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Duality and socle generators for residual intersections

David Eisenbud and Bernd Ulrich

Abstract

We prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.

Suppose that I is an ideal of codimension g in a Gorenstein ring, and JI is an ideal with s=g+t generators such that K:=J:I has codimension s. Let I¯ be the image of I in R¯:=R/K.

In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring R¯I¯I¯t+1 is a Gorenstein ring, and that the modules I¯u and I¯t+1-u are dual to one another via the multiplication pairing into I¯t+1ωR¯.

In the second part of the paper we study the analogue of residue theory, and prove that, when R/K is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of It+1/JItωR/K is generated by a Jacobian determinant.

Funding statement: This paper reports on work begun during the Commutative Algebra Program, 2012-13, at MSRI. We are grateful to MSRI for providing such an exciting environment, where a chance meeting led to the beginning of the work described here. Both authors are grateful to the National Science Foundation for partial support. The second author was also supported as a Fellow of the Simons Foundation.

A Differents and socles for Gorenstein rings

In this section we provide self-contained expositions of the classical results on differents and socles that we have used, mostly for complete intersections in characteristic 0. More generally than is usually stated, these yield a formula for the socle of a 0-dimensional Gorenstein ring. The results of this section are known, some in greater generality, but not easily available. Classic references are by Noether [26], Berger [3], Tate [25, Appendix], Scheja and Storch [28, 29], and Kunz [21, 22].

Let A be a Noetherian ring, let R be an A-algebra that is essentially of finite type, and write Re=RAR. Let 𝔻 be the kernel of the multiplication map μ:ReR, so that we have an exact sequence

0𝔻Re𝜇R0.

We want to compare three measures of ramification:

  1. The Kähler different𝔇K(R/A), introduced in a different case in Section 7, is defined to be Fitt0R(ΩR/A).

  2. The Noether different𝔇N(R/A) is defined to be μ(annRe𝔻).

  3. The Dedekind different𝔇D(R/A) is defined, for instance, when AR is a ring extension, A is a Noetherian normal domain, R is reduced and a finitely generated torsion free A-module, and R/A is separable. The complementary module (R/A) is the fractional R-ideal such that

    HomA(R,A)=(R/A)TrL/K,

    where K=Q(A) and L=Q(R) are the total rings of quotients of A and R, respectively. The Dedekind different is defined to be the inverse of the complementary module, 𝔇D(R/A)=(R/A)-1.

Because ΩR/A𝔻ReR and Fitt0Re(𝔻)annRe𝔻, it follows that

𝔇K(R/A)𝔇N(R/A).

The Dedekind different is an ideal because A is normal. We also have

𝔇N(R/A)𝔇D(R/A),

which implies that

𝔇N(R/A)HomA(R,A)RTrR/A.

For a short proof see [24, formula (3.3) proved in Lemma 3.4]. The last containment can be an equality even when the Dedekind different is not defined:

Theorem A.1.

Let A be a Noetherian ring and let R be an A-algebra that is finitely generated and free as an A-module. If HomA(R,A) is cyclic as an R-module, then

𝔇N(R/A)HomA(R,A)=RTrR/A.

Proof.

We will divide the proof into several parts:

Step (1). Because R is a free A-module, the natural map

Φ:RARHomA(HomA(R,A),R)

given by

st(φφ(s)t)

is an isomorphism of R-R-bimodules. The annihilator of 𝔻 is the unique largest R-R-submodule of RAR on which the left and the right R-module structures coincide, and the subset HomR(HomA(R,A),R) has the same property in HomA(HomA(R,A),R). It follows that Φ carries the annihilator of 𝔻 onto HomR(HomA(R,A),R).

Since R is a finitely generated free A-module and HomA(R,A) is cyclic as an R-module, we have HomA(R,A)R. It follows that annRe𝔻 is cyclic as an R-module.

Step (2). Let Γ be a generator of annRe𝔻. As Φ(Γ) generates HomR(HomA(R,A),R) and HomA(R,A)R, we see that Φ(Γ) is an R-isomorphism. Let

σ=Φ(Γ)-1(1)HomA(R,A).

It follows that σμ:RARA is a symmetric, nonsingular A-bilinear form.

Step (3). Let {vi} be an A-basis of R, and suppose that Γ=ivivi. We claim that σ(vivj)=δi,j – that is, {vi} is the dual basis of {vi} with respect to σμ. Indeed, since Φ(Γ) is R-linear, we have Φ(Γ)(rσ)=r for every rR. Thus, for each j,

vj=Φ(Γ)(vjσ)=Φ(ivivi)(vjσ)=i(vjσ)(vi)vi=iσ(vjvi)vi.

Since the vi form an A-basis, we see that σ(vivj)=δi,j as required.

Step (4). Finally, we claim that TrR/A=μ(Γ)σ. Let r be an element of R, regarded as an A-endomorphism of R by multiplication. We have

μ(Γ)σ(r)=σ(μ(Γ)r)=σ(ivirvi).

Since {vi} and {vi} are dual bases with respect to σμ, this sum is equal to TrR/A(r). Since 𝔇N(R/A)=μ(annRe𝔻)=Rμ(Γ), we see that

𝔇N(R/A)HomA(R,A)=𝔇N(R/A)σ=Rμ(Γ)σ=RTrR/A

as required. ∎

Theorem A.2.

In addition to the assumptions in the definition of the Dedekind different above, suppose that A is a regular local ring. If R is Gorenstein, then DD(R/A)=DN(R/A).

Proof.

We first verify that the assumptions of Theorem A.1 are satisfied. Recall that AR and R is a finitely generated A-module. For all maximal ideals 𝔪 of R, the rings R𝔪 have the same dimension as A, as can be seen for instance by tensoring with the completion of A, so that R splits as a product of local rings, and using the torsion freeness of R over A. Thus, since the rings R𝔪 are Cohen–Macaulay, it follows that R is a maximal Cohen–Macaulay A-module, hence a free A-module. Moreover, as the rings R𝔪 are Gorenstein and have the same dimension as A, the R-module HomA(R,A) is locally free of rank 1. Therefore we have HomA(R,A)R because R is semilocal.

Thus we may apply Theorem A.1. Since HomA(R,A)=(R/A)TrR/A by the definition of the complementary module, the theorem shows that 𝔇N(R/A)(R/A)=R, which gives 𝔇N(R/A)=(R/A)-1=𝔇D(R/A). ∎

Theorem A.3.

Let AR be a ring extension, where A is regular local and R is finitely generated and torsion free as an A-module. If R is locally a complete intersection, then

RA[x1,,xn]W/(F1,,Fn)

for some regular sequence F1,,Fn of length n in the polynomial ring A[x1,,xn] and some multiplicatively closed subset W. Write Δ for the image in R of the Jacobian determinant of F1,,Fn with respect to x1,,xn. One has

𝔇N(R/A)=𝔇K(R/A)=RΔ.

Proof.

Write RA[x1,,xn]/𝒥. As R is a finitely generated torsion free A-module, it follows as in the previous proof that every maximal ideal 𝔪 of R has the same codimension d:=dimA. Since 𝔪 must contain the maximal ideal of A, its preimage 𝔐 in A[x1,,xn] has codimension d+n. Hence the ideal 𝒥𝔐 has codimension n. Thus it is generated by n elements because R is locally a complete intersection. Write W for the complement in A[x1,,xn] of the union of the finitely many maximal ideals 𝔐. By basic element theory, 𝒥W is again generated by n elements F1,,Fn that can be chosen to form a regular sequence in A[x1,,xn].

To prove the claim about differents, first notice that R is a Cohen–Macaulay ring and hence a free A-module, as shown in the previous proof. As before, let 𝔻 be the kernel of the multiplication map μ:Re=RARR. The preimage 𝔻~ of 𝔻 in A[x1,,xn]WAR is the kernel of the natural map to R, so it is generated by a regular sequence G=G1,,Gn of length n. The ideal 𝔻~ also contains the sequence F1:=F11,,Fn1, which is still a regular sequence because R is flat over A.

Notice that 𝔻=𝔻~/(F1)=(G)/(F1). The preimage in A[x1,,xn]WAR of the annihilator of 𝔻 may thus be written as (F1):(G). This ideal quotient is generated by F1 and the determinant of any matrix Θ expressing the elements of F1 as linear combinations of the elements of G, see [35] or [6]. It follows that 𝔇N(R/A) is generated by the image in R of detΘ.

On the other hand, since G is a regular sequence and since

𝔻ReR=𝔻/𝔻2ΩR/A,

the image in R of Θ is a presentation matrix of ΩR/A. Thus the image of detΘ also generates the ideal 𝔇K(R/A)=RΔ. ∎

Corollary A.4.

If the assumptions in the definition of the Dedekind different and the hypotheses of Theorem A.3 are satisfied, then

(R/A)=𝔇K(R/A)-1=RΔ-1.

Proof.

One uses Theorem A.2, Theorem A.3, and the fact that the fractional ideal (R/A) is invertible, hence reflexive. ∎

Theorem A.5.

If R is a local Gorenstein algebra over a field k with dimkR finite and not divisible by the characteristic of k, then DN(R/k) is equal to the socle of R. If, moreover, R is a complete intersection, then the socle of R is generated by the Jacobian determinant.

Proof.

Since the trace of any nilpotent element is 0, it follows that the trace lies in the socle of Homk(R,k) and generates it if the characteristic of k does not divide dimkR. Thus Proposition A.1 implies that 𝔇N(R/k)Homk(R,k) is the socle of Homk(R,k). Therefore 𝔇N(R/k) is the socle of R since Homk(R,k)R as R-modules.

Finally, if R is a complete intersection, then 𝔇N(R/k)=𝔇K(R/k) is generated by the Jacobian determinant, by Theorem A.3. ∎

Proof of Theorem 7.1.

One implication is a special case of Theorem A.5. To prove the opposite implication, we must show that the Kähler different 𝔇K(R/k) is 0 when the ring R=k[[x1,,xd]]/(a1,,as) is not a 0-dimensional complete intersection.

First suppose that R is 0-dimensional and not a complete intersection. Replacing the ai by general linear combinations, we may assume that any d of the ai form a regular sequence. By the previous theorem, the Jacobian determinant of ai1,,aid generates the socle modulo (ai1,,aid) and is thus contained in (a1,,as) as required.

Now suppose that R is not 0-dimensional. To simplify the notation, set 𝔪=(x1,,xd) and 𝒥=(a1,,as) and suppose that s is the minimal number of generators of 𝒥. We may assume that R is not a complete intersection since otherwise 𝔇K(R/k)=0. For any sufficiently large integer n, the Artin–Rees Lemma and the Principal Ideal Theorem together imply that 𝒥+𝔪n requires at least s+dim𝒥 generators. Thus, R/𝔪nR is not a complete intersection.

We conclude from the 0-dimensional argument that, for any n0,

𝔇K((R/𝔪nR)/k)=0.

In particular, 𝔇K(R/k) is in 𝔪nR. By the Krull Intersection Theorem, 𝔇K(R/k)=0. ∎

Acknowledgements

The results of the present paper owe a great deal to the program Macaulay2 [36], which enabled us to determine the limits of validity of many of the assertions below; some of those computations are represented by examples in the current paper. We are also grateful to Craig Huneke, whose work on residual intersections inspired and guided the whole subject.

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Received: 2016-05-29
Revised: 2017-09-04
Published Online: 2018-01-24
Published in Print: 2019-11-01

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