# 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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