Background
Results
Example
Torsion Divisor Classes
Sandra Spiroff
Joint with Sean Sather-Wagstaff, North Dakota State Univ.
University of Mississippi
AMS Fall Eastern Sectional Meeting
Halifax, Nova Scotia, October 18, 2014
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
Outline
Background
Results
Example
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
Background
Let A → B be a homomorphism of Noetherian normal integral
domains. Under certain circumstances, one can show that such a
map induces a group homomorphism Cl(A) → Cl(B).
• (faithfully) flat ring homomorphisms;
The map Cl(A) → Cl(B) is given by [a] 7→ [a ⊗A B].
• integral extensions;
• (Danilov, 1970) the natural surjection A[[T ]] → A;
• (Lipman, 1979) any surjection of the form A → A/tA,
assuming that A and A/tA are both Noetherian normal integral
domains.
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Torsion Divisor Classes
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Example
• (S-W, S, 2009) homomorphisms of finite flat dimension
between normal domains.
e.g., A → A/I , where I is a prime ideal of A with finite
projective dimension, A and A/I are normal domains.
The map Cl(A) → Cl(B) is given by [a] 7→ [(a ⊗A B)BB ].
• (S-W, S, 2013) A → (A/I )0 , where A is a normal domain, I a
prime ideal of A such that A/I is excellent and satisfies (R1 ), and
(A/I )0 the integral closure of A/I .
The kernel of this induced map has been studied by V.I. Danilov,
P. Griffith, J. Lipman, C. Miller, D. Weston, and others.
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Torsion Divisor Classes
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Example
Some Definitions
The dual of an A-module a is aA := HomA (a, A).
There is a map σ : a → aAA , defined by σ(x) = ϕx , where
ϕx (f ) := f (x), for f ∈ HomA (a, A). If σ is an isomorphism, then a
is said to be reflexive.
The divisor class group of A, denoted Cl(A), is the group of
isomorphism classes of reflexive ideals of A, or equivalently,
reflexive A-modules of rank one. An element [a] in Cl(A) is called
a divisor class.
• multiplication is defined by [a] · [b] = [(a ⊗A b)AA ];
• the identity element is [A];
• [a]−1 = [aA ] = HomA (a, A).
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Example
FACT: A Noetherian integral domain is a unique factorization
domain if and only if every height one prime is principal.
Example
• Let A = C[X ,Y ,Z ,W ]/(XY −ZW). Then Cl(A) ∼
= Z.
• Let B = C[X ,Y ,Z ]/(XY −Z 2 ). Then Cl(B) ∼
= Z/2Z.
There is a map Cl(A) → Cl(B) induced from A →
→ B.
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Torsion Divisor Classes
Background
Results
Example
Theorem (Griffith, Weston 1994*)
Let (A, m) be an excellent, local, normal domain and let t be a
principal prime element in m such that A/tA satisfies the condition
(R1 ). Let e > 1 be an integer which represents a unit in A. Then
the kernel of the homomorphism Cl(A) → Cl((A/tA)0 ) contains no
element of order e.
Recall: The map Cl(A) → Cl(B) is given by [a] 7→ [(a ⊗A B)BB ],
where B = (A/I )0 .
*P. Griffith and D. Weston, Restrictions of torsion divisor classes to
hypersurfaces, J. Algebra 167 (1994), no. 2, 473–487.
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Torsion Divisor Classes
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Results
Example
Outline of their proof
• Suppose [a] is an element of order e in kernel of
Cl(A) → Cl((A/tA)0 ); let α ∈ A be such that a(e) = αA.
• They describe an A-algebra structure on the direct sum
R = A ⊕ a ⊕ a(2) ⊕ · · · ⊕ a(e−1) that makes R isomorphic to the
integral closure of A[T ]/(T e − α).
• They then show that the integral closure of R ⊗A (A/tA)0 is
étale over (A/tA)0 and conclude that e = 1.
• Subtle Detail: That the integral closure of R ⊗A (A/tA)0 is
local.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
The best way to be certain of this claim is to use a result of
M. Hochster and C. Huneke, interpreting the integral closure of
R ⊗A (A/tA)0 as the S2 -ification of R ⊗A (A/tA)0 .
M. Hochster and C. Huneke, Indecomposable canonical modules and
connectedness, in: Commutative Algebra: Syzygies, Multiplicities, and
Birational Algebra, South Hadley, MA, 1992, AMS, 1994, pp. 197-208.
Fact (Hochster, Huneke)
Let (B, n) be an excellent equidimensional local ring. If B is (S2 )
and x1 , . . . , xk is a part of a system of parameters, then the
S2 -ification of B/(x1 , . . . , xk )B is local.
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Torsion Divisor Classes
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Example
Definition
For a Noetherian integral domain B, a subring S of the total ring
of quotients of B is an S2 -ification of B if:
• S is module finite over B;
• S satisfies the Serre condition (S2 ) over B; and
• Coker(B → S) has no prime ideal of B of height less than two
in its support.
Recall: S satisfies the (S2 ) condition over B if for every prime ideal
p of B, depth Sp ≥ min{2, ht p}.
FACT: If A is a local Noetherian normal domain that satisfies (R1 ),
then the S2 -ification S of A coincides with the integral closure of A
in its field of fractions.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Results
Example
Theorem One [Sather-Wagstaff, S-]
Let (A, m) be an excellent, local, normal domain and I a prime
ideal of A with finite projective dimension such that:
(i) A = A/I is normal;
(ii) µ(I ) ≤ dim A − 2, where µ(I ) is the minimal number of
generators of I ; and
(iii) I is a complete intersection on the punctured spectrum of A,
i.e., for each prime ideal p 6= m, the localization Ip is either
equal to Ap or generated by a regular sequence in Ap .
Then for any integer e > 1 which represents a unit in A, the kernel
of the homomorphism Cl(A) → Cl(A) contains no elements of
order e.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Example
Fact (Hochster, Huneke 1992**)
For (T , n) a complete local equidimensional ring such that (0) has
no embedded primes, the following conditions are equivalent:
• Hndim T (T ) is indecomposable;
• The canonical module of T is indecomposable;
• The S2 -ification of T is local;
• For every ideal J of height at least two, Spec(T ) − V (J) is
connected;
• Given any two distinct minimal primes p, q of T , there is a
sequence of minimal primes p = p0 , . . . , pr = q such that for
0 ≤ i < r , ht(pi + pi+1 ) ≤ 1.
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Torsion Divisor Classes
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Results
Example
A significant tool in the proof of theorem one is the next result.
Theorem Two [Sather-Wagstaff, S-] Let (A, m) be a complete,
local, normal domain such that A/m is separably closed and let I
be a prime ideal of A with finite projective dimension such that
A = A/I is normal. Let e > 1 be an integer which represents a
unit in A and assume that A contains a primitive e-th root of
unity. Let [a] be an element in Cl(A) with order e. Set
R = A ⊕ a ⊕ a(2) ⊕ · · · ⊕ a(e−1) . If any of the five equivalent
conditions of the HH Fact hold for R/IR, then
[a] ∈
/ Ker(Cl(A) → Cl(A)).
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Results
Example
Lemma: Let P ∈ Spec(R) and set p = P ∩ A. Then ht p = ht P.
Furthermore, if Ap is regular, then the extensions Ap → Rp and
Ap → RP are étale. In particular, the extension A ,→ R is étale in
codimension one.
Definition
We say that R is étale in codimension one (or more accurately, in
codimension less than or equal to one) over A if, for every prime
ideal P of S with height less than or equal to one and p = P ∩ A,
the ring RP is étale over Ap .
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Example
Lemma:The ring R is a complete local normal domain.
Lemma: Set R̄ = R ⊗A Ā = R/IR. The ring R̄ is equidimensional,
complete, and local. In particular, Fact HH applies with T = R̄.
Lemma: The extension Ā → R̄ is étale in codimension one.
Moreover, the ring R̄ satisfies the Serre condition (R1 ).
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Example
Lemma: For σ : R̄ → R̄ ĀĀ , we have Ker σ = j(R̄).
Definition
Denote by j(R̄) the largest ideal which is a submodule of R̄ of
dimension smaller than dim R̄. Specifically,
j(R̄) = {r̄ ∈ R̄ : dim(R̄/ annR̄ (r̄ )) < dim R̄}.
Lemma: The ring R̄/j(R̄) satisfies (R1 ). Its integral closure
(R̄/j(R̄))0 in its total ring of quotients is its S2 -ification.
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Torsion Divisor Classes
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Example
Lemma: The Ā-module R̄ ĀĀ is free of rank e and thus satisfies
(S2 ) over Ā.
Lemma: (R̄/j(R̄))0 satisfies (S2 ) as an Ā-module and as a ring.
Lemma: (R̄/j(R̄))0 ∼
= R̄ ĀĀ as Ā-isomorphisms.
= (R̄/j(R̄))ĀĀ ∼
Remark
Upshot: (R̄/j(R̄))0 is the S2 -ification of R̄.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
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Example
Lemma: The ring (R̄/j(R̄))0 is a normal domain.
Lemma: The map Ā → (R̄/j(R̄))0 is étale in codimension one.
Lemma: The extension Ā ,→ (R̄/j(R̄))0 is étale.
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Torsion Divisor Classes
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Results
Example
Conclusion of proof of Theorem Two.
Let S denote (R̄/j(R̄))0 . The extension Ā ,→ S is étale and local
and S is free over Ā of rank e. There are isomorphisms
Ā/m̄ ∼
= S/m̄S ∼
= (Ā/m̄)e .
It follows that e = 1. Contradiction.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
Theorem One Hypotheses:
(A, m) is an excellent, local, normal domain and I a prime ideal of
A with finite projective dimension such that:
(i) A = A/I is normal;
(ii) µ(I ) ≤ dim A − 2, where µ(I ) is the minimal number of
generators of I ; and
(iii) I is a complete intersection on the punctured spectrum of A,
i.e., for each prime ideal p 6= m, the localization Ip is either
equal to Ap or generated by a regular sequence in Ap .
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
Example
Let (A, m) be an excellent normal local integral domain, and let
the positive integer e represent a unit in A. Assume that A has an
A-regular sequence f1 , . . . , f6 ∈ m such that A/(f1 , . . . , f6 )A is a
normal domain. (Examples of such rings are constructed in S-W, S
2009.) In particular, we have 6 ≤ depth A ≤ dim A.
Localize at a minimal prime of the ideal J = (f1 , . . . , f6 )A, if
necessary, to assume that J is m-primary. It follows that A is
Cohen-Macaulay of dimension
6. Arrange the sequence f1 , . . . , f6 in
f1 f2 f3
a 2 × 3 matrix F = f4 f5 f6 , and consider the ideal I = I2 (F )
generated by the 2 × 2 minors d1 = f1 f5 − f2 f4 , d2 = f2 f6 − f3 f5 ,
d3 = f1 f6 − f3 f4 .
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
Thank You, eh.
Merci, eh.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
Let A be a ring, and let R be an A-algebra.
Definition
For a prime ideal P of R and p = P ∩ A, we say that P is
unramified over A if pRP = PRP and RP /pRP is a separable field
extension of Ap /pAp .
Definition
We say that R is unramified in codimension i (or more accurately,
in codimension less than or equal to i) over A if every prime ideal
of R with height less than or equal to i is unramified over A.
Definition
For R to be unramified over A means that (1) every prime ideal of
R is unramified over A, and (2) for every p ∈ Spec(A) there are
only finitely many prime ideals of R lying over p.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
• A satisfies the hypotheses of the first Theorem (excellent,
local, normal domain);
• I is a height-2 prime ideal of A of finite projective dimension
generated by 3 elements (so µ(I ) < dim A − 2) such that A/I is a
normal domain;
• I is a complete intersection on the punctured spectrum; i.e.,
for p ∈ Spec◦ (A) such that I ⊆ p, Ip is generated by an Ap -regular
sequence.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
(n1 )
• a = p1
(en1 )
• a(e) = p1
(nr )
∩ · · · ∩ pr
, pi height one primes in A
(enr )
∩ · · · ∩ pr
= αA and vpi (a) = eni
• ∃b ∈ A such that vpi (b) = ni for i = 1, . . . , r , else vq (b) ≥ 0
• Set u = b e /a ∈ K . Then vpi (u) = eni − eni = 0 and
vq (u) = vq (b e ) ≥ 0.
√
• Let e u be a fixed e-th root of u in an algebraic closure of K .
√
√
Then e a := b/ e u is an e-th root of a. Also, au = b e .
S. Spiroff, University of Mississippi
Torsion Divisor Classes
Background
Results
Example
The ring
#
"
#
√
√
"
√
e
e
e
2
e−1
u
u
u
R =A⊕ a·
⊕ a(2) · 2 ⊕ · · · ⊕ a(e−1) · e−1
b
b
b
√
√
is the integral closure of A in the field extension K [ e a] = K [ e u];
i.e., R is a domain.
The ring structure on R: for ai ∈ a(i) and aj ∈ a(j) , we have

√
e
!
! 
u i+j
√
√

e
e

i
j
if i + j < e
a
a
i j
u
u
b i+j
√
√
ai i
aj j
=
e
e

b
b
ai aj u i+j−e
u i+j

ai aj
=
if i + j ≥ e.
b i+j
a b i+j−e
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Example
Key References
1. P. Griffith, Some results in local rings on ramification in low
codimension, J. Algebra 137 (1991), no. 2, 473–490.
2. P. Griffith and D. Weston, Restrictions of torsion divisor
classes to hypersurfaces, J. Algebra 167 (1994), no. 2,
473–487.
3. M. Hochster and C. Huneke, Indecomposable canonical
modules and connectedness, in: Commutative Algebra:
Syzygies, Multiplicities, and Birational Algebra, South Hadley,
MA, 1992, AMS, 1994, pp. 197-208.
4. S. Sather-Wagstaff and S. Spiroff, Maps on divisor class
groups induced by ring homomorphisms of finite flat
dimension, J. Comm. Algebra 1 (2009), no. 3, 567–589.
S. Spiroff, University of Mississippi
Torsion Divisor Classes
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Example
Proposition
Let C ,→ D be a module finite extension of local normal domains
which is unramified in codimension one. If t is an element in the
maximal ideal of C such that C̄ = C /tC satisfies the condition
(R1 ), then D̄ = D/tD satisfies the condition (R1 ) as well, and
C̄ ,→ D̄ is unramified in codimension one.
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Example
Corollary
Under the same hypotheses as in the above Proposition, if the
rings are complete, then the integral closures (C̄ )0 and (D̄)0 are
local integral domains.
Corollary
Under the same hypotheses as in the previous corollary, the
extension (C̄ )0 → (D̄)0 is module finite and unramified in
codimension one.
S. Spiroff, University of Mississippi
Torsion Divisor Classes