Q. Let I be a right (left) R-submodule
of Q. We say that I is a fractional right (left) R-ideal if there are units a and b of Q such
that aI ⊂ R (Ia ⊂ R) and bR ⊂ I (Rb ⊂ I).
We immediately get another description of duals:
105
Proposition 4.1.4 Let R be an order in a quotient ring Q, and let I be a fractional
right (left) R-ideal. Then we have an isomorphism of left (right) R-modules I ∗ ∼
=
{q ∈ Q|qI ⊂ R} (I ∗ ∼
= {q ∈ Q|Iq ⊂ R}).
Proof:
This is [15, Proposition 3.1.15(iv)]
2
Having this proposition, we will from now on identify the objects therein. We use it
in the next lemma
Lemma 4.1.5 Let R be an order in a quotient ring Q. If I is a fractional right (left)
R-ideal, then I ∗ is a fractional left (right) R-ideal.
Proof:
We carry the proof out for fractional right R-ideals. Then by definition there are
units a, b of Q such that aI ⊂ R and bR ⊂ I. This implies that b ∈ I, and thus I ∗ b ⊂ R.
Since I ∗ = {q ∈ Q|qI ⊂ R} it follows especially that a ∈ I ∗ . We recall (or observe) that
I ∗ is a left R-module and hence Ra ⊂ I ∗ .
2
This lemma, along with the previous proposition applied again, shows that doubleduals of fractional (left or right) R-ideals may also be identified with fractional R-ideals,
and from now on we will do so without further comment. Moreover the results show directly that I ⊂ I ∗∗ whenever I is a fractional R-ideal. We should note that the definition
of fractional ideals implies immediately that they are torsionless R-modules.
At this point we issue a warning. On several occasions in the sequel we will be considering fractional twosided R-ideals, i.e., twosided R-submodules of Q that are fractional
ideals on both sides. When taking duals of such fractional ideals, we must be careful to
specify on which side we are doing it, since we have no guarantee that the end result is the
106
same. (However compare [15, Proposition 5.1.8] for a case where there is no ambiguity.)
If there is a danger of confusion, we will thus follow the example of [4, 15] and write
RI
∗
= {q ∈ Q|qI ⊂ R}
and
IR∗ = {q ∈ Q|Iq ⊂ R}.
At last we warn that a fractional twosided R-ideal may well be reflexive on one side
without being so on the other side.
We now apply the general results above to rings of differential operators. Let k be a
field of characteristic zero, and A be a commutative k-algebra, reduced and essentially
of finite type over k. Let K be the total ring of fractions of A. From theorem 4.1.2 we
then have a chain of inclusions D(A) ⊂ D(K) ⊂ QD(K) , where QD(K) is the total ring
of fractions of D(K). Moreover we have that D(A) is an order in QD(K) . We now recall
the notion of essential submodules from section 3.2.2. The next lemma shows that many
ideals are in fact fractional:
Lemma 4.1.6 Essential twosided ideals of D(A) are fractional twosided D(A)-ideals.
Proof:
Let I be an essential twosided ideal of D(A). By lemma 3.2.3 it contains a
regular element of A, which in turn is a regular element of D(A).
Let a ∈ A be such
an element. Then aD(A), D(A)a ⊂ I and hence we have shown that I is a fractional
twosided D(A)-ideal.
2
Therefore we have that all essential ideals of I are fractional, and we may assume
that their duals and doubleduals have the same form as in proposition 4.1.4:
107
Proposition 4.1.7 Let I be an essential twosided ideal of D(A). Then I ∗ is a twosided
D(A)-submodule of QD(K) such that D(A) ⊂ I ∗ ⊂ D(K). Moreover I ∗∗ is a twosided
ideal of D(A).
Proof:
We do the proof for the right side. Then I ∗ = {q ∈ QD(K) |qI ⊂ D(A)}. Since I is
a twosided ideal of D(A), it follows that D(A) ⊂ I ∗ and that I ∗ is a twosided submodule
of QD(K) . By lemma 3.2.3 we have that I contains a regular element a ∈ A. Thus if
q ∈ I ∗ , we have that qa ∈ D(A), so q ∈ D(A)a−1 ⊂ D(K). Finally, since 1 ∈ D(A) ⊂ I ∗ ,
and I ∗∗ = {q ∈ QD(K) |I ∗ q ⊂ D(A)}, it follows immediately that I ∗∗ ⊂ D(A).
2
This proposition is what we have been aiming for. It tells us that if I is an essential
twosided ideal of D(A), then the right and left duals and doubleduals of I are contained
in D(K), and hence we have the formulas
I ∗ = {D ∈ D(K)|DI ⊂ D(A)}
and
I ∗∗ = {D ∈ D(A)|I ∗ D ⊂ D(A)}
(and similarly for the left side).
We note that the general theory at the beginning of this section applies especially to
commutative rings. We will establish some connections between reflexive ideals of commutative algebras and reflexive twosided ideals of their rings of differential operators. To
do so we need to show some additional properties of reflexive modules over commutative
rings. The definitions of duals and reflexive modules are as before. First we need a result
from homological algebra:
108
Lemma 4.1.8 Duals commute with flat extensions on finitely presented modules. That
is, if A is a commutative ring, f : A → B is a flat homomorphism and M is a finitely
presented A-module, then there is a natural isomorphism
τM
B ⊗A M ∗ −→ (B ⊗A M )∗
given by the formula τM (b⊗φ)(b0 ⊗m) = bb0 f (φ(m))
Proof:
In order to simplify the notation, we let F (−) and G(−) denote the two functors
B ⊗A HomA (−, A) and HomB (B ⊗A −, B) from A − Mod into B − Mod. Since B is flat
over A, we have that F and G are both left exact contravariant. Moreover if we define
τM as in the statement of the lemma it is trivial to verify that τ : F → G is a natural
transformation of functors. If M is finitely generated free A-module, say M = Ar for
some integer r, then both F (M) and G(M) are isomorphic to B r in an obvious way, and
τM becomes the identity under these identifications. Finally if M is finitely presented,
we have an exact sequence
0 ←− M ←− Ar ←− As
for some integers r and s, and then the result follows from the diagram
0
- F (M)
- F (Ar )
∼
=
τM
0
?
- G(M)
?
- G(Ar )
- F (As )
∼
=
?
- G(As )
2
For a commutative, noetherian and reduced ring A, the the ideal ΓA was defined
in section 3.1.1. If furthermore A has an étale covering B which is a componentwise
geometrical unibranch, then by theorem 3.2.11 we have that A contains an ideal χA such
109
that B ⊗A χA = ΓB . We have:
Lemma 4.1.9 Let A be a commutative, noetherian and reduced ring. Then ΓA is a
reflexive ideal of A.
Proof:
Write A ⊂ Ā ⊂ K, as at the end of section 3.1.1. Since ΓA contains a regular
element of A, it is an essential ideal of A, and thus we have that ΓA is a fractional
ideal of A. Hence by proposition 4.1.4 and lemma 4.1.5 it follows immediately that
ΓA ∗ = {x ∈ K|xΓA ⊂ A} ⊃ A and ΓA ∗∗ = {x ∈ K|xΓA ∗ ⊂ A} ⊂ A. By lemma 3.1.4 we
have that ΓA is the conductor of Ā into A, and hence it follows that Ā ⊂ ΓA ∗ . Thus if
a ∈ ΓA ∗∗ ⊂ A, then aĀ ⊂ aΓA ∗ ⊂ A, so a ∈ ΓA , again by lemma 3.1.4. This shows that
ΓA ⊂ ΓA ∗∗ . The other inclusion is always true.
2
Proposition 4.1.10 Let A be a commutative, noetherian and reduced ring, and assume
that A has an étale covering B which is a componentwise geometrical unibranch. Then
the ideal χA is reflexive.
Proof:
We have by definition that ΓB = B ⊗A χA , and by lemma 3.2.5 it follows that
χA contains a regular element of A. Therefore is χA a fractional ideal of A. Since B
is flat over A, lemma 4.1.8 implies that duals on finitely presented modules commute
with the extension A → B, and since ideals of noetherian rings are finitely presented,
this implies that ΓB ∗ = B ⊗A χA ∗ . Fractional ideals of commutative noetherian rings are
obviously isomorphic (as modules) to ideals, and are hence finitely presented. Therefore
we can use lemma 4.1.8 on χA ∗ to get that ΓB ∗∗ = B ⊗A χA ∗∗ . The ring B is noetherian
and reduced, so the ideal ΓB is reflexive by lemma 4.1.9. Hence we get that B ⊗A χA =
ΓB = B ⊗A χA ∗∗ . Since A → B is faithfully flat, we thus have that χA ∗∗ = χA .
110
2
4.2
4.2.1
Rings satisfying the condition S2
Minimal reflexive modules
We now introduce the condition S2 for commutative noetherian domains. If A is a
commutative noetherian domain, it is clear that we may identify A with a subring of K, its
field of fractions. Moreover, if S is any multiplicatively closed subset of A (not containing
zero), then we may identify S −1 A with a subring of K such that the localization map
A → S −1 A is simply the inclusion. We use this convention in what follows:
Definition Let A be a commutative noetherian domain and let X denote the set of all
height 1 prime ideals of A. We say that A satisfies the condition S2 (or the condition 2
of Serre) if A =
T
p∈X
Ap .
An immediate task is to show that if A is a k-algebra which satisfies the condition
S2 , then D(A) possesses a similar property (cf. [4]):
Lemma 4.2.1 Let A be a commutative noetherian k-algebra. Assume that A is a domain
which satisfies the condition S2 , and let K denote the field of fractions of A. Then as
subrings of D(K),
D(A) =
\
D(Ap )
p∈X
Proof:
From a remark following theorem 2.2.5 it follows that
D(A) = {D ∈ D(K)|D ∗ A ⊂ A}.
Let p ∈ Spec A. Since K is also the field of fractions of Ap , a similar formula holds for
D(Ap ). Furthermore, since A → Ap is injective and formally étale, we have by the same
111
remark that D(A) ⊂ D(Ap ). This is true for all p ∈ X , and hence the inclusion “⊂”
follows.
Conversely assume that D ∈
T
p∈X
D(Ap ). Then if a ∈ A, we have that D ∗ a ∈ Ap
for all p ∈ X , and since A satisfies the condition S2 , this implies that D ∗ a ∈ A. Hence
is D ∗ A ⊂ A and we have that D ∈ D(A).
2
Next we show a connection between reflexive ideals of certain commutative algebras,
and reflexive ideals in their rings of differential operators:
Lemma 4.2.2 Let k be a field of characteristic zero and let A be a commutative, reduced
k-algebra, and essentially of finite type over k. If I ⊂ A is an essential D(A)-submodule
of A, and a reflexive ideal of A, then D(A, I) is a reflexive right ideal of D(A).
Proof:
Lemma 3.2.4 implies that D(A, I) is an essential twosided ideal of D(A). Hence
by the remarks following proposition 4.1.7 we have that
D(A, I)∗ = {D ∈ D(K)|DD(A, I) ⊂ D(A)},
and
D(A, I)∗∗ = {D ∈ D(K)|D(A, I)∗ D ⊂ D(A)}.
By proposition 4.1.4 we have that I ∗ = {x ∈ K|xI ⊂ A}. We also recall that in this
case we can identify D(A) with the ring {D ∈ D(K)|D ∗ A ⊂ A}. From all this it follows
immediately that I ∗ ⊂ D(A, I)∗ . Thus if D ∈ D(A, I)∗∗ , then we have especially that
I ∗ D ⊂ D(A) and hence I ∗ (D ∗ A) ⊂ A. Since I is reflexive, this implies that D ∗ A ⊂ I,
that is, D ∈ D(A, I).
2
112
Lemma 4.2.3 Let k be a field of characteristic zero, let A be a commutative algebra, essentially of finite type over k, and assume that A is a domain which satisfies the condition
S2 . If J is a twosided nonzero ideal of D(A), then
\
J∗ =
\
(J ∗ )p =
p∈X
(Jp )∗
p∈X
where −∗ means the dual either as a right or as a left D(A)-module.
Proof:
Let K denote the field of fractions of A. We may assume that all localizations
of A are subrings of K, and their respective rings of differential operators are contained
in D(K). This implies especially that if I is any left D(A)-submodule of D(K), and
p ∈ Spec A, then we may identify the left D(Ap )-module D(Ap )⊗D(A) I with the set
Ip = Ap I ⊂ D(K). Since J is a twosided essential ideal of D(A), we may use the
description of J ∗ following proposition 4.1.7. We will prove the theorem by showing a
chain of inclusions:
J∗ ⊂
\
(J ∗ )p ⊂
p∈X
\
(Jp )∗ ⊂ J ∗
p∈X
From the comment above it is clear that J ∗ ⊂ (J ∗ )p for all prime ideals p of A, so
the first inclusion certainly holds. Likewise assume that D ∈
DJp ⊂ D(Ap ) for all p ∈ X , and hence DJ ⊂
T
p∈X
T
p∈X (Jp )
∗
. Then DJ ⊂
D(Ap ) = D(A) by lemma 4.2.1. The
other side is similar. Therefore the last inclusion also holds.
Thus it only remains to show that (J ∗ )p ⊂ (Jp )∗ for all p ∈ X . Fix a p ∈ Spec A.
By proposition 2.3.10, Jp is a twosided ideal of D(Ap ), so (Jp )∗ is a twosided D(Ap )submodule of D(K). Since (J ∗ )p = Ap J ∗ it therefore suffices to show that J ∗ ⊂ (Jp )∗ .
I.e., we must show that if D ∈ D(K) is such that DJ ⊂ D(A), then DJp ⊂ D(Ap ),
and the same for the other side. Since Jp is a twosided ideal of D(Ap ), and Jp =
Ap J , it follows immediately that J D ⊂ D(A) implies that Jp D ⊂ D(Ap ). Similarly if
DJ ⊂ D(A) and s−1 D 0 is an element of Jp (with s−1 ∈ Ap and D 0 ∈ J ), then we use
proposition 2.3.4 to move D to the right of s−1 and the result follows.
113
2
Remark We have not been able to prove that duals commute with localizations in
D(A). We only have that (J ∗ )p ⊂ (Jp )∗ . However if A is also a geometrical unibranch,
equality follows from [4].
Theorem 4.2.4 Let A be a local domain of krull dimension 1, a localization of a finitely
generated k-algebra, where k is a field of characteristic zero. Then
(i) A has a unique minimal nonzero left D(A)-submodule χA ,
(ii) The ideal D(A, χA ) is the unique minimal nonzero twosided ideal of D(A),
(iii) χA is reflexive as an A-module,
(iv) D(A, χA ) is a reflexive right ideal of D(A).
Proof:
Parts (i)–(ii) are just restatements of corollary 3.2.16, part (iii) follows di-
rectly from proposition 4.1.10 combined with corollary 3.2.13, and part (iv) follows from
2
part (iii) and lemma 4.2.2.
Remark The existence of a submodule and ideal as in parts (i) and (ii) in the theorem
was previously known, at least in the special case when A is the localization of an affine
curve (cf. [17, 25]). However, our description of these objects and the fact that they are
reflexive is new.
We use this to get a result which holds for domains that satisfy the condition S2 :
Theorem 4.2.5 Let k be a field of characteristic zero, and let A be a commutative kalgebra, essentially of finite type over k. Let X denote the set of all height 1 prime ideals
of A. If A is a domain which satisfies the condition S2 , then
(i) the ideal I =
T
p∈X
χAp , where the ideals χAp are as in theorem 4.2.4, is the unique
minimal element of the set
{J|J is a reflexive nonzero ideal of A and a D(A)-submodule of A };
114
(ii) The twosided ideal D(A, I) is reflexive as a right D(A)-module and it is contained
in all twosided nonzero ideals of D(A) which are reflexive on either side.
Proof:
(i) For every p ∈ X , Ap satisfies the conditions of theorem 4.2.4 and hence it has
a left D(Ap )-submodule χAp as prescribed there. Since χAp = Ap for every p at which A is
regular, χAp 6= Ap for at most finitely many p ∈ X . Therefore we have that I :=
T
p∈X
χAp
is a nonzero ideal of A. Moreover, the description of D(A) in lemma 4.2.1 implies that I
is a left D(A)-submodule of A. For every p ∈ X we then have that Ip ⊂ χAp is a nonzero
left D(Ap )-submodule of Ap , and hence Ip = χAp by minimality. Every χAp is reflexive,
so by [2, theorem VII.4.1] it follows that
I ∗∗ =
\
Ip∗∗ =
p∈X
\
χAp ∗∗ =
p∈X
\
χAp = I
p∈X
I.e., I is a nonzero reflexive ideal of A.
Now let 0 6= J ⊂ A be a D(A)-submodule of A. Then for every p ∈ X , Jp is a
nonzero D(Ap )-submodule of Ap , so by minimality, Ip = χAp ⊂ Jp . Hence is Jp∗ ⊂ Ip∗
for all p ∈ X . Since A satisfies the condition S2 , we have, again by [2, theorem VII.4.1],
that J ∗ =
T
p∈X
Jp∗ , and therefore J ∗ ⊂ I ∗ . By taking duals again we thus have that
I = I ∗∗ ⊂ J ∗∗ . This implies that if J is a nonzero left D(A)-submodule of A, and a
reflexive ideal, then I ⊂ J.
(ii). Lemma 4.2.2 implies that D(A, I) is a reflexive right ideal of D(A). Let J be a
nonzero twosided ideal of D(A). Then for all p ∈ X , Jp is a nonzero twosided ideal of
D(Ap ) and hence by corollary 3.2.16 and the remark after theorem 2.2.10 we have that
D(A, I)p = D(Ap , Ip ) = D(Ap , χAp ) ⊂ Jp . Taking duals and using lemma 4.2.3 we thus
have that
J∗ =
\
p∈X
(Jp )∗ ⊂
\
(D(A, I)p )∗ = D(A, I)∗
p∈X
115
By taking duals again we then have in any case that
D(A, I) ⊂ D(A, I)∗∗ ⊂ J ∗∗ .
Hence if J is reflexive on either side, it contains D(A, I)∗∗ , which in turn is equal to
D(A, I) if it is the doubledual of D(A, I) as a right ideal.
116
2
Chapter 5
Algebras related to rings of
differential operators
Let A be a commutative, noetherian and reduced k-algebra and assume that A has an
étale covering which is a componentwise geometrical unibranch. By theorem 3.2.2 it is
then possible to define the essential D(A)-submodule χA of A, and the essential twosided
ideal D(A, χA ) of D(A). In particular, χA is an ideal of A, and it is not contained in any
of the minimal prime ideals of A. Hence if A is of finite krull dimension it follows that
A/χA is of krull dimension strictly less than A.
This implies especially that if k is a field, and A is a finitely generated k-algebra of
krull dimension 1 which satisfies the conditions above, then A/χA is a finite dimensional
vector space over k. The same is true if we replace A with its completion at a maximal
ideal. From this, the definition of rings of differential operators, and lemma 1.2.2 it
follows that the ring D(A)/D(A, χA ) is isomorphic to a subalgebra of Mn (k), the algebra
of all n × n-matrices over k, where n = dimk (A/χA ).
In the case that k is an algebraically closed field of characteristic zero and A is
reduced and a finitely generated k-algebra of krull dimension 1, the algebras A/χA and
D(A)/D(A, χA ) have been studied in [3, 17, 23, 24, 25] and they have been given the
117
names
C(A) := A/χA
and
H(A) := D(A)/D(A, χA )
We use these names whenever χA can be defined, i.e., when A is a commutative, noetherian and reduced k-algebra which has an étale componentwise geometrical unibranch
covering.
In this chapter we prove two results. We give a formula for the dimension of C(A)
if A is the completion at a maximal ideal of an affine plane curve over an algebraically
closed field of characteristic zero. This is a partial answer to a question asked in [24].
We also identify the canonical image of the algebra H(A) in the ring D(A/χA ) if A is
a reduced algebra, essentially of finite type over a field of characteristic zero, and with
irreducible components that are regular. This is an analog of [3, theorems 4.5, 4.6], and
the proof is similar.
5.1
5.1.1
The algebra C(A)
A dimension formula related to affine plane curves
Let A be a commutative, noetherian and reduced k-algebra. For such an algebra we
defined the ideal ΓA in section 3.1.1. It is clear from the definition that ΓA is not
contained in any of the minimal prime ideals of A. Thus if k is a field and A is a finitely
generated and reduced k-algebra of krull dimension 1, or a completion thereof, it follows
that A/ΓA is a finite dimensional vector space over k. In this section we will give a
formula for the dimension of this vector space (in terms of intersection multiplicities) in
some special cases.
118
For power series f1 , . . . , fn in k[[X, Y ]] and i = 1, . . . , n we let
fbi := f1 · · · fi−1 fi+1 · · · fn ,
and for 0 ≤ i < j ≤ n we let
fbij := f1 · · · fi−1 fi+1 · · · fj−1 fj+1 · · · fn
(where the product of zero number of factors is defined as 1).
Lemma 5.1.1 Let k be a field and let f1 ,. . . , fn be in k[[X, Y ]]. Assume that for all i,
the power series fi and fbi have no common components. Then there is an exact sequence
ψ
0 −→ k[[X, Y ]]/(fb1,n , . . . , fbn−1,n ) −→ k[[X, Y ]]/(fb1 , . . . , fbn )
π
−→ k[[X, Y ]]/(fn , fbn ) −→ 0
where the map ψ is defined by ψ(z) := fn z, and π is the natural projection (we note that
(fb1 , . . . , fbn ) ⊂ (fn , fbn )).
Proof:
First of all we observe that the map ψ is indeed well defined. The map π is
surjective, and exactness in the middle follows immediately from the definitions of the
maps. Hence we only need to show that ψ is injective. For that we imitate the proof
of [5, p. 76]. Namely, let z ∈ k[[X, Y ]] be such that ψ(z) = 0, i.e., fn z = u1 fb1 + . . . + un fbn
for some u1 , . . . , un ∈ k[[X, Y ]]. Then
un fbn = fn z − u1 fb1 − . . . − un−1 fbn−1
= fn (z − u1 fb1,n − . . . − un−1 fbn−1,n )
119
and since by assumption fn and fbn have no common factors, and k[[X, Y ]] is a unique
factorization domain, this implies that fbn divides the power series z − u1 fb1,n − . . . −
un−1 fbn−1,n , say
ufbn = z − u1 fb1,n − . . . − un−1 fbn−1,n
where u ∈ k[[X, Y ]]. Hence
z = ufbn + u1 fb1,n + . . . + un−1 fbn−1,n
∈ (fb1,n , . . . , fbn−1,n )
2
For an algebraically closed field k and power series f and g in k[[X, Y ]], we let i(f ∩ g)
denote the intersection multiplicity of f and g at the maximal ideal, defined as in [5, 6].
If gcd(f, g) = 1 we have that
i(f ∩ g) = dimk (k[[X, Y ]]/(f, g))
Lemma 5.1.2 Let k be an algebraically closed field and let f1 , . . . , fn ∈ k[[X, Y ]] be
such that gcd(fi , fj ) = 1 whenever i 6= j. Then
dimk (k[[X, Y ]]/(fb1 , . . . , fbn )) =
X
i(fi ∩ fj )
1≤i<j≤n
Proof:
First we observe that the assumptions in this lemma imply the assumptions of
lemma 5.1.1.
We prove the lemma by induction on n, and note that for n = 2 this is just the formula
given above. From the definition of fbn and the properties of intersection multiplicities
120
(cf. [6, examples 1.1.1, 1.2.5d]) we have that
dimk (k[[X, Y ]]/(fn , fbn )) =
n−1
X
i(fi ∩ fn ).
i=1
By the induction hypothesis we may assume that k[[X, Y ]]/(fb1,n , . . . , fbn−1,n ) is a finite
dimensional vector space over k and that
dimk (k[[X, Y ]]/(fb1,n , . . . , fbn−1,n )) =
X
i(fi ∩ fj ).
1≤i<j≤n−1
Finally by using the exact sequence of k-vectorspaces from lemma 5.1.1 we deduce that
dimk (k[[X, Y ]]/(fb1 , . . . , fbn ))
= dimk (k[[X, Y ]]/(fb1,n , . . . , fbn−1,n )) + dimk (k[[X, Y ]]/(fn , fbn ))
=
X
i(fi ∩ fj ) +
X
i(fi ∩ fn )
i=1
1≤i<j≤n−1
=
n−1
X
i(fi ∩ fj )
1≤i<j≤n
2
With this we can now prove:
Theorem 5.1.3 Let A be the k-algebra
A := k[[X, Y ]]/(f1 f2 · · · fn )
where k is an algebraically closed field, and f1 , . . . , fn are irreducible power series such
that gcd(fi , fj ) = 1 whenever i 6= j. Then
dimk (A/ΓA ) =
X
1≤i<j≤n
121
i(fi ∩ fj )
Proof:
Let π denote the natural projection k[[X, Y ]] → A. The minimal prime ideals
of A are precisely the ideals (π(fi )), i = 1, . . . , n, and then from lemma 3.1.2 we have
that Γ(π(fi )) =
T
j6=i (π(fj ))
= (π(fbi )) (where the last equality follows from the fact that
k[[X, Y ]] is a unique factorization domain). This, and the definition of ΓA implies that
ΓA = (π(fb1 ), . . . , π(fbn )), and then we have that
A/ΓA ∼
= k[[X, Y ]]/(fb1 , . . . , fbn )
2
Thus the result follows directly from lemma 5.1.2.
Let A be a complete local ring with maximal ideal m and residue field k. Assume
that A is noetherian, reduced, of krull dimension 1 and of embedding dimension 2 (i.e.,
m is generated by two elements). Furthermore assume that k is algebraically closed and
of characteristic 0. Then A ∼
= k[[X, Y ]]/I for some ideal I of k[[X, Y ]] (cf. [14, §29]).
Since we assume that A reduced and of krull dimension 1, it follows that I =
Tn
i=1
pi
where p1 , . . . , pn are (distinct) height 1 prime ideals of k[[X, Y ]]. By [14, theorem 20.1]
we have that all the ideals pi are principal, say pi = (fi ) for some irreducible power series
fi ∈ k[[X, Y ]]. This shows that I =
Tn
i=1 (fi )
= (f1 f2 · · · fn ), and therefore theorem 5.1.3
applies to the ring A.
We observe especially the following. Let R be the coordinate ring of a reduced affine
plane curve over an algebraically closed field k of characteristic 0, i.e., R = k[X, Y ]/I
where I is a radical ideal. If A = R̂, the completion of R at some maximal ideal, then A
is as above (A is reduced since R is excellent, cf. [7]) and hence theorem 5.1.3 applies.
122
5.2
The algebra H(A)
5.2.1
A description when the irreducible components of A are
regular
Let A be a commutative noetherian and reduced k-algebra. By the remark following
lemma 3.2.4 we have that ΓA is an essential D(A)-submodule of A, and so D(A, ΓA ) is
an essential ideal of D(A). By lemma 1.2.2 we then have an injective morphism
φ : D(A)/D(A, ΓA ) −→ D(A/ΓA ).
In general we do not expect this morphism to be surjective. For example we observe
that by lemmas 3.1.2 and 3.2.1 every p ∈ Ass(A) is a D(A)-submodule of A, and thus it
follows that φ(D) ∗ (p + ΓA )/ΓA ⊂ (p + ΓA )/ΓA . I.e., we have that
Im φ ⊂ {D ∈ D(A/ΓA )|D ∗ ((p + ΓA )/ΓA ) ⊂ (p + ΓA )/ΓA for all p ∈ Ass(A)}
We will show that sometimes this inclusion is in fact an equality (cf. [3, 25]).
Let I be an ideal of a commutative k-algebra A. If I is not a left D(A)-submodule of
A, then D(A, I) is only a right ideal of D(A) but not a left ideal. Hence we introduce
the following notation. Let ID(A) (D(A, I)) denote the idealizer of D(A, I) in D(A). That
is, ID(A) (D(A, I)) is the largest subring of D(A) in which D(A, I) is a twosided ideal. It
follows immediately that
ID(A) (D(A, I)) = {D ∈ D(A)|D ∗ I ⊂ I}
123
and it is clear that order of differential operators defines a filtration on ID(A) (D(A, I)). If
I is a left D(A)-submodule, by lemma 1.2.2 the natural projection A → A/I induces an
homomorphism π : D(A) → D(A/I) of filtered rings, and its kernel is equal to D(A, I).
If I is not a D(A)-submodule of A, this is of course no longer true. However, by using
the idealizer, we can state a similar result which holds for all ideals of A.
Lemma 5.2.1 Let A be a commutative k-algebra, and let I be an ideal of A. Let π :
A → A/I be the natural projection. Then there is a homomorphism π : ID(A) (D(A, I)) →
D(A/I) of filtered rings, defined for D ∈ ID(A) (D(A, I)) by the formula π(D) ∗ π(a) :=
π(D ∗ a) for all a ∈ A. The kernel of this map is precisely D(A, I), so we have an exact
sequence
π
0 −→ D(A, I) −→ ID(A) (D(A, I)) −→ D(A/I)
Proof:
Same as the proof of lemma 1.2.2, after recalling that ID(A) (D(A, I)) is equal to
{D ∈ D(A)|D ∗ I ⊂ I}.
2
Let A be a commutative, noetherian and reduced k-algebra, and as in section 3.1.1,
let A :=
Q
p∈Ass(A)
A/p. For simplicity of notation we will identify A with its image in A,
and by lemma 3.2.7 we may also identify all the ideals Γp with their canonical images
in the rings A/p. With these identifications we have that ΓA =
shows that A/ΓA =
Q
p∈Ass(A) (A/p)/Γp ,
Q
p∈Ass(A)
and thus
Y
D(A) =
D(A/p)
p∈Ass(A)
Y
D(A, ΓA ) =
D(A/p, Γp )
p∈Ass(A)
Y
D(A/ΓA ) =
D((A/p)/Γp ).
p∈Ass(A)
With this description it is now clear that
ID(A) (D(A, ΓA )) =
Y
p∈Ass(A)
124
ID(A/p) (D(A/p, Γp)),
Γp ⊂ A. This
and that the canonical morphism ID(A) (D(A, ΓA )) → D(A/ΓA ) of lemma 5.2.1 is equal
to the direct product of the canonical morphisms
ID(A/p) (D(A/p, Γp )) −→ D((A/p)/Γp).
Using this we can now prove:
Lemma 5.2.2 Let k be a field of characteristic zero, and let A be a commutative kalgebra, essentially of finite type and reduced. Assume that A/p is regular for all p ∈
Ass(A). Then the natural morphism
ID(A) (D(A, ΓA )) −→ D(A/ΓA )
is surjective.
Proof:
By using the descriptions above, it is sufficient to prove that the morphisms
ID(A/p) (D(A/p, Γp)) −→ D((A/p)/Γp )
are surjective for all p ∈ Ass(A). However, since we assume that the rings A/p are regular
for all p ∈ Ass(A), this follows directly from [25, proposition 1.6].
2
We also recall that the inclusion A ⊂ A induces an inclusion D(A) ⊂ D(A) in such a
way that D(A) = {D ∈ D(A)|D ∗ A ⊂ A}. We have:
Lemma 5.2.3 Let A be a commutative, noetherian and reduced k-algebra. Then
D(A, A) = D(A, ΓA ) = D(A, ΓA )
125
Proof:
We identify A and D(A) with their images in A and D(A). As before propo-
sition 3.1.4, we let ei be the indecomposable idempotents of A corresponding to the
minimal prime ideals of A. Since A is a direct product of rings, D(A) is also a direct
product of rings by lemma 1.2.5, and then it is clear that all the ei s commute with all
elements of D(A). We recall also that A =
P
i ei A
and that ΓA is in fact the conductor
of A into A.
Let D be in D(A). Having the above it follows that
D ∈ D(A, A) ⇔ D ∗ ei A ∈ A for all i
⇔ ei D ∗ A ∈ A for all i
⇔ D ∗ A ⊂ ΓA
⇔ D ∈ D(A, ΓA ).
This shows that D(A, A) = D(A, ΓA ). For the third equality we first observe that
D(A, ΓA ) is obviously contained in D(A, ΓA ). Conversely let D ∈ D(A, ΓA ). Since
ΓA is an ideal of A, we have that D ∗ ei A = ei D ∗ A ⊂ ei ΓA ⊂ ΓA for all ei and hence it
follows that D ∈ D(A, ΓA ).
2
Having established these lemmas, we can now prove the main result of this section:
Theorem 5.2.4 Let k be a field of characteristic zero, and let A be a commutative
noetherian and reduced k-algebra, essentially of finite type. Assume that A/p is regular
for all p ∈ Ass(A). Then there are canonical isomorphisms
H(A) := D(A)/D(A, ΓA )
∼
= {D ∈ D(A/ΓA )|D ∗ (A/ΓA ) ⊂ A/ΓA }
∼
= {D ∈ D(A/ΓA )|D ∗ ((p + ΓA )/ΓA ) ⊂ (p + ΓA )/ΓA for all p ∈ Ass(A)}
126
Proof:
Let
T := ID(A) (D(A, ΓA )).
It is clear that D(A) = {D ∈ T |D ∗ A ⊂ A}, and D(A, ΓA ) is an ideal of T by definition.
We let π : A → A/ΓA denote the natural projection, and for D ∈ D(A), we let D denote
its image in H(A). Having set this notation it now follows from our assumptions on A
and all the lemmas in this section that
D(A)/D(A, ΓA ) ⊂ T /D(A, ΓA ) = T /D(A, ΓA ) ∼
= D(A/ΓA )
Furthermore it is clear that the (injective) morphism D(A)/D(A, ΓA ) → D(A/ΓA ) is
given by D 7→ π(D). From this it follows that if D ∈ D(A), then π(D) ∗ (A/ΓA ) =
π(D) ∗ π(A) = π(D ∗ A) ⊂ π(A) = A/ΓA . This shows that we have a well defined and
injective canonical homomorphism
H(A) −→ {D ∈ D(A/ΓA )|D ∗ (A/ΓA ) ⊂ A/ΓA }.
For surjectivity let D ∈ D(A/ΓA ) be such that D ∗ (A/ΓA ) ⊂ A/ΓA . By lemma 5.2.2
there is a D0 ∈ T with π(D 0 ) = D, and then we have for all a ∈ A that
π(D 0 ∗ a) = π(D 0 ) ∗ π(a) = D ∗ π(a) ∈ A/ΓA .
This shows that D0 ∗ A ⊂ A, so D 0 ∈ D(A), and hence D is the image of D 0 ∈
D(A)/D(A, ΓA ). This proves the first isomorphism
To prove the second isomorphism, let
S := {D ∈ D(A/ΓA )|D ∗ ((p + ΓA )/ΓA ) ⊂ (p + ΓA )/ΓA for all p ∈ Ass(A)}
and
S 0 := {D ∈ D(A/ΓA )|D ∗ A ⊂ A}
127
As mentioned earlier, we have the canonical injective homomorphism
φ : D(A)/D(A, ΓA ) → S.
We have already established the isomorphism H(A) → S 0 , so in order to prove that φ is
an isomorphism, it suffices to provide an injective morphism S → S 0 making the diagram
∼
=
H(A)
- S0
@
@
φ@
@
R
@
6
S
commutative.
It is clear that A/ΓA ⊂ A/ΓA , so our first problem is to show that differential operators
in S extend to differential operators on A/ΓA .
Fix a p ∈ Ass(A) and let πp0 : A/ΓA → A/(p + ΓA ) denote the canonical projection.
By definition of S it is contained in the ring
ID(A/ΓA ) (D(A/ΓA , (p + ΓA )/ΓA )),
so it follows immediately from lemma 5.2.1 that we have a canonical ring homomorphism
ψp : S → D(A/(p + ΓA )) such that ψp (D) ∗ πp0 (a) = πp0 (D ∗ a) for all D ∈ S and all
a ∈ A/ΓA .
We now observe that (p + ΓA )/p = Γp , so it follows immediately that A/(p + ΓA ) is
canonically isomorphic to (A/p)/Γp . We therefore get a ring homomorphism ψp0 : S →
D((A/p)/Γp ), and if we let πp00 denote the composite A/ΓA → A/(p + ΓA ) → (A/p)/Γp ,
then it is trivial to check that ψp0 (D) ∗ πp00 (a) = πp00 (D ∗ a).
The above is true for all p ∈ Ass(A), and since ΓA =
ψ0 : S →
Q
p∈Ass(A)
Q
p∈Ass(A)
Γp , we get a morphism
D((A/p)/Γp) ∼
= D(A/ΓA ). Furthermore we observe that under the
128
identification
Y
(A/p)/Γp ∼
= A/ΓA ,
p∈Ass(A)
the map (πp00 )p∈Ass(A) : A/ΓA → A/ΓA is simply the natural inclusion. Therefore it follows
that ψ 0 (D) ∗ a = D ∗ a for all D ∈ S and all a ∈ A/ΓA . Thus we have that ψ 0 is injective,
ψ0
and Im ψ 0 ⊂ S 0 . Finally it is easy to verify that the composite H(A) → S → S 0 is equal
∼
=
S 0 constructed in the first half of the proof.
to the isomorphism H(A) →
129
2
Bibliography
[1] J. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Differential operators on the cubic
cone, Russian Math. Surveys 27 (1972), 169-174.
[2] N. Bourbaki, Commutative Algebra, Chapters 1-7, Springer-Verlag, Berlin, Heidelberg and New York, 1989.
[3] K. A. Brown, The Artin Algebras Associated with Differential Operators on Singular
Affine Curves, Preprint, University of Glasgow, 1989.
[4] M. Chamarie and J. T. Stafford, When rings of differential operators are maximal
orders, Math. Proc. Camb. Phil. Soc. 102 (1987), 399-410.
[5] W. Fulton, Algebraic Curves, Benjamin/Cummings, Menlo Park California, 1969.
[6] W. Fulton, Intersection Theory, Ergebnisse der Mathematik, 3. Folge, Band 2,
Springer-Verlag, Berlin, Heidelberg, 1984.
[7] A. Grothendieck, Éléments de Géométrie Algébrique, Publications Mathématiques
No. 4, 8, 11, 17, 20, 24, 28 and 32, I.H.E.S., Paris, 1960 – 1967.
[8] R. Hart, Differential operators on affine algebras, J. London Math. Soc. (2) 28
(1983), 470-476.
[9] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.
130
[10] R. G. Heyneman and M. E. Sweedler, Affine hopf algebras, J. Algebra 13 (1969),
192-241.
[11] B. Iversen, Generic Local Structure in Commutative Algebra, Lecture Notes in Mathematics, Vol. 310, Springer-Verlag, Berlin and Heidelberg, 1973.
[12] M.-A. Knus and M. Ojanguren, Théorie de la Descente et Algèbres d’Azumaya,
Lecture Notes in Mathematics, Vol. 389, Springer-Verlag, Berlin and Heidelberg,
1974.
[13] T. Levasseur and J. T. Stafford, Rings of differential operators on classical rings of
invariants, Mem. Amer. Math. Soc. 81 No. 412 (1989).
[14] H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge,
1986.
[15] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, WileyInterscience, Chichester and New York, 1987.
[16] J. S. Milne, Étale Cohomology, Princeton University Press, Princeton, 1980.
[17] J. L. Muhasky, The differential operator ring of an affine curve, Trans. Amer. Math.
Soc. (2) 307 (1988), 705-723.
[18] I. M. Musson, Some rings of differential operators which are Morita equivalent to
Weyl algebra A1 , Proc. Amer. Math. Soc. 98 (1986), 29-30.
[19] I. M. Musson, Rings of differential operators and zero divisors, J. Algebra (2) 125
(1989), 489-501.
[20] Y. Nakai, High order derivations 1, Osaka J. Math 7 (1970), 1-27.
[21] M. Raynaud, Anneaux Locaux Henséliens, Lecture Notes in Mathematics Vol. 169,
Springer-Verlag, Berlin and Heidelberg, 1970.
131
[22] J. J. Rotman, An Introduction to Homological Algebra, Academic Press, Orlando
Florida, 1979.
[23] S. P. Smith, Curves, Differential Operators and Finite Dimensional Algebras,
Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, 158-176, Lecture Notes
in Mathematics Vol. 1296, Springer-Verlag, Berlin and Heidelberg, 1987.
[24] S. P. Smith, preprint, May 1988.
[25] S. P. Smith and J. T. Stafford, Differential operators on an affine curve, Proc.
London Math. Soc. (3) 56 (1988), 229-259.
[26] M. E. Sweedler, Groups of Simple Algebras, Publications Mathématiques No. 44,
79-189, I.H.E.S., Paris, 1974.
132