i
NOTES ON LOCAL COHOMOLOGY
C. W. WILKERSON
Lecture One
These notes began as an attempt to understand the statement of the
Benson-Carlson theorem in group cohomology. The information was
mostly gleaned from the book ”Twenty-four hours of local cohomolgy”
by Iyengar, Leuschke, Leykin, Miller, Miller, Singh, and Walther.
0.1. Theorem. Let G be a finite group, k a field, and M a k[G] module.
Then there is a spectral sequence
∗,∗
Hm
(H ∗ (G, M)) =⇒ Σ−d H∗ (G, M) .
∗,∗
Here Hm
is the (graded) local cohomology with respect to the homogeneous augmentation ideal and d is the Krull dimension of H ∗ (BG, k).
The convention here is that homology modules have negative gradings while cohomology modules have positive gradings.
0.2. Corollary. If H ∗ (G, k) is Cohen-Macaulay, the spectral sequence
∗,∗
above collapses, Hm
(H ∗ (G, k)) = Σ−d H∗ (G, k), and H ∗ (G, k) is Gorenstein.
These notes will not discuss the construction of the spectral sequence
but should allow us to see how Cor. 0.2 follows from the main theorem.
In particular, local cohomology and its connections to Cohen-Macaulay
and Gorenstein rings will be discussed.
Let R be a commutative ring with unit, M a left R-module, and a
an ideal of R. In the applications R is often Noetherian. This is also
required for many of the constructions. The a-torsion torsion elements
Γa(M) = {m ∈ M|an m = 0 for n >> 0}
is the set of elements in M annihilated by some power of a.
0.3. Lemma. Γa(−) is a left exact additive functor from R-modules to
R-modules. That is
1
2
C. W. WILKERSON
a) f : M → N induces Γa(f ) : Γa(M) → Γa(N) .
b) If M is isomorphic to A ⊕ B, then Γa(M) is isomorphic to Γa(A) ⊕
Γa(B).
c) If 0 → A → B → C → 0 is exact then 0 → Γa(A) → Γa(B) → Γa(C)
is exact.
This may fail to be exact on the right. For example R = Z, a = (2),
and 0 → Z → Z → Z/2Z → 0 .
Since Γa(−) is left exact, it has right derived functors {Rj Γa(−)}. Here
R0 Γa(M) = Γa(M). These R-modules are denoted as {Haj (M)} and
for each short exact sequence as above produce a long exact sequence
0 → Γa(A) = Ha0 (A) → . . . Ha0 (C) → Ha1 (A)
→ . . . → Ha1 (C) → . . . → Haj (A) → Haj (B) → Haj (C) → . . .
More explicitly, take a resolution {I ∗ } of M by injective R-modules, a
chain complex {I ∗ = 0 → I 0 → I 1 → I 2 → . . .} with H 0 (I ∗ ) = M and
H j (I ∗ ) = 0 for j > 0. Then Haj (M) = H j (Γa(I ∗ )).
In the context of sheaves over schemes, let X = Spec(R), Y = V (a)
the subscheme associeted to a, and M̃ the sheaf corresponding to the
R-module M. Then Γa(M) = ΓY (X, M̃ ) and the local cohomology
modules are isomorphic to the sheaf cohomology modules with support in Y , HY∗ (X, M̃ ).
Calculating from first principles is only practical for small examples
for which the injective resolutions are apparent. Recall that if I is an
injective R-module, then it is divisible. That is, if a ∈ I, r ∈ R a
non-zero-divisor, there exists b ∈ I with rb = a. Conversely, if R is a
PID, then divisible modules are injective. Our first examples treat this
case.
0.4. Example. Let R = Z and a = (p), for some prime p.
a) Let M = Z/nZ where n is a nonzero integer. Then 0 → Q/Z →
Q/Z → 0 is an injecxtive resolution, where the nonzero map is multiplication by n. Then Γp(Q/Z) = Z/p∞ Z and the induced map is
NOTES ON LOCAL COHOMOLOGY
3
multiplication by n. Hence Hp0 (Z/dZ) = Z/pe Z and Hpj (Z/dZ) = 0 if
j > 0. Here pe is the largest power of p that divides n.
b) Let R = Z, p = (p), and M = Z. Then 0 → Q → Q/Z → 0 is
an injective resolution, since Q and Q/Z are divisible abelian groups.
Γp(Q) = 0 and Γp(Q/Z) = Z/p∞ Z = Z[1/p]/Z. Z/p∞ Z is generated
over Z by {1/p, 1/p2, . . . 1/pn , . . .}, or indeed any infinite subsequence
of these. Thus Hp0 (Z) = 0 and Hp1 (Z) = Z/p∞ Z.
c) Let R = Z, a = (0), and M any abelian group. Since Γ(0) (A) = A
j
0
for all A, H(0)
(M) = M and H(0)
(M) = 0 for j ≥ 1. That is, Γ(0) (−)
is the identity functor, hence exact, and therefore has trivial higher
derived functors.
d) Let R = Z, a = (1), and M any abelian group. Then Γ(1) (A) = 0 for
j
all A, and H(1)
(M) = 0 for j ≥ 0. That is, Γ(1) (−) is the zero functor,
hence exact, and therefore has trivial higher derived functors.
Note that in example b) the nonzero local cohomology group is not
finitely generated. This non-finite generation is common for local cohomology. Note also that Z/p∞ Z is Artinian (descending chains of
submodules terminate). This too is typical for local cohomology.
The next example of a polynomial algebra on one variable is directly
applicable to the main theorem in the case G = Z/2Z and k = F2 . It
is also analogous to the calculations above with Z.
0.5. Example. Let R = k[x] where k is a field and m is the maximal
ideal generated by x. Then L, the fraction field of R,i k(x) is divisible
and hence injective.
a) Let M = k = R/m. Then 0 → L/R → L/R → 0 is an injective
resolution of M, where the nonzero map is multiplication by x. Then
Γm(L/R) = R[1/x]/R, which is generated over R by any infinite subsequence of {1/x, 1/x2 , . . . 1/xn , . . .}. The kernel of multiplication by
x is generated by the equivalence class of {1/x} and is isomorphic to
0
k with the trivial action of k[x] on it. Thus Hm
((k) = k. On the other
hand multiplication by x is a surjective endomorpism of R[1/x]/R, so
1
Hm
(k) = 0.
4
C. W. WILKERSON
b) Let M = R. Then 0 → L → L/R → 0 is an injective resolution of
0
R. Γm(L) = 0 and as before Γm(L/R) = R[1/x]/R. Thus Hm
(k[x]) = 0
1
and Hm(k[x]) = k[x][1/x]/k[x]. This can be thought of as the sub-k[x]module of k(x)/k[x] generated by the classes {1/x, 1/x2 , . . . , 1/xn , . . .}.
0.6. Example. The constructions involved in defining local cohomology can be done in the context of graded algebras and modules, with
homogeneous ideals. So let’s consider the graded polynomial algebra
k[x], where the grading of x is 1. Take m = (x) and M = R = k[x].
Now in the graded category the injective hull of k[x] is k[x, x−1 ], so one
has the injective resolution 0 → k[x, x−1 ] → k[x, x−1 ]/k[x] → 0. These
modules are now Z-graded. Γm(k[x, x−1 ]) = 0 and Γm(k[x, x−1 ]/k[x]) =
0
1
k[x, x−1 ]/k[x]. Hence Hm
(k[x]) = 0 and Hm
(k[x]) = k[x, x−1 ]/k[x]. It
can be visualized as the ideal yk[y] where k[y] is the polynomial algebra on a generator of degree −1. The k[x]-action is xy a = y a−1 if
a − 1 > 0 and 0 otherwise. Thus the local cohomology has a bigrading.
If bidegree is (i, j) and i 6= 1, the component is 0. The component for
(1, −j) is k for j ≥ 1 and 0 otherwise.
0.7. Example. (The example G = Z/2Z, k = F2 ) H ∗ (Z/2Z, F2 ) = F2 [x]
where x has degree 1. The previous example calculated the local cohomology as yk[y] where y has degree −1. Thus the Benson-Carlson
theorem holds by calculation in this example.
NOTES ON LOCAL COHOMOLOGY
5
Lecture Two : Computations with Koszul and Čech complexes
If R is a Noetherian ring, then each injective is a direct sum of indecomposable injectives, {E(R/p)}, where p ranges over the prime ideals
of R, and E(R/p) is the injective hull. This is a minimal essential
injective super-module and is unique up to isomorphism. Thus the
injective resolution of an R-module M is formed from copies of these
basic modules.
0.8. Theorem. Let R be a Noetherian ring and a an ideal of R. Then
for each prime ideal p of R,
Haj (E(R/p)) = 0
if j ≥ 1 and
Ha0 (E(R/p)) = E(R/p)
if a ⊂ p and is zero otherwise.
The multiplicities of the minimal injectives in a resoltion of a module
M are called the Bass numbers of M and are subject to much research.
Fortunately, one does not always need a resolution to calculate the
local cohomology. This lecture will discuss some other methods of calculation.
The sequence of maps
. . . R/an → R/an−1 → . . . R/a2 → R/a
gives rise to a direct limit
HomR (R/a, M) → HomR (R/a2 , M) . . . HomR (R/an , M) → . . .
n
and one has Γa(M) = −
lim
→n HomR (R/a , M) in a functorial manner.
Applying this to an injective resolution of M and using the exactness
of filtered direct limits yields the following:
∗
n
0.9. Theorem. Ha∗ (M) = −
lim
→ ExtR (R/a , M).
n
One can regard this as a hint of constructions we’ll see later in which
M is replaced by its R/a-approximation, CellR/a(M) which has as its
homotopy groups the local cohomology groups {Haj (M)}.
In the direct limit above the sequence {an } can be replaced with any
cofinal sequence of ideals. For example if n is a nilpotent ideal, in the
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C. W. WILKERSON
limit one gets the zero ideal and hence Γn(−) is the identity functor.
Likewise in general, an ideal can be replaced by its radical for the purposes of local cohomology.
0.10. Corollary. If R is Noetherian, any element of Haj (M) is annihilated by some power of a.
By the previous theorem, each element in local cohomology is in the
image of one from ExtjR (R/an , ), which is annihilated by an .
We need methods of calculating local cohomology that are more directly related to the structure of a and R.
0.11. Definition. a) Let x ∈ R. The Koszul complex K ∗ (x, R) is the
sequence
x
0→R−
→R→0
, where the first R is in homological degree −1, the second in degree 0,
and the map is multiplication by x. H j (x, M) is H j (K ∗ (x, R) ⊗R M).
b) Let x = {x1 , x2 , . . . xr } be a sequence of elements in R.
O
K ∗ (x, R) =
K ∗ (xi , R)
R,i
is the Koszul complex on x.
c) H j (x, M) = H j (K ∗ (x, R) ⊗R M).
0.12. Definition. a) x is a regular element in R if it is not a zerodivisor and xR 6= R. If M is an R-module, x is a M-regular element if
xm = 0 implies m = 0 in M and xM 6= M.
b) x = {x1 , x2 , . . . xr } is a regular sequence in R if x1 is regular and
each xk is regular for R/(x1 , . . . , xk−1 )R.
0.13. Corollary. x is M-regular if and only if H −1 (x, M) = 0 and
H 0 (x, M) 6= 0. Note that H 0 (x, M) = M/xM. In general, for
x = {x1 , . . . , xd }, H −d (x, M) = (0 :M x), the elements in M annihilated by the x and H 0 (x, M) = M/xM.
0.14. Theorem. Let (R, m) be a local ring, {x1 , . . . xn } ⊂ m, and M a
finitely generated R-module. Then H j (x, M) = 0 for j < n if and only
if x is a regular M-sequence.
NOTES ON LOCAL COHOMOLOGY
7
The second complex needed is a limit of Koszul complexes, in a sense
to be explained later.
0.15. Definition. The Čech complex Č ∗ (x, R) is the complex
0 → R → Rx = R[1/x] → 0
where the non-trivial map is localization at the multiplicative set S =
{1, x, x2 . . .}. The leftmost R is in homological degree 0 and the next
in degree 1. We also have
Č ∗ (x, R) ⊗R M = {0 → M → Mx → 0} .
N
∗
Similarly, for x = {x1 , x2 , . . .}, define Č ∗ (x, R) =
R Č (xi , R) with
the appropriate sign convention on the differentials.
Notice that we have M → ⊕i Mxi at the 0-stage of Č ∗ (x, R) ⊗R M.
0.16. Lemma. Let M be a R-module and a generated by {x1 , x2 , . . .}.
Then H 0 (Č ∗ (x, R) ⊗R M) is naturally isomorphic to Γa(M).
This lemma together with showing that the higher Čech cohomology
vanishes on injective modules shows that it must coincide with the local cohomology. This requires the Noetherian condition in general.
0.17. Theorem. Let R be a Noetherian ring and a = (x1 , x2 , . . .). Then
Haj (M) is natually isomorphic to H j (Č ∗ (x, R) ⊗R M).
Finally, we wish to relate the Čech complex to the Koszul complex. First, for a given x ∈ R we have the chain map K ∗ (xt+1 , R) →
K ∗ (xt , R) given by
0 −−−→


y
xt+1
R −−−→

x
y
R −−−→

=
y
0


y
0 ←−−− R ←−−− R ←−−−
x
x
x



=
x

0
x


xt
0 −−−→ R −−−→ R −−−→ 0
Applying HomR (−, R), we get
xt+1
xt
0 ←−−− R ←−−− R ←−−− 0
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C. W. WILKERSON
That is, we get, after observing that
x
x
Rx = R[1/x] = −
lim
→ ... −
→ . . .},
→{R −
∗ t
that Č(x, R) = Σ1 lim
−→t K (x , R). This argument generalizes to a sequence x of elements. In general, the KoszulNcomplex has a duality, in
that HomR (K ∗ (x, R), M) = Σ+c (K ∗ (x, R) R M), where c is the cardinality of x. Thus local cohomology inherits the properties of Koszul
homology for regular sequences.
0.18. Corollary. If a contains a M-regular sequence of length d, then
Haj (M) = 0 for 0 ≤ j ≤ d − 1. If a is generated by n elements, then
Haj (M) = 0 for j > n.
0.19. Example. Let R = k[x1 , . . . , xd ] be the polynomial algebra on d
generators of degree 1 over a field k and m the maximal ideal generated
j
by the {x′i s}. Let M = R. Then Hm
(R) = 0 for j < d , for j > d, and
d
Hm
(R) = (y1 y2 . . . yd )k[y1 , y2, . . . yd ]
where each yk has degree −1. The R-action is given by
xi y1a1 . . . yiai . . . ydad = y1a1 . . . yiai −1 . . . ydad
if ai ≥ 2 and 0 otherwise.
NOTES ON LOCAL COHOMOLOGY
9
Lecture Three: Cohen-Macaulay, Gorenstein, and Duality
In the hierarchy of commutative local rings, regular local rings occupy
the highest rung. These are the local analogue of polynomial algebras
in the graded world. One step down are the complete intersections,
that is, quotient rings of the form polynomial or regular local mod an
ideal generated by a regular sequence. There are two useful lower rungs
that are not so well known outside of commutative algebra. We discuss
the most general case first.
0.20. Definition. Let (R, m, K) be a local ring. If m contains a regular
sequence {x1 , . . . , xd } of length d = dim(R), R is said to be CohenMacaulay. In general, the maximal length of an R-regular or M-regular
sequence is the depth of R or M
We use the analogous definitions in the graded case.
0.21. Example. For k a field, the polynomial algebra k[x1 , . . . , xd ] is a
graded Cohen-Macaulay ring. Any regular local ring is Cohen-Macaulay,
as is any complete intersection.
0.22. Theorem. (Noether Normalization) Let R be a connected graded
finitely generated commutative algebra over the field k. There exists a
sequence of homogeneous elements {x1 , . . . , xd } of R such that
k[x1 , . . . , xd ] is a polynomial subalgebra of R and R is a finitely generated module over k[x1 , . . . , xd ]. These can be chosen so that R is a free
module over k[x1 , . . . , xd ] if and only if R is a Cohen-Macaulay ring.
0.23. Corollary.
Let R∗ be a graded connected algebra over the field k
P
∞
and Pt (R∗ ) = i=0 dimk (Ri ) the Poincare-Hilbert series for R∗ . If R∗
is Cohen-Macaulay, then
Y
Pt (R∗ ) = g(t)/ (1 − tni )
where g(t) is a polynomial with non-negative integer coefficients.
0.24. Example. Let R = k[x, y/(xy). dim(R) = 1, and taking z = x+y,
R is freely generated over k[z] by {1, y}. The regular sequence can be
taken to be {z}. Pt (R) = (1 + t)/(1 − t).
Cohen-Macaulay also has a characterization in terms of local cohomology:
10
C. W. WILKERSON
0.25. Theorem. Let (R, m) be a local ring. R is Cohen-Macaulay if
j
and only if Hm
(R) = 0 for all j 6= dim(R).
In particular, this implies that in the Benson-Carlson spectral sed
quence, there is only one non-zero vertical line, Hm
.
For local rings, the rung between Cohen-Macaulay rings and complete intersections is occupied by Gorenstein local rings. Identified first
by Bass and Serre, Gorenstein rings have many characterizations. The
standard one is somewhat non-obvious.
0.26. Definition. A local ring (R, m, K) is Gorenstein if and only if it
has a finite injective resolution over itself.
The first case to understand is when R has zero Krull dimension.
0.27. Theorem. Let (R, m, K) be a zero-dimensional local ring. The
following conditions are equivalent:
a) R is Gorenstein;
b) R is injective as an R-module;
c) dimK soc R = 1 ;
d) ER (R/m) = R;
Recall that socR M = HomR (R/m, M) = (0 :M m).
For example, if X is a compact closed oriented manifold, its cohomology
algebra H ∗ (X, Q) satisfies Poincaré duality. That is, the cohomology
is
i=n
M
R=
Ri
i=0
and the cup product gives a perfect pairing Ri ⊗K Rn−i → Rn for all
relevant i.
0.28. Lemma. If R is a graded commutative algebra over the field K,
finite dimensional as a K-vector space, and satifies Poincaré duality,
then R is a Gorenstein ring.
This follows immediately from the theorem above, since the socle is
generated by the top class of R.
NOTES ON LOCAL COHOMOLOGY
11
We can understand Gorenstein rings inductively, via the next theorem:
0.29. Theorem. Let (R, m, K) be a local ring and {x1 , . . . , xc } a regular sequence. Then R is Gorenstein if and only if R/(x1 , . . . , xc ) is
Gorenstein.
0.30. Theorem. Let (R, m, K) be a Noetherian local ring and M a
finitely generated R-module. Then
a) depthR M = min{i|ExtiR (K, M) 6= 0}.
b) injdimR M = sup{i|ExtiR (K, M) 6= 0}.
c) depthR M ≤ dim M and injdimR M ≥ dim M.
d) if injdimR M is finite, injdimR M = depthR R.
Thus by c) and d), if injdimR R is finite, depthR R = dim R, so Gorenstein implies Cohen-Macaulay.
0.31. Example.
a) Regular local rings are Gorenstein.
b) Complete interection rings are Gorenstein.
c) Gorenstein rings are Cohen-Macaulay, but Cohen-Macaulay rings
need not be Gorenstein.
0.32. Theorem. If (R, m, K) is Gorenstein, then it is Cohen-Macaulay.
j
In particular (R, m, K) is Gorenstein if and only if Hm
(R) = 0 for
d
j 6= d = dim(R) and Hm(R) = ER (K).
ER (K) can be used to form a type of dual for R-modules:
0.33. Definition. Let (R, m, K) be a local ring and M an R-module.
Then M ˇ= HomR (M, ER (K)).
Note that M 6= Mˇˇ in general. Rather, it is the m-adic completion
of M. For example, with R = Z and m = (p), EZ (Fp ) = Z/p∞ Z, and
Zˇˇ= HomZ (Z/p∞ Z, Z/p∞ Z) = Zpˆ, the p-adic integers.
0.34. Theorem. (Local Duality) Let (R, m, K) be a d-dimensional local
Gorenstein ring and M a finitely generated R-module. Then
i
Hm
(M) ≈ Extd−i
R (M, R)ˇ .
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C. W. WILKERSON
Alternately,
i
R
Hm
(M) ≈ T ord−i
(ER (K), M) .
The second form of the duality follows directly from realizing that the
∗
Čech complex Č(x, R) gives a shifted flat resolution of Hm
(R) = ER (K)
since R is Gorenstein. Hence tensoring with M and taking cohomology
computes a shifted T or module.
If X is a non-singular affine variety, it has a cotangent vector bundle T ∗ X and associated line bundle ∧n T ∗ X. The sections of this line
bundle are a locally free rank one module over the coordinate ring of
the variety. It has remarkable properties and has been generalized to
the singular case.
0.35. Definition. Let (R, m) be a Cohen-Macaulay local ring. A canonical module is a finitely generated R-module ω for which
a) depth ω = dim(R)
b) typeR ω = 1. Alternately, HomR (ω, ω) = R.
c) injdimR ω is finite.
These require some explanation. The depth of a module M is the
length of the longest M-regular sequence. The type of a module M is
dimK ExttR (K, M), where K = R/m and t is the depth of M. This is
analogous to the rank one projective module in the geometric case.
Note that if R is Gorenstein, then R itself is a canonical module. For local rings, the canonical module is essentially unique, in that any two are
isomorphic. There is a form of local duality for Cohen-Macaulay rings
with canonical modules: Not every Cohen-Macaulay local ring has a
canonical module. Those that are the surjective image of a Gorenstein
local ring do. In particular, complete local rings and finitely generated
graded K-algebras qualify.
0.36. Theorem. Let R, m, K) be a d-dimensional local Cohen-Macaulay
ring with canonical module ω and let M be a finitely generated Rmodule. Then for 0 ≤ i ≤ d there are isomorphisms
i
Hm
(M) ≈ Extd−i
R (M, ω)ˇ .
d
In particular, Hm
(R) ≈ ωˇ.
NOTES ON LOCAL COHOMOLOGY
13
In the graded case, there are some small changes. Recall that if M
is graded, M[a] is the module with M[a]i = Ma+i . In the topological
world, M[a] = Σ−a M. In this context the dual HomR (M, ER (K))
becomes HomK (M, K).
L
0.37. Theorem. Let R = n≥0 Rn be a finitely generated graded algebra over R0 = K, a field.Then the graded injective hull of K is
M
ER (K) = HomK (R, K) =
HomK (Rn , K) .
n≥0
ER (K)−i = HomK (R, K)−i = HomK (Ri , K). If R is Gorenstein and
j
d-dimensional then Hm
(R) = 0 for j 6= d and
d
Hm
(R) = HomK (R, K)[a] = Σ−a HomK (R, K)
for some integer a.
Here the action of R on HomK (R, K) is by r(f )(m) = f (rm). We
think of HomK (R, K) as being negatively graded. We have seen examples of this already in the polynomial algebra case. The point is that
K → HomK (R, K) is an essenital extension. That is, any non-zero Rsubmodule of HomK (R, K) must intersect K non-trivially. Also, since
HomK (R, K) is dual to the free module R, it is injective.
0.38. Example. Let R = K[x1 , . . . , xd ] where the {xi } have degree 1.
Then
d
Hm
(R) = HomK (R, K)[d] = Σ−d HomK (R, K) .
0.39. Theorem. Let R be a graded ring as above and R Gorenstein.
Then the Poincaré-Hilbert series Pt (R) satifies the functional equation
Pt (R) = (−1)d tN P1/t (R)
for d = dim(R) and some integer N.
Notice that the Corollary to the Benson-Carlson theorem now follows
by calculation, since the spectral sequence of the theorem has only one
d
line, and must therefore collapse. The conclusion is that Hm
(R) =
ER (k)[d], and hence R is Gorenstein.
14
C. W. WILKERSON
Section Four - Residues
I’d like to go now a little beyond the Benson-Carlson material to
residues. I’ll interpret this widely to mean mappings or pairings from
cohomology and homology modules to the ground field. Let’s start
with the nicest example. Let R be the polynomial algebra K[z] in the
variable z over the field K. Think of R as being a regular local ring,
either by localizing so that (x) becomes the only maximal ideal, or by
putting the grading on R with degree z = 1 and using only homogeneous ideals and maps.
0.40. Definition. ΩR/K is the module of Kahler differentials of R to
R over K. It’s generated over R by the symbols {dr}, subject to the
relations d(rr ′) = rdr ′ + r ′ dr for r, r ′ ∈ R and dk = 0 for k ∈ K.
0.41. Proposition. For R = K[z] as above, ΩR/K is the free R-module
generated by dz, i.e. R dz.
P
1
0.42. Proposition. For R = K[z] and m = (z), Hm
(ΩR/K ) = i<0 ai z i dz,
ai ∈ K, subject to the relations z(z −1 ) = 0 and z(z −i ) = z −i+1 if i > 1.
This is basically the same proof as earlier calculations that showed
1
Hm
(R) = R[1/z]/R. One can use the Čech complex.
0.43. Definition. For R = K[z] and m = (z), the residue map
1
res : Hm
(ΩR/K ) → K
X is
res(
ai z i dz) = a−1 .
i<0
Similar definitions and calculations can be done for the cases of R
the formal power series ring in one variable, or, over C, the convergent
power series ring. Thus one sees the connection to the residues of
complex variable theory. There are also similar calculations for R a
regular local ring of dimension d. Here ΩR/K is replaced by its d-fold
d
exterior power and one has res : Hm
(∧d (ΩR/K )) → K.
Department of Mathematics, Purdue University
Department of Mathematics, Texas A & M University
E-mail address: cwilkers@purdue.edu
E-mail address: cwilkers@math.tamu.edu