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. S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results 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. BB The map Cl(A) ! Cl(B) is given by [a] 7! [(a ⊗A B) ]. • (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. S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Some Definitions A The dual of an A-module a is a := HomA(a; A). AA There is a map σ : a ! a , defined by σ(x) = 'x , where 'x (f ) := f (x), for f 2 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. AA • multiplication is defined by [a] · [b] = [(a ⊗A b) ]; • the identity element is [A]; −1 A • [a] = [a ] = HomA(a; A). S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results 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. 2 ∼ • Let B = C[X ;Y ;Z]=(XY −Z ). Then Cl(B) = Z=2Z. There is a map Cl(A) ! Cl(B) induced from A !! B. S. Spiroff, University of Mississippi 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. BB Recall: The map Cl(A) ! Cl(B) is given by [a] 7! [(a ⊗A B) ], 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. S. Spiroff, University of Mississippi Torsion Divisor Classes Background 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 2 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). 0 • They then show that the integral closure of R ⊗A (A=tA) is ´etaleover (A=tA)0 and conclude that e = 1. 0 • Subtle Detail: That the integral closure of R ⊗A (A=tA) 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 0 0 R ⊗A (A=tA) as the S2-ification of R ⊗A (A=tA) . 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. S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results 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 ≥ minf2; ht pg. 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 Background 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 Background Results 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: dim T • Hn (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. S. Spiroff, University of Mississippi Torsion Divisor Classes Background 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] 2= Ker(Cl(A) ! Cl(A)). S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Lemma: Let P 2 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 ´etale.In particular, the extension A ,! R is ´etalein codimension one. Definition We say that R is ´etalein 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 ´etaleover Ap. S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Lemma:The ring R is a complete local normal domain. Lemma: Set R¯ = R ⊗A A¯ = R=IR. The ring R¯ is equidimensional, complete, and local. In particular, Fact HH applies with T = R¯. Lemma: The extension A¯ ! R¯ is ´etalein codimension one. Moreover, the ring R¯ satisfies the Serre condition (R1). S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Lemma: For σ : R¯ ! R¯ A¯A¯ , we have Ker σ = j(R¯). Definition Denote by j(R¯) the largest ideal which is a submodule of R¯ of dimension smaller than dim R¯. Specifically, ¯ ¯ ¯ ¯ j(R) = f¯r 2 R : dim(R= annR¯ (¯r)) < dim Rg: Lemma: The ring R¯=j(R¯) satisfies (R1). Its integral closure 0 (R¯=j(R¯)) in its total ring of quotients is its S2-ification. S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Lemma: The A¯-module R¯ A¯A¯ is free of rank e and thus satisfies (S2) over A¯. 0 Lemma:(R¯=j(R¯)) satisfies (S2) as an A¯-module and as a ring. ¯ ¯ ¯ ¯ Lemma:(R¯=j(R¯))0 =∼ (R¯=j(R¯))AA =∼ R¯ AA as A¯-isomorphisms. Remark 0 Upshot: (R¯=j(R¯)) is the S2-ification of R¯. S. Spiroff, University of Mississippi Torsion Divisor Classes Background Results Example Lemma: The ring (R¯=j(R¯))0 is a normal domain.