Maximal Cohen-Macaulay Modules and Gorenstein Algebras

Jan O. Kleppe and Chris Peterson

Abstract

Let B be a graded Cohen-Macaulay quotient of a Gorenstein ring, R. It is known that sections of the dual of the canonical module, K_{B}, can be used to construct Gorenstein quotients of R. The purpose of this paper is to place this method of construction into a broader context. If M is a maximal Cohen-Macaulay B-module whose sheafified top exterior power is a twist of K̃_{B} and if M satisfies certain additional homological conditions then regular sections of M^{∗} can again be used to construct Gorenstein quotients of R. On Cohen-Macaulay quotients, the normal module, the first Koszul homology module and several other associated modules all have sheafified top exterior power equal to a twist of K̃_{B}. If additional restrictions are placed on the Cohen-Macaulay quotients then these modules will satisfy the required additional homological conditions. This places the canonical module within a broad family of easily manipulated maximal Cohen-Macaulay modules whose sections can be used to construct Gorenstein quotients of R. AMS Subject Classification. 13C14, 13D02, 14C99, 14F05, 14M05. Keywords. Gorenstein algebra, Cohen-Macaulay algebra, maximal Cohen-Macaulay module, sections of modules, conormal module, Koszul homology, canonical module, strongly Cohen-Macaulay, licci

Contents

1 Introduction 1 1.1 Notation, definitions and background . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Sections of Modules over Cohen-Macaulay Quotients 4

3 Families of Maximal Cohen-Macaulay Modules 8 3.1 Canonical modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Normal modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 First Koszul homology modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Further applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1 Introduction

Let R be a Gorenstein ring. If B is a graded Cohen-Macaulay quotient of R, then degeneracy loci of sections of the dual of the canonical module of B in R define Gorenstein algebras. Several authors have made use of the Gorenstein algebras arising from such degeneracy loci in Gorenstein liaison theory as well as in several other contexts. If M is a free Rmodule of rank r then the degeneracy loci of regular sections of M^{∗} again define Gorenstein algebras (in fact complete intersection algebras). Such degeneracy loci as a source of complete intersections is commonplace in complete intersection liaison theory and other areas. A large part of the success of these methods is due to the fact that these modules can be constructed in abundance . In addition, these modules and their sections are particularly easy to manipulate. For instance, by utilizing tensor operations, it is relatively easy to find sections whose degeneracy loci lie within prespecified ideals. There are many other easily manipulated, readily available modules associated to Cohen-Macaulay quotients of a Gorenstein ring. Several of these modules have the same property with respect to degeneracy loci of their regular sections (i.e they can be used to construct Gorenstein algebras). The present paper is an attempt to identify these modules and place them in the more general framework of maximal Cohen-Macaulay modules with “good sections". Section 1 of the paper provides a brief glossary of definitions and recurring notation. Section 2 consists of a theorem on maximal Cohen-Macaulay modules and its proof. Section 3 is filled with several examples and applications identifying important classes of modules which fit into our framework. The examples deal primarily with the conormal module, the first Koszul homology module and several other related modules.

1.1 Notation, definitions and background

Throughout this paper we will use boldface type like K_{B}, B, R, M, I to denote algebraic objects such as modules, ideals and algebras. R will be a graded Gorenstein quotient of a polynomial ring over an algebraically closed field, k, with the usual grading. B will be a graded Cohen-Macaulay quotient of R, F_{i} will be a graded free R-module and I, J, J′ will be homogeneous ideals in R. M will denote a finitely generated graded module over B and A will be a graded quotient of B. K_{B} = Ext_{R}^{c}(B, R(e)) will denote the canonical module of B in R (where c denotes the codimension of B in R and R(e) denotes the canonical module of R). N_{B} denotes the normal module of B in R. H_{i} = H_{i}(I) denotes the i^{th} Koszul homology module of the ideal I in R. If B = R/I then the B-dual modules, Hom_{B}(−, B), to M, N_{B}, H_{i} will be denoted by M^{∗}, N_{B}^{∗}, H_{i}^{∗} resp. We denote by F_{i}^{∗} the R-dual module Hom_{R}(F_{i}, R). The sheafification of a module will be denoted with a tilde overbar such as K̃_{B}, M̃, Ñ_{B} and H̃_{i}. Sheaves will be given a calligraphic type such as M, Z, O_{S} etc.

We will make reference to the notions of a quotient, R/J, being licci or strongly Cohen-Macaulay. Definitions of these terms are given below.

Definition 1. Two quotients of R, R/J and R/J′ (equidimensional and without embedded components), are said to be geometrically complete intersection linked (resp. geometrically Gorenstein linked) if J and J′ do not share common components and if R/(J ∩ J′) is a complete intersection (resp. Gorenstein). Two quotients of R, R/J and R/J′, are said to be algebraically complete intersection linked (resp. algebraically Gorenstein linked) if there exists a complete intersection (resp. Gorenstein) ideal K (with K ⊆ J ∩ J′) such that J = K : J′ and J′ = K : J. The relationship of being linked generates an equivalence relation. The resulting equivalence classes are called linkage classes. For more details see [19].

Definition 2. The Cohen-Macaulay quotient B = R/I is said to be licci if it is in the linkage class of a complete intersection.

Definition 3. The quotient B = R/I is said to be strongly Cohen-Macaulay if all the non trivial Koszul homology modules H_{1}(I), H_{2}(I), … of I are Cohen-Macaulay.

We will need the following definition, statements and theorems. For a proof of Theorem 5, see [6], Theorem 21.21 and Proposition 21.12. For a proof of Theorem 6, see [14]. For a proof of Theorem 7, see [4], Theorem 4.1, cf. [17], Corollary 6.5 for the case i = c − 1.

Definition 4. If M is a finitely generated B-module then

M is a maximal Cohen−Macaulay B−module ⟺ depth M = dim B.  (1)

Here, depth_{J} M is the length of a maximal M-sequence in J and depth M = depth_{𝔪} M where 𝔪 is the irrelevant maximal ideal. If H_{J}^{i}(−) is the right derived functor of Γ_{J}(−), sections with support in Proj(R/J), then for M ≠ 0

M is a maximal Cohen−Macaulay B−module ⟺ H_{𝔪}^{i}(M) = 0 for i < dim B. (2)

The previous fact is useful to see when one module in a short exact sequence is maximal Cohen-Macaulay provided the other modules of the short exact sequence are maximal Cohen-Macaulay (or have sufficient depth) (cf. [6], Corollary 18.6 for another description). For maximal Cohen-Macaulay B-modules, we have

depth_{J} M = depth_{J} B.  (3)

If S = Proj(B) and Z is (say closed) in S and U = S−Z, we let H_{∗}^{0}(U, M̃) = ⊕_{v} H^{0}(U, M̃(v)). If depth_{I(Z)} M ≥ r, then H_{I(Z)}^{i}(M) = 0 for i < r, and conversely [11]. This implies

H_{∗}^{i}(U, M̃) = 0 for 1 ≤ i ≤ r − 2.  (4)

Moreover, M ≃ H_{∗}^{0}(U, M̃) if r ≥ 2. Since B is Cohen-Macaulay, we remark [6]

depth_{J} B = dim B − dim B/J.  (5)

Theorem 5. If M is a maximal Cohen-Macaulay B-module then Hom_{B}(M, K_{B}) is a maximal Cohen-Macaulay B-module and Ext_{B}^{i}(M, K_{B}) = 0 for i > 0.

Theorem 6. If B = R/I is licci, then B is strongly Cohen-Macaulay.

Theorem 7. If B = R/I is licci of codimension c, then every nontrivial Tor_{i}^{R}(B, K_{B}) is a maximal Cohen-Macaulay B-module for 0 ≤ i ≤ c.

If B = R/I, if S = Proj(B) is Gorenstein on U = S − Z and if depth_{I(Z)} B ≥ 2 (resp. depth_{I(Z)} B ≥ 1), then

Ext_{B}^{i}(I/I^{2}, B) ≃ Ext_{B}^{i}(I/I^{2} ⊗ K_{B}, K_{B}) for i ≤ 2 (resp. i ≤ 1)  (6)

(this follows from a spectral sequence argument [13]). Since Tor_{1}^{R}(B, K_{B}) ≃ I/I^{2} ⊗ K_{B}, it follows from (6), Theorem 5 and Theorem 7 that

Ext_{B}^{i}(I/I^{2}, B) = 0 for 1 ≤ i ≤ 2 (resp. i = 1)  (7)

provided B is licci and depth_{I(Z)} B ≥ 2 (resp depth_{I(Z)} B ≥ 1), cf [17], Proposition 6.15. Let F_{1} ↠ I be surjective. Then we have a complex

0 → H_{1} →^{α} F_{1} ⊗_{R} B → I/I^{2} → 0.  (8)

I is called syzygetic if α is injective (or equivalently if (8) is exact). If B is strongly Cohen-Macaulay (or much weaker, H_{1}(I) is B-torsion free) and generically a complete intersection then I is syzygetic, cf. [15].

2 Sections of Modules over Cohen-Macaulay Quotients

In this section we prove a general theorem on sections of maximal Cohen-Macaulay modules. The theorem will be applied throughout the paper.

Theorem 8. Let R be a graded Gorenstein quotient of a polynomial ring and assume R has canonical module R(e). Let B be a codimension c graded Cohen-Macaulay quotient of R. Let K_{B} = Ext_{R}^{c}(B, R(e)) be the canonical module of B. Let M be a finitely generated maximal Cohen-Macaulay B-module. Let S = Proj B, let Z be a closed scheme such that dim(B) − dim(B/I(Z)) ≥ max(r, 2) and let U = S − Z. Let M_{i} = H_{∗}^{0}(U, ∧^{i}M̃) for i ≥ 0. Suppose the following conditions are satisfied:

1. M̃|_{U} is locally free (of rank r)

2. ∧^{r}M̃|_{U} ≃ K̃_{B}(t)|_{U} (for some integer t)

3. M_{i} is a finitely generated maximal Cohen-Macaulay B-module for 2 ≤ i ≤ r/2

then we can make the following conclusions:

1. M ≃ M_{1}, M_{2}, …, M_{r} are maximal Cohen-Macaulay B-modules

2. M_{r−i} ≃ Hom_{B}(M_{i}, K_{B})(t) for 0 ≤ i ≤ r

3. If s is an integer, then any regular section, σ ∈ H^{0}(U, M̃^{∗}(s)), defines a Gorenstein quotient R ↠ A of codimension r + c given by the exact sequence

0 → M_{r}(−rs) → M_{r−1}((1 − r)s) → ⋯ → M_{2}(−2s) → M(−s) →^{σ^{∗}} B → A → 0. (9)

Proof. Conclusion 2 follows from the isomorphisms

M_{r−i} := H_{∗}^{0}(U, ∧^{r−i}M̃) ≃ H_{∗}^{0}(U, (∧^{i}M̃)^{∗} ⊗ ∧^{r}M̃) ≃ H_{∗}^{0}(U, (∧^{i}M̃)^{∗} ⊗ K̃_{B}(t)) ≃
H_{∗}^{0}(U, { \atop Hom_{B}(M_{i}, K_{B}(t)))} ≃ Hom_{B}(M_{i}, K_{B}(t)).

Conclusion 1 follows if it is noted that Hom_{B}(M_{i}, K_{B}) is a maximal Cohen-Macaulay module if M_{i} is a maximal Cohen-Macaulay module (cf. Theorem 5).

To prove conclusion 3, we start with a regular section σ ∈ H^{0}(U, M̃^{∗}(s)) i.e. a map

σ : O_{U} → M̃^{∗}(s)|_{U}.

If we dualize and take global sections then we get

M(−s) = H_{∗}^{0}(U, M̃^{∗∗})(−s) → H_{∗}^{0}(U, O_{S}) = B.

We define A to be the cokernel of this map. Since M̃|_{U} is locally free and since σ is a regular section, the associated Koszul complex

0 → ∧^{r}M̃(−rs)|_{U} → ∧^{r−1}M̃(−(r − 1)s)|_{U} → ⋯ → ∧^{2}M̃(−2s)|_{U} → M̃(−s)|_{U} →^{σ^{∗}} B̃|_{U} (10)

is exact. We claim that the global section functor, H_{∗}^{0}(U, −) is exact on (10). To see this, define

Z_{i} = ker(∧^{r−i}M̃(−(r − i)s)|_{U} → ∧^{r−i−1}M̃(−(r − i − 1)s)|_{U}) for 1 ≤ i ≤ r − 1.

In particular, Z_{1} = ∧^{r}M̃(−rs)|_{U} ≃ K̃_{B}(t − rs)|_{U}. Split (10) into short exact sequences and apply the global section functor to each of them. It suffices to show that

H_{∗}^{1}(U, Z_{i}) = 0 for 1 ≤ i ≤ r − 2.

To use an induction on i, we instead prove the more general statement

H_{∗}^{j}(U, Z_{i}) = 0 for 1 ≤ i ≤ r − 2 and 1 ≤ j ≤ r − i − 1.  (11)

Since Z_{1} = K̃_{B}(t − rs)|_{U} and since depth_{I(Z)} K_{B} = depth_{I(Z)} B ≥ r, we obtain (11) for i = 1 if we use the fact

H_{I(Z)}^{j+1}(K_{B}) = 0 for j + 1 < r.

To use induction, we now assume that (11) holds for i = k where 1 ≤ k < r − 2. Since

0 → Z_{k} → ∧^{r−k}M̃((−r + k)s)|_{U} → Z_{k+1} → 0  (12)

is exact and since depth_{I(Z)} M_{r−k} ≥ r we get

H_{∗}^{j}(U, Z_{k+1}) ≃ H_{∗}^{j+1}(U, Z_{k}) for 1 ≤ j ≤ r − 3

and we conclude by the induction assumption. Thus we have proven the exactness of

0 → M_{r}(−rs) → M_{r−1}((1 − r)s) → ⋯ → M_{2}(−2s) → M(−s) →^{σ^{∗}} B → A → 0. (13)

To see that A is Gorenstein, we first prove that A is Cohen-Macaulay by showing that H_{𝔪}^{i}(A) = 0 for i < dim A. Note that since σ is a regular section on U and dim B/I(Z) ≤ dim B − r, it is clear that dim A = dim B − r. Let Z_{i} = H_{∗}^{0}(U, Z_{i}) and Z_{r} = ker(B → A). Then the global version of (12) yields

H_{𝔪}^{i}(A) ≃ H_{𝔪}^{i+1}(Z_{r}) ≃ H_{𝔪}^{i+2}(Z_{r−1}) ≃ ⋯ ≃ H_{𝔪}^{i+r}(Z_{1})

since i < dim A implies i + r < dim B = depth M_{j}. Since Z_{1} ≃ K_{B}(t − rs), we also have H_{𝔪}^{i+r}(Z_{1}) = 0 and A is Cohen-Macaulay. It remains to prove

Ext_{B}^{r}(A, K_{B}) ≃ A(ν)

for some integer ν. The isomorphisms in the proof of Conclusion 2 of the theorem are given by the determinant and the section, σ. Since Z_{1} ≃ M_{r}(−rs) and Ext^{1}(M_{r−1}, K_{B}) = 0 (cf. Theorem 5 for the vanishing) we have in particular a commutative diagram

Hom_{B}(M_{r−1}((−r + 1)s), K_{B}(t)) → Hom_{B}(Z_{1}, K_{B}(t)) → Ext_{B}^{1}(Z_{2}, K_{B}(t)) → 0
↓  ∘  ↓
M_{1}((r − 1)s)  →  B(rs)

where the down arrows are isomorphisms. The vanishing of Theorem 5 combined with the global version of (12) imply

Ext_{B}^{r}(A, K_{B}) ≃ Ext_{B}^{r−1}(Z_{r}, K_{B}) ≃ Ext_{B}^{r−2}(Z_{r−1}, K_{B}) ≃ ⋯ ≃ Ext_{B}^{1}(Z_{2}, K_{B}).

Hence we get

Ext_{B}^{r}(A, K_{B}) ≃ A(rs − t)

as required.

Remark 9. Conclusion 2 of Theorem 8, slightly extended, imposes a self duality on the long exact sequence (9) of Theorem 8 and we get a self duality on M_{r/2} if r is even. Note that if M̃ is locally free (or weaker, Ext^{i}(M̃_{j}, K̃_{B}) = 0 for i > 0, cf. Theorem 5) on S, then Serre Duality and ∧^{r−i}M̃ ≃ (∧^{i}M̃)^{∗} ⊗ K̃_{B}(t) yields

H_{∗}^{j}(S, M̃_{r−i}) ≃ H_{∗}^{dim S−j}(S, M̃_{i})^{∨}(t).  (14)

This explains (and essentially reproves) conclusion 1 of Theorem 8. More importantly, if r is even, we see immediately from (14) that we can weaken the Cohen-Macaulay assumption on M_{r/2} to

depth_{𝔪} M_{r/2} ≥ \frac{1}{2} dim B + 1  (15)

provided that M̃ is locally free on S.

Utilizing Serre duality carefully, we can weaken condition 3 of Theorem 8 by replacing the maximal Cohen-Macaulay assumption on M_{r/2} with

depth_{𝔪} M_{r/2} ≥ \frac{1}{2} dim B + 1 + \frac{1}{2} dim B/I(Z)

provided M̃ is locally free on U = S − Z. Indeed, to conclude as in Theorem 8, it suffices to prove that this implies that M_{r/2} is a maximal Cohen-Macaulay B-module. Since

depth_{𝔪} M_{r/2} ≥ \frac{1}{2} dim B + 1 + \frac{1}{2} dim B/I(Z) = dim B + 1 − \frac{1}{2} depth_{I(Z)} B,

it suffices to prove H_{∗}^{dim S−j}(S, M̃_{r/2}) = 0 for dim S − j ≥ dim B − \frac{1}{2} depth_{I(Z)} B (j > 0). This is equivalent to proving

H_{∗}^{j}(U, M̃_{r/2}) = 0 for j + 1 ≤ \frac{1}{2} depth_{I(Z)} B

since depth_{I(Z)} K_{B} ≥ j + 2 and since Ext_{O_{S}}^{i}(M̃_{r/2}, K̃_{B}(v)) is dual to H^{dim S−j}(S, M̃_{r/2}(−v)) implies

Ext_{O_{S}}^{j}(M̃_{r/2}, K̃_{B}(v)) ≃ Ext_{O_{U}}^{j} (M̃_{r/2}|_{U}, K̃_{B}(v)|_{U}) ≃ H^{j}(U, Hom(M̃_{r/2}, K̃_{B}(v)))

by [11], exp. VI. Now, using [22], Lemma 1.8a), it is easy to see that depth_{𝔪} M_{r/2} ≥ dim B + 1 − \frac{1}{2} depth_{I(Z)} B implies

depth_{I(Z)} M_{r/2} ≥ depth_{I(Z)} B + 1 − \frac{1}{2} depth_{I(Z)} B = 1 + \frac{1}{2} depth_{I(Z)} B

and we conclude by (4).

Remark 10. If we, in addition to the assumptions of the theorem, suppose that M_{i} has finite projective dimension for 0 ≤ i ≤ r/2 then all M_{i} have finite projective dimension over R. We can then find an R-free resolution of A. Indeed, we have Ext_{R}^{i}(M_{i}, R) = 0 for i ≠ c due to the fact that M_{i} is a maximal Cohen-Macaulay B-module and due to

Ext_{R}^{c}(M_{i}, R(e)) ≃ Hom_{B}(M_{i}, K_{B}) ≃ M_{r−i}(−t).

Applying (−)^{∗} = Hom_{R}(−, R) to the R-free resolution

0 → P_{c} → P_{c−1} → ⋯ → P_{1} → P_{0} → M_{i} → 0

we get the resolution

0 → P_{0}^{∗} → P_{1}^{∗} → ⋯ → P_{c−1}^{∗} → P_{c}^{∗} → Ext_{R}^{c}(M_{i}, R) ≃ M_{r−i}(−e − t) → 0.

We conclude by applying an iterated mapping cone construction to (9). Note that the Cohen-Macaulay type of A is the same as the Cohen-Macaulay type of M_{r}(−rs) = K_{B}(t − rs). Hence A is Gorenstein since the Cohen-Macaulay type of K_{B} is 1.

Remark 11. In order to further generalize Theorem 8, we can look precisely at which depth_{𝔪}- condition is needed to prove the exactness of (9) as well as the Gorenstein property of A. This leads us to replace assumption 3 of the Theorem with the apparently weaker assumption

3′. depth_{𝔪} M_{r−i} ≥ dim B − i + 1 for 1 ≤ i ≤ r − 1.

A careful use of Serre Duality, along the lines of Remark 9, gives the maximal Cohen-Macaulay property for all modules M_{i} under the assumption of 3′. In a forthcoming paper, where we consider sections of a certain class of sheaves, we have been able to find conditions which guarantees that the saturation, Ā, of A is Gorenstein (so Proj A is arithmetically Gorenstein). Under additional assumptions (but still weaker than the Cohen-Macaulay assumption) we have A is Gorenstein as well.

3 Families of Maximal Cohen-Macaulay Modules

In this section, we apply Theorem 8 to several families of maximal Cohen-Macaulay modules. There is a unique module, up to twist, with rank r = 1 which satisfies the conditions of Theorem 8 (namely K_{B}). In contrast, there are many modules which have rank r ≥ 2 on an open set, U, and which satisfy the conditions of Theorem 8. If M is to be a candidate for such a module, it must certainly satisfy

⋀^{r} M̃|_{U} ≃ K̃_{B}(t)|_{U}.  (16)

With this in mind, we concentrate on the module of Kaehler differentials Ω_{B}, the normal module N_{B} and the first Koszul homology module H_{1} (where the latter is built on some minimal set of generators of I = ker(R → B)). It is well known that these modules satisfy the isomorphism given in (16) on some naturally defined U. For instance, for N_{B} and H_{1} we let U be the open set of S = Proj(B) ↪ P = Proj(R) where U ↪ P is a local complete intersection. Then N_{B} satisfies (16) on U by [12], III, Theorem 7.11. Moreover, since the sheafification of (8) is exact on U, the top exterior powers of (I/I^{2})^{∗} = N_{B} and H_{1} therefore correspond on U, up to twist (F_{1} is a free R-module). As a result, H_{1} satisfies (16).

However, Theorem 8 does not apply to Ω_{B}. Indeed consulting [5], we see that while many of the cohomology groups H_{∗}^{p}(S, ∧^{q}Ω̃_{B}) vanish (e.g. for p ≠ q if S is smooth), not every intermediate cohomology group vanishes as the maximal Cohen-Macaulay property of Theorem 8 requires. For the modules N_{B} and H_{1}, we shall see that there are large classes of Cohen-Macaulay quotients over which all of the conditions of Theorem 8 are satisfied.

3.1 Canonical modules

In this subsection we briefly discuss a consequence of Theorem 8 with respect to sections of K_{B}^{∗}(s). If S is Gorenstein outside a closed set Z of codimension at least 2 in S, then we easily see that the canonical module satisfies the conditions of Theorem 8. The Gorenstein quotients constructed via these modules are given by an exact sequence of the form

0 → K_{B}(−s) → B → A → 0.  (17)

Any quotient, R/(J+J′), obtained from two geometrically Gorenstein linked Cohen-Macaulay quotients, R/J and R/J′, is an example of a Gorenstein quotient, A, as given by exact sequence (17). Indeed, since they are geometrically Gorenstein linked, D := R/J ∩ J′ is Gorenstein by definition. Hence K_{D} ≃ D(s) and Hom_{D}(R/J, D) ≃ K_{(R/J)}(−s). Considering the exact sequence

0 → (J + J′)/J → R/J → R/(J + J′) → 0  (18)

and observing (J + J′)/J ≃ J′/J ∩ J′ ≃ Hom_{D}(R/J, D) ≃ K_{(R/J)}(−s), we can transform sequence (18) into the form of sequence (17) by letting B = R/J. If we have an R-free resolution of B then we easily get an R-free resolution of A by applying the mapping cone construction to sequence (17). The construction of Gorenstein algebras in this manner has been utilized by several authors, cf. [9], [10], [17], [18].

Example 12. Let R = k[a, b, c, d, e, f] and let I = (bc − ad, be − af, de − cf) be the ideal of 2 × 2 minors of the 2 × 3 matrix of indeterminants of R. In this case, B = R/I is Cohen-Macaulay of codimension 2, S = Proj B is a local complete intersection (so we can take Z = ∅) and the minimal R-free resolution of B is

0 → R(−3)^{2} → R(−2)^{3} → R → B → 0.

A regular section, σ ∈ H^{0}(S, K̃_{B}^{∗}(−1)), defines a Gorenstein algebra, A, of codimension 3 with R-free resolution

0 → R(−5) → R(−3)^{5} → R(−2)^{5} → R → A → 0.

An example of such a Gorenstein algebra is

A = R/(bc − ad, be − af, de − cf, a^{2} + ad + ce, ab + bd + cf).

3.2 Normal modules

In this subsection, we apply Theorem 8 to the case where M = N_{B} and B is a codimension c licci quotient of R. Our result overlaps work of Ellingsrud-Peskine [7], which treats the Artin Gorenstein case with c = 2 more generally.

Proposition 13. Let R be a graded Gorenstein quotient of a polynomial ring and assume R has canonical module R(e). Let B be a codimension c graded licci quotient of R (for conclusions 3 and 4, we will assume B has finite projective dimension over R). Let K_{B} = Ext_{R}^{c}(B, R(e)) be the canonical module of B and let N_{B} be the normal module of B. Let S = Proj(B), let Z be a closed scheme such that dim(B) − dim(B/I(Z)) ≥ max(c, 2) and let U = S − Z. If U is a local complete intersection in Proj(R) then

1. Every regular section, σ ∈ H^{0}(U, Ñ_{B}^{∗}(s)), defines a Gorenstein quotient R ↠ A of codimension 2c.

2. H_{∗}^{0}(U, ∧^{i}Ñ_{B}) is a maximal Cohen-Macaulay B-module for every 0 ≤ i ≤ c

3. H_{∗}^{0}(U, ∧^{i}Ñ_{B}) ≃ Ext_{R}^{i}(B, B) ≃ Tor_{c−i}^{R}(B, K_{B})(−e)

4. There is an exact sequence

0 → K_{B}(−e − cs) → Ext_{R}^{c−1}(B, B)((1 − c)s) → ⋯ → Ext_{R}^{i}(B, B)(−is)

→ ⋯ → Ext_{R}^{1}(B, B)(−s) ≃ N_{B}(−s) →^{σ^{∗}} B → A → 0

Remark 14. A consequence of the maximal Cohen-Macaulay property of N_{i} := H_{∗}^{0}(U, ∧^{i}Ñ_{B}) of Proposition 13 is that for every i ≥ 0, we have

H_{∗}^{p}(U, ∧^{i}Ñ_{B}|_{U}) = 0 for 1 ≤ p ≤ c − 2
H_{∗}^{p}(S, Ñ_{i}) = 0 for 1 ≤ p ≤ dim B − 2.

Proof. If c = 1, then Proposition 13 is trivial to prove. Indeed R ↠ A is in this case a complete intersection. Suppose c ≥ 2. We claim that

H_{∗}^{0}(U, ∧^{i}Ñ_{B}) ≃ Ext_{R}^{i}(B, B) ≃ Tor_{c−i}^{R}(B, K_{B})(−e).

To show it, we consider a minimal R-free resolution

0 → F_{c} → F_{c−1} → ⋯ → F_{1} → F_{0} = R → B → 0.  (19)

If we apply (−)^{∗} = Hom_{R}(−, R), we get the exact sequence

0 → F_{0}^{∗} → F_{1}^{∗} → ⋯ → F_{c−1}^{∗} → F_{c}^{∗} → K_{B}(−e) → 0.  (20)

It follows that the homology groups of the complex

0 → F_{0}^{∗} ⊗_{R} B → F_{1}^{∗} ⊗_{R} B → ⋯ → F_{c−1}^{∗} ⊗_{R} B → F_{c}^{∗} ⊗_{R} B → 0  (21)

are Tor_{i}^{R}(B, K_{B})(−e). On the other hand since F_{i}^{∗} ⊗_{R} B ≃ Hom_{R}(F_{i}, B), we get the same complex (21) by applying Hom_{R}(−, B) to (19). The homology groups of (21) are therefore also given by Ext_{R}^{c−i}(B, B). This proves the right isomorphism of the claim.

To see the other isomorphism, we remark that

{ \atop Tor_{c−i}^{R}(B, K_{B})}|_{U} ≃ (Tor_{c−i}^{R}(B, B) ⊗_{B} K_{B})|_{U} ≃ (∧^{c−i}Ñ_{B}^{∗} ⊗_{O_{S}} K̃_{B})|_{U}  (22)

since ∧^{i}Ñ_{B}^{∗}|_{U} is locally free, cf. [1], lem.17, p.169. Moreover, ∧^{c}Ñ_{B}|_{U} ≃ K̃_{B}(−e)|_{U} by [12], Chapter III, Theorem 7.11. Combining with

∧^{i}Ñ_{B}|_{U} ≃ ((∧^{c−i}Ñ_{B})^{∗} ⊗_{O_{S}} ∧^{c}Ñ_{B})|_{U} ≃ (∧^{c−i}Ñ_{B}^{∗} ⊗_{O_{S}} K̃_{B}(−e))|_{U}

and the fact that all nontrivial Tor_{c−i}^{R}(B, K_{B}) are maximal Cohen-Macaulay B-modules (by Theorem 7), we get

Tor_{c−i}^{R}(B, K_{B})(−e) ≃ H_{∗}^{0}(U, { \atop Tor_{c−i}^{R}(B, K_{B})(−e))} ≃ H_{∗}^{0}(U, ∧^{i}Ñ_{B})  (23)

Indeed since the depth_{I(Z)} of a maximal Cohen-Macaulay B-module is equal to depth_{I(Z)} B (which is ≥ 2 by assumption), we get the left isomorphism in (23), and the claim is proved.

Moreover (23) shows that H_{∗}^{0}(U, ∧^{i}Ñ_{B}) is a maximal Cohen-Macaulay module for every 0 ≤ i ≤ c since Tor_{c−i}^{R}(B, K_{B}) is a maximal Cohen-Macaulay module for every 0 ≤ i ≤ c. Finally once having proved this maximal Cohen-Macaulay property and the isomorphism ∧^{c}Ñ_{B}|_{U} ≃ K̃_{B}(−e)|_{U}, Proposition 13 follows directly from Theorem 8, and we are done.

Remark 15. We see from the proof of Proposition 13 that we can weaken the licci assumption slightly. Instead of assuming that B is licci, we can instead assume that Tor_{i}^{R}(B, K_{B}) is a maximal Cohen-Macaulay B-module for each 0 ≤ i ≤ c. This extends Proposition 13 because it is known that there are nice classes of non-licci algebras B for which every nontrivial Tor_{i}^{R}(B, K_{B}) is a maximal Cohen-Macaulay module [15], [23].

Remark 16. Suppose that the codimension of B is 2 or that B is Gorenstein with codimension 3. In this case, we can explicitly write down an R-free resolution of the Gorenstein algebra A of Proposition 13 in terms of the free R-modules, F_{i}, in the resolution of I = ker(R → B) (as given in exact sequence (19)).

If the codimension of B is 2 then (19) is the short exact sequence

0 → F_{2} → F_{1} → I → 0.  (24)

To determine an R-free resolution of A we apply Hom_{R}(−, I) to (24) (and use N_{B} ≃ Ext_{R}^{1}(I, I)) to get

0 → R → F_{1}^{∗} ⊗_{R} I → F_{2}^{∗} ⊗_{R} I → N_{B} → 0.  (25)

The map R → F_{1}^{∗} ⊗ I in (25) lifts to the map R → F_{1}^{∗} ⊗ F_{1} ≃ Hom(F_{1}, F_{1}) given by 1 → id_{F_{1}} and further to the map Δ : R → (F_{1}^{∗} ⊗ F_{1}) ⊕ (F_{2}^{∗} ⊗ F_{2}) defined by Δ(1) = (id_{F_{1}}, −id_{F_{2}}). Using this lifting, the short exact sequences we get by tensoring (24) with F_{1}^{∗} and F_{2}^{∗} and the mapping cone construction, we get a minimal R-free resolution of N_{B}

0 → F_{1}^{∗} ⊗_{R} F_{2} → ((F_{1}^{∗} ⊗_{R} F_{1}) ⊕ (F_{2}^{∗} ⊗_{R} F_{2}))/Δ(R) → F_{2}^{∗} ⊗_{R} F_{1} → N_{B} → 0.  (26)

Finally, we use the R-free resolutions of B and K_{B}(−e) (as given in (19) and (20)) together with the exact sequences

0 → K_{B}(−e − 2s) → N_{B}(−s) → I_{B/A} → 0

and

0 → I_{B/A} → B → A → 0,

and apply the mapping cone construction to yield the R-free resolution

0 → R(−2s) → F_{1}^{∗}(−2s) ⊕ (F_{1}^{∗} ⊗_{R} F_{2})(−s) →
F_{2}^{∗}(−2s) ⊕ (((F_{1}^{∗} ⊗_{R} F_{1}) ⊕ (F_{2}^{∗} ⊗_{R} F_{2}))/Δ(R))(−s) ⊕ F_{2} →
(F_{2}^{∗} ⊗_{R} F_{1})(−s) ⊕ F_{1} → R → A → 0.

The resolution is minimal for most values of s and for all values of s ≫ 0.

If the codimension of B is 3 and if B is Gorenstein, then (19) is the exact sequence

0 → R(−f) = F_{3} → F_{2} → F_{1} → I → 0.

In this case, N_{B} ≃ (∧^{2}I)(f) provided that chark ≠ 2, cf. [16], Proposition 2.4. Twisting the exact sequence

0 → F_{2} ⊗ F_{3} → (F_{1} ⊗ F_{3}) ⊕ S_{2}F_{2} → F_{1} ⊗ F_{2} → ∧^{2}F_{1} → ∧^{2}I → 0  (27)

of Lebelt-Weyman by f, we obtain at once a minimal R-free resolution of N_{B}. Recalling that the canonical module, K_{B}, satisfies K_{B}(−e) ≃ B(f) , cf. (19) and (20), and that the canonical module of R is R(e), we get

Ext_{R}^{3}(N_{B}, R) ≃ Ext_{R}^{3}(N_{B}, R(e))(−e) ≃ Hom_{B}(N_{B}, K_{B})(−e) ≃ (I/I^{2})(f).  (28)

We have Ext_{R}^{i}(N_{B}, R) = 0 for i ≠ 3 since N_{B} is a maximal Cohen-Macaulay module. We have F_{3−i}(f)^{∗} ≃ F_{i} for 0 ≤ i ≤ 3 since B is Gorenstein. Noting these isomorphisms, the R-dual sequence of (27) yields the R-free resolution

0 → ∧^{2}F_{2} → (F_{1} ⊗ F_{2}) → S_{2}F_{1} ⊕ F_{2} → F_{1} → I/I^{2} → 0.  (29)

Splitting the exact sequence

0 → B(f − 3s) → (I/I^{2})(f − 2s) → N_{B}(−s) → I_{B/A} → 0

of Proposition 13 into two short exact sequences and using the mapping cone construction, the resolution of I/I^{2} and the resolution of N_{B}, we obtain the following R-free resolution

0 → R(−3s) → F_{2}(f − 3s) ⊕ ∧^{2}F_{2}(f − 2s) →
F_{1}(f − 3s) ⊕ (F_{1} ⊗ F_{2})(f − 2s) ⊕ F_{2}(−s) →
R(f − 3s) ⊕ S_{2}F_{1}(f − 2s) ⊕ F_{2}(f − 2s) ⊕ S_{2}F_{2}(f − s) ⊕ F_{1}(−s) ⊕ R(−f)
→ F_{1}(f − 2s) ⊕ (F_{1} ⊗ F_{2})(f − s) ⊕ F_{2} → ∧^{2}F_{1}(f − s) ⊕ F_{1} → R → A → 0.

Again, the resolution is minimal for most values of s and for all s ≫ 0.

Example 17. Let R = k[a, b, c, d, e, f] and let I = (bc − ad, be − af, de − cf) be the ideal of 2 × 2 minors of the 2 × 3 matrix of indeterminants of R. In this case, B = R/I is Cohen-Macaulay of codimension 2, S = Proj B is a local complete intersection (so we can take Z = ∅) and the minimal R-free resolution of B is

0 → R(−3)^{2} → R(−2)^{3} → R → B → 0.

A regular section, σ ∈ H^{0}(S, Ñ_{B}^{∗}(3)), defines a Gorenstein algebra, A, of codimension 4 with R-free resolution

0 → R(−6) → R(−4)^{9} → R(−3)^{16} → R(−2)^{9} → R → A → 0.

An example of such a Gorenstein algebra is

A = R/(bc − ad, be − af, de − cf, ab − c^{2}, a^{2} + ce, b^{2} − cd, ac + af, c^{2} + cf, ad + bf).

3.3 First Koszul homology modules

Huneke [14] has proved that the Koszul homology modules H_{i}(I), built on some minimal set of generators of I, are Cohen-Macaulay provided B = R/I is licci. We shall see that this result provides us with a class of Cohen-Macaulay quotients of R to which we may apply Theorem 8 (with M = H_{1}(I)). If all H_{i}(I) are Cohen-Macaulay, then B = R/I is called strongly Cohen-Macaulay. We have the following result.

Proposition 18. Let R be a graded Gorenstein quotient of a polynomial ring and assume R has canonical module R(e). Let B = R/I be a codimension c graded strongly Cohen-Macaulay quotient of R. Let K_{B} = Ext_{R}^{c}(B, R(e)) be the canonical module of B and let H_{1} = H_{1}(I), H_{2} = H_{2}(I), … be the Koszul homology modules of I. Let μ(I) denote the number of minimal generators of I and let r = μ(I) − c. Let d = Σn_{1i} where F_{1} = ⊕R(−n_{1i}) minimally surjects to I. Let S = Proj(B), let Z be a closed scheme such that dim(B) − dim(B/I(Z)) ≥ max(r, 2) and let U = S − Z. If U is a local complete intersection in Proj(R) then 1. Every regular section, σ ∈ H^{0}(U, H̃_{1}^{∗}(s)), defines a Gorenstein quotient R ↠ A of

codimension μ(I).

2. H_{∗}^{0}(U, ∧^{i}H̃_{1}) is a maximal Cohen-Macaulay B-module for every 1 ≤ i ≤ r

3. H_{∗}^{0}(U, ∧^{i}H̃_{1}) ≃ H_{i}

4. Letting a = e + d + rs, there is an exact sequence

0 → K_{B}(−a) → H_{r−1}((1 − r)s) → ⋯ → H_{2}(−2s) → H_{1}(−s) → B → A → 0.

Proof. Looking to (8), and taking maximal exterior powers, we see that

∧^{r}H̃_{1}|_{U} ≃ (∧^{c}Ñ_{B} ⊗_{O_{S}} ∧^{μ(I)}(⊕B̃(−n_{1i})))|_{U} ≃ K̃_{B}(−e − d)|_{U}  (30)

since (∧^{c}Ñ_{B})|_{U} ≃ K̃_{B}(−e)|_{U} and d = Σn_{1i}. To apply Theorem 8 to M = H_{1}, it suffices to prove that H_{∗}^{0}(U, ∧^{i}H̃_{1}) is a maximal Cohen-Macaulay B-module for each 1 ≤ i ≤ r. Since H_{1}, H_{2}, …, H_{r} are maximal Cohen-Macaulay modules (cf. [14], Remark 1.3 to see dim H_{i}(I) = dim B), it is enough to prove

H_{∗}^{0}(U, ∧^{i}H̃_{1}) ≃ H_{i}.

However, since H̃_{1} is locally free on U, one knows that ∧^{i}H̃_{1}|_{U} ≃ H̃_{i}|_{U}, cf. [20] (the inaccuracy of [20] does not effect our conclusion because B contains a field, see [21]). We conclude the proof using depth_{I(Z)} H_{i} = depth_{I(Z)} B ≥ 2.

To construct Gorenstein algebras of small codimension, we use Proposition 13 and Proposition 18 for small c and small r = μ(I) − c. We have in particular

Corollary 19. Let R ↠ B = R/I be a graded Cohen-Macaulay quotient of codimension c ≥ 1 and a local complete intersection outside a set Z of codimension ≥ max(μ(I) − c, 2) in S = Proj B.

1. If μ(I) ≤ c + 2 or

2. if B is licci or

3. if c = 2 or

4. if c = 3 and B is Gorenstein

then any regular section, σ ∈ H^{0}(S − Z, H̃_{1}^{∗}(s)), defines a Gorenstein quotient R ↠ A of codimension μ(I).

Proof. B is strongly Cohen-Macaulay if μ(I) − c ≤ 2 [24], Corollary 3.3.14. If B is licci then B is strongly Cohen-Macaulay [14]. If c = 2 and B is Cohen-Macaulay then B is licci [8]. If c = 3 and B is Gorenstein then B is licci [25].

Remark 20. If the codimension of B is 2 or if B is Gorenstein with codimension 3 and r ≤ 3, we can explicitly write down an R-free resolution of the Gorenstein algebra A of Proposition 18 in terms of the free modules, F_{i}, in the resolution of I = ker(R → B) as given in (19).

If the codimension of B is 2 then exact sequence (19) becomes the short exact sequence

0 → F_{2} → F_{1} → I → 0.  (31)

By [2], we have an exact sequence

0 → ∧^{i+1}F_{2} → ∧^{i+1}F_{1} → ∧^{i}F_{2} → H_{i} → 0

for every i ≥ 1. Splitting the exact sequence

0 → K_{B}(−a) → ⋯ → H_{2}(−2s) → H_{1}(−s) → B → A → 0  (32)

of Proposition 18 into short exact sequences and applying the mapping cone construction, we obtain an R-free resolution of A.

If B is Gorenstein with codimension 3 and if 1 ≤ r ≤ 3 then we remark that r = 2 since μ(I) = r + c must be odd. In view of (32) it suffices to find an R-free resolution of H_{1}. If we combine the isomorphism of conclusion (2) of Theorem 8 with Proposition 18, noting that t = −e − d by (30) and K_{B}(−e) ≃ B(f), we get H_{i} ≃ H_{r−i}^{∗}(f − d). Hence it suffices to find an R-free resolution of (the maximal Cohen-Macaulay module) H_{1}^{∗}. Now since B is Gorenstein, sequence (8) is exact (cf. Section 1). In addition, its B-dual

0 → N_{B} → F_{1}^{∗} ⊗ B → H_{1}^{∗} → 0  (33)

is exact since Ext_{B}^{1}(I/I^{2}, B) = 0 by (7). The R-free resolution of N_{B} is given by twisting sequence (27) by f (noting that N_{B} ≃ (∧^{2}I)(f) if char k ≠ 2). The mapping cone construction applied to (33) yields

0 → F_{1} ⊕ S_{2}F_{2}(f) → (F_{2} ⊗ F_{2}(f)) ⊕ (F_{1} ⊗ F_{2}(f)) →
∧^{2}F_{1}(f) ⊕ (F_{1} ⊗ F_{2}(f)) → F_{2}(f) → H_{1}^{∗} → 0.

This reduces to

0 → F_{1} → ∧^{2}F_{2}(f) → ∧^{2}F_{1}(f) → F_{2}(f) → H_{1}^{∗} → 0.

Finally, combining with H_{1} ≃ H_{1}^{∗}(f − d) and (32) and using that 2f = d by [3] (since μ(I) = 5) we readily obtain the following R-free resolution of A

0 → R(−2f − 2s) → F_{2}(−f − 2s) ⊕ F_{1}(−f − s) → F_{1}(−f − 2s) ⊕ ∧^{2}F_{2}(−s) ⊕ R(−f) → R(−f − 2s) ⊕ ∧^{2}F_{1}(−s) ⊕ F_{2} → F_{2}(−s) ⊕ F_{1} → R → A → 0.

Example 21. Let R = k[a, b, c, d, e, f] and let I = (bc − ad, be − af, de − cf, a^{2} + ad + ce, ab + bd + cf) be the ideal defining the Gorenstein algebra constructed in Example 12. In this case, B = R/I is Gorenstein of codimension 3, S = Proj B is a local complete intersection (but not smooth) and the minimal R-free resolution of B is

0 → R(−5) → R(−3)^{5} → R(−2)^{5} → R → B → 0.

A regular section, σ ∈ H^{0}(S, H̃_{1}^{∗}(−1)) defines a Gorenstein algebra, A, of codimension μ(I) = 5 with R-free resolution

0 → R(−8) → R(−6)^{10} → R(−5)^{16} → R(−3)^{16} → R(−2)^{10} → R → A → 0.

An example of such a Gorenstein algebra is

A = R/(I + (b^{2} − cd + df, c^{2} + bd, ac + cd − e^{2} + f^{2}, bd − ae + bf, d^{2} − ce + df)).

3.4 Further applications

In this subsection we give some ideas and examples of how we can use Theorem 8 to construct other Gorenstein algebras, especially when the rank of M is small. This will lead to Gorenstein quotients R ↠ B = R/I of small codimension. Throughout this subsection, we assume that B is of finite projective dimension over R.

Indeed, one may hope to construct Gorenstein quotients by applying Theorem 8 to the left term of a short exact sequence provided the (dual of the) right term is “good” for the Theorem (as we did in (30) using (8)). For instance looking also to (19), we have a surjection F_{2} ⊗ B ↠ H_{1} if H_{1} → F_{1} ⊗ B is injective (cf. (8)). If we define G_{2} by the exact sequence

0 → G_{2} → F_{2} ⊗ B → H_{1} → 0  (34)

and if U is a local complete intersection in Proj(R), then a regular section σ ∈ H^{0}(U, G̃_{2}(s)) defines a Gorenstein quotient in good cases. Indeed, sheafifying (34) and taking maximal exterior powers, we get (16). The module G_{2}^{∗} is maximal Cohen-Macaulay in the licci case by the next lemma. However, we have not been able to to show the maximal Cohen-Macaulayness of all H_{∗}^{0}(U, ∧^{i}G̃_{2}^{∗}) except in the low rank case of Remark 27.

To analyze this idea further, it seems more natural to consider the dual of (19), or more precisely the complex

0 → F_{0}^{∗} ⊗_{R} B ⟶^{ψ_{0}} F_{1}^{∗} ⊗_{R} B ⟶^{ψ_{1}} ⋯ → F_{c−1}^{∗} ⊗_{R} B ⟶^{ψ_{c−1}} F_{c}^{∗} ⊗_{R} B → 0  (35)

where coker ψ_{c−1} = K_{B}(−e) and F_{0}^{∗} ⊗_{R} B = B leads to ψ_{0} = 0. As we have seen when we considered (21), the homology of (35) is given by the modules Tor_{i}^{R}(B, K_{B})(−e), and N_{B} ≃ Tor_{c−1}^{R}(B, K_{B})(−e) = ker ψ_{1}.

Lemma 22. If R ↠ B = R/I is licci of codimension at least 2 and if S ↪ P is a local complete intersection on a non-empty open set U = S−Z, then coker ψ_{i}, ker ψ_{i} and nontrivial im ψ_{i} are maximal Cohen-Macaulay B-modules for 1 ≤ i ≤ c − 1. Moreover, when dim B − dim B/I(Z) ≥ 2, we have H_{1}^{∗} ≃ im ψ_{1}, G_{2}^{∗} ≃ coker ψ_{1} and H_{1}^{∗} (if nontrivial) and G_{2}^{∗} are maximal Cohen-Macaulay B-modules.

Proof. Since coker ψ_{c−1} = K_{B}(−e), we suppose by induction that coker ψ_{j} are maximal Cohen-Macaulay modules for j > i. Looking to the exactness of the horizontal sequences in

0 → im ψ_{i} → F_{i+1}^{∗} ⊗_{R} B → coker ψ_{i} → 0
↓ ↓ ↓
0 → ker ψ_{i+1} → F_{i+1}^{∗} ⊗_{R} B → im ψ_{i+1} → 0

we see by (2) that nontrivial im ψ_{j} and ker ψ_{j} are maximal Cohen-Macaulay modules for j > i. Now ker ψ_{i+1}/ im ψ_{i} ≃ Tor_{c−i−1}^{R}(B, K_{B})(−e) is a maximal Cohen-Macaulay module (cf. Theorem 7 and note that U ≠ ∅ shows that the Tor-group is non-vanishing by (22)). Hence, im ψ_{i} (if non-vanishing), ker ψ_{i} and by the snake lemma, coker ψ_{i}, are maximal Cohen-Macaulay modules.

To finish the proof of the lemma, it suffices to prove

H_{1}^{∗} = im ψ_{1} and G_{2}^{∗} ≃ coker ψ_{1}.

Indeed, for i = 1 and 2 we get Ext_{B}^{i}(I/I^{2}, B) = 0 by (7). Hence we have the short exact sequence (33) in this case as well (i.e. we get H_{1}^{∗} = im ψ_{1}). Since Ext_{B}^{1}(H_{1}, B) ≃ Ext_{B}^{2}(I/I^{2}, B) by (8), the conclusion follows by applying (−)^{∗} to (34).

Proposition 23. Let R ↠ B = R/I be licci of codimension c ≥ 2 and S = P roj(B) ↪ P roj(R) a local complete intersection on a non-empty open U = S − Z. Then N_{B}, H_{1} ≠ 0, ker ψ_{c−1}, and K_{B} are each maximal Cohen-Macaulay B-modules whose sheafified maximal exterior power is a twist of K̃_{B} on U. The same statement holds for G_{2}^{∗} and a nontrivial (im ψ_{c−1})^{∗} provided dim B − dim B/I(Z) ≥ 2.

Proof. By (17) and the proofs of Proposition 13 and Proposition 18, K_{B}, N_{B} and H_{1} are as claimed. Next, splitting the right part of (35) into short exact sequences (resp. using (34)), we see that the sheafified maximal exterior powers of a nontrivial (im ψ_{c−1})^{∗} and ker ψ_{c−1} (resp. G_{2}^{∗}) are a twist of K̃_{B} on U. Moreover, ker ψ_{c−1} and G_{2}^{∗} are maximal Cohen-Macaulay modules by Lemma 22. To prove that a nontrivial (im ψ_{c−1})^{∗} is a maximal Cohen-Macaulay module, we remark that there is an exact sequence

0 → K_{B}(−e)^{∗} → F_{c}^{∗} ⊗_{R} B → (im ψ_{c−1})^{∗} → 0.  (36)

Indeed Ext_{B}^{1}(K_{B}, B) ≃ Ext_{B}^{1}(K_{B} ⊗ K_{B}, K_{B}) = 0 because Ext_{B}^{1}(∧^{2}K_{B}, K_{B}) = 0 by the the codimension assumption and Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 by the Cohen-Macaulayness of S_{2}(K_{B}) [24], thm 4.2.6 (if the characteristic of k is 2 then the argument above works if we replace ∧^{2}K_{B} by ker(K_{B} ⊗ K_{B} → S_{2}(K_{B}))) . The Cohen-Macaulayness implies also that K_{B}^{∗} ≃ Hom_{B}(K_{B} ⊗ K_{B}, K_{B}) ≃ Hom_{B}(S_{2}(K_{B}), K_{B}) is a maximal Cohen-Macaulay module. Moreover (36) above and (2) imply that a nontrivial (im ψ_{c−1})^{∗} is a maximal Cohen-Macaulay module provided we can prove Ext_{B}^{1}((im ψ_{c−1})^{∗}, K_{B}) = 0. Since the dual of (36) is short exact, and since we at least have right exactness after applying (−) ⊗_{R} K_{B} to this dual sequence, we get precisely this last mentioned Ext-group to vanish, and we are done.

In the codimension 2 case (c = 2), then (35) leads to the exact sequence

0 → N_{B} → F_{1}^{∗} ⊗_{R} B ⟶^{ψ_{1}} F_{2}^{∗} ⊗_{R} B → K_{B}(−e) → 0  (37)

and im ψ_{1} ≃ H_{1}^{∗} and G_{2}^{∗} ≃ K_{B}(−e) by Lemma 22. Hence (17), Proposition 13 and Proposition 18 take care of all Gorenstein quotients to be constructed by this idea of applying Theorem 8 to the “syzygy" modules of (35) and its dual.

If the codimension c = 3, then we have a complex

0 → N_{B} → F_{1}^{∗} ⊗_{R} B ⟶^{ψ_{1}} F_{2}^{∗} ⊗_{R} B ⟶^{ψ_{2}} F_{3}^{∗} ⊗_{R} B → K_{B}(−e) → 0  (38)

which is exact except in the middle. Indeed ker ψ_{2}/ im ψ_{1} ≃ Tor_{1}^{R}(B, K_{B})(−e) ≃ I/I^{2} ⊗ K_{B}(−e). By Proposition 23 we see that, in addition to the modules K_{B}, N_{B} and H_{1}, there are precisely 3 more modules to which we might apply Theorem 8, using this idea.

To apply Theorem 8 to these 3 modules, we must prove the maximal Cohen-Macaulayness of H_{∗}^{0}(U, ∧^{i}(−)) of the 3 modules for all 2 ≤ i ≤ rank/2. We have not been able to prove this extra assumption in general, and we leave it as an interesting question for further investigations.

Question 24. Let R be a Gorenstein ring. Let B be a codimension 3 graded licci quotient of R. Let S = Proj(B), let Z be a scheme such that dim(B) − dim(B/I(Z)) ≥ max(r, 2) and let U = S − Z. If S is a local complete intersection on U, do regular sections on U of twists of { \atop (ker ψ_{2}})^{∗}, G̃_{2} and { \atop im ψ_{2}} define Gorenstein quotients of R (where r denotes the rank of each of these modules)?

The extra assumption above is fulfilled if the rank of the module is at most 3. Remarking that the part of the proof of Proposition 23 which is concerned with the module im ψ_{c−1}, works with a local Gorenstein assumption instead of the local complete intersection assumption, we have in particular

Corollary 25. Let R ↠ B = R/I be licci of codimension c ≥ 2 and Cohen-Macaulay type g and suppose it is locally Gorenstein outside some closed set Z of codimension ≥ max(g−1, 2) in S = Proj(B). If g ≤ 4, then any regular section of H^{0}(S − Z, { \atop im ψ_{c−1}}(s)) defines a Gorenstein quotient R ↠ A of codimension c + g − 1.

Remark 26. Under reasonable additional assumptions (which we expect a licci quotient, B, to have) we can extend Corollary 25 further. For instance, if

depth_{𝔪} S_{3}(K_{B}) ≥ dim B − 1  (39)

and S = Proj(B) is locally Gorenstein (so U = S and I(Z) = 𝔪) and licci, then the conclusion of Corollary 25 holds also for g = 5. This follows from Theorem 8 if we can prove that (39) and the assumption dim B ≥ 4 implies that H_{∗}^{0}(S, ∧^{2}(ĩm ψ_{c−1})^{∗}) is a maximal Cohen-Macaulay B-module. To indicate a proof, let N = im ψ_{c−1}. We have the exact sequence (36) and its B-dual

0 → N → F_{c} ⊗_{R} B ⟶^{β} K_{B}(−e) → 0.  (40)

Let \overline{N} be the kernel of the composition of β ⊗ id : F_{c} ⊗_{R} S_{2}(K_{B}) → K_{B}(−e) ⊗ S_{2}(K_{B}) and the natural surjection K_{B}(−e) ⊗ S_{2}(K_{B}) ↠ S_{3}(K_{B})(−e). By (39), \overline{N} is a maximal Cohen-Macaulay B-module as is (N ⊗ K_{B})^{∗} (by Theorem 5). Indeed, since the sheafification of \overline{N} and N ⊗ K_{B} ⊗ K_{B} are isomorphic, we get

(N ⊗ K_{B})^{∗} ≃ Hom(N ⊗ K_{B} ⊗ K_{B}, K_{B}) ≃ Hom(\overline{N}, K_{B}).

Hence, we get H_{∗}^{1}(S, { \atop (N ⊗ K_{B})^{∗}}) = 0. Now using exact sequence (36) and [12], II, exc 5.16, we have the exact sequence

0 → (K̃_{B}(−e) ⊗ Ñ)^{∗} → ∧^{2}{ \atop F_{c}^{∗} ⊗ B} → ∧^{2}Ñ^{∗} → 0

which is still exact after applying H_{∗}^{0}(S, −). The maximal Cohen-Macaulayness of (K_{B} ⊗N)^{∗} implies depth_{𝔪} H_{∗}^{0}(S, ∧^{2}Ñ^{∗}) ≥ dim B − 1 since the rank of N^{∗} is 4. We conclude by (15) of Remark 9.

Remark 27. Suppose that B has codimension 3. If B is an almost complete intersection (i.e. μ(I_{B}) = rank F_{1} = 4) and if the Cohen-Macaulay type of B is g = 2, we see from (8) and (34) that rank G̃_{2} = 4. If (39) holds and if S ↪ Proj R is a local complete intersection of dimension dim S ≥ 3, we can argue exactly as in Remark 26 to see that H_{∗}^{0}(S, ∧^{2}G̃_{2}^{∗}) is a maximal Cohen-Macaulay B-module. Indeed, (8) shows that rank H̃_{1} = 1 and we have H̃_{1} = K̃_{B}(t) by (30). Hence, H_{1} ≃ K_{B}(t) and we conclude by the arguments of Remark 26 by replacing exact sequence (40) (resp. exact sequence (36)) by the sequence (34) (resp. its short exact dual sequence, cf. Lemma 22). Such almost complete intersections are licci [3]. Hence, by Proposition 23 and Theorem 8, each regular section of H^{0}(S, G̃_{2}(s)) defines a Gorenstein quotient R → A of codimension 7.

We finish this subsection by giving a few more examples of low rank modules where we can apply Theorem 8. If B = R/I is licci, it is known that S_{2}(K_{B}) is a maximal Cohen-Macaulay B-module, as is K_{B}^{∗} ≃ Hom_{B}(S_{2}(K_{B}), K_{B}) (cf. the proof of Proposition 23). Theorem 8 therefore immediately applies to M = H_{∗}^{0}(U, S_{2}(K̃_{B}) ⊕ K̃_{B}^{∗}(t_{1})) and to M = H_{∗}^{0}(U, K̃_{B} ⊕ K̃_{B}(t_{2}) ⊕ K̃_{B}^{∗}(t_{3})), i.e.

Proposition 28. Let S = Proj B and let B = R/I be licci of codimension c and locally Gorenstein outside some closed set Z ⊆ S.

i) If dim B−dim B/I(Z) ≥ 2, then any regular section of H^{0}(S−Z, S_{2}(K̃_{B})^{∗}(s_{1})⊕K̃_{B}(s_{2})) defines a Gorenstein quotient R ↠ A of codimension c + 2.

ii) If dim B − dim B/I(Z) ≥ 3, then any regular section of H^{0}(S − Z, K̃_{B}^{∗}(s_{1}) ⊕ K̃_{B}^{∗}(s_{2}) ⊕ K̃_{B}(s_{3})) defines a Gorenstein quotient R ↠ A of codimension c + 3.

Example 29. Let R = k[a, b, c, d, e, f] and let I = (bc − ad, be − af, de − cf) be the ideal of 2 × 2 minors of the 2 × 3 matrix of indeterminants of R (as in Example 12). In this case, B = R/I is Cohen-Macaulay of codimension 2, S = Proj B is a local complete intersection and the minimal R-free resolution of B is

0 → R(−3)^{2} → R(−2)^{3} → R → B → 0.

A regular section, σ ∈ H^{0}(S, K̃_{B}^{∗}(−1) ⊕ K̃_{B}^{∗}(−1) ⊕ K̃_{B}(4)) defines a Gorenstein algebra, A, of codimension 5. An example of such a Gorenstein algebra is

A = R/(I + (a^{2} + ad + ce, ab + bd + cf, ad + ce + ef, bd + cf + f^{2}, ce + 2ef, ac − cd, ae − cf))

with R-free resolution

0 → R(−8) → R(−6)^{10} → R(−4)^{3} ⊕ R(−5)^{16} → R(−3)^{16} ⊕ R(−4)^{3} → R(−2)^{10} → R ↠ A.

This should be compared with the Gorenstein algebra constructed in Example 21.

Remark 30. Assume L is an O_{S}-module.

i) If L is invertible on the set U = S − Z of Proposition 28 (or some subset of U, no licci assumption on B is needed) and H_{∗}^{0}(U, L^{∗}) (resp. H_{∗}^{0}(U, L ⊕ L^{∗})) is a maximal Cohen-Macaulay module, then Theorem 8 also applies to M = H_{∗}^{0}(U, (L ⊗ K̃_{B}) ⊕ L^{∗}(t_{1})) (resp. to M = H_{∗}^{0}(U, L ⊕ K̃_{B}(t_{2}) ⊕ L^{∗}(t_{3}))) provided the codimension of Z is ≥ 2 (resp. ≥ 3) in S = Proj(B). Indeed, H_{∗}^{0}(U, L ⊗ K̃_{B}) ≃ H_{∗}^{0}(U, Hom(L^{∗}, K̃_{B})) ≃ Hom_{B}(H_{∗}^{0}(U, L^{∗}), K_{B}) is a maximal Cohen-Macaulay B-module by Theorem 5.

ii) If L is an n^{th} root of a twist of the canonical bundle and L = H_{∗}^{0}(S, L), S_{2}L, …, S_{n}L ≃ K_{B}(t) are all maximal Cohen-Macaulay modules then there are many possible choices for M in terms of direct sums of various symmetric powers of L. This occurs in the case of determinantal ideals of codimension n+1 (see [6] page 595 and the discussion of the Buchsbaum-Rim complex).

Example 31. Let R = k[a, b, c, d, e, f, g, h, i, j] and let

I = (bc − ad, be − af, bg − ah, bi − aj, de − cf, dg − ch, di − cj, fg − eh, fi − ej, hi − gj)

be the ideal of 2 × 2 minors of the 2 × 5 matrix of indeterminants, N, of R. In this case, B = R/I is Cohen-Macaulay of codimension 4, S = Proj B is a local complete intersection and the minimal R-free resolution of B is

0 → R(−5)^{4} → R(−4)^{15} → R(−3)^{20} → R(−2)^{10} → R → B → 0.

The cokernel of the map defined by the matrix N is a B-module, L. The minimal R-free resolution of L is

0 → R(−5)^{3} → R(−4)^{10} → R(−3)^{10} → R(−1)^{5} → R^{2} → L → 0.

The sheafification of L is a line bundle, L, on Proj B. Exactness of the complexes given in [6], page 595, show that L, S^{2}L and S^{3}L = K_{B}(t) are all maximal Cohen-Macaulay modules. Therefore, a regular section, σ ∈ H^{0}(S, L^{∗}(2) ⊕ L^{∗}(2) ⊕ L^{∗}(2)) defines a Gorenstein algebra, A, of codimension 7. An example of such a Gorenstein algebra is

A = R/(I + (bh + aj + dj, ah + ai + cj, cf + aj, ce + ai, d^{2} + gh + hj, cd + g^{2} + gj))

with R-free resolution

0 → R(−11) → R(−9)^{16} → R(−7)^{9} ⊕ R(−8)^{35} → R(−5)^{4} ⊕ R(−6)^{60} ⊕ R(−7)^{15} →
R(−4)^{15} ⊕ R(−5)^{60} ⊕ R(−6)^{4} → R(−3)^{35} ⊕ R(−4)^{9} → R(−2)^{16} → R → A → 0.

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Jan O. Kleppe

Faculty of Engineering Oslo University College Cort Adelers gt. 30 N-0254, Oslo, Norway JanOddvar.Kleppe@iu.hio.no http://www.iu.hio.no/∼jank

Chris Peterson

Department of Mathematics Colorado State University Fort Collins, CO 80523-1874, U.S.A. peterson@math.colostate.edu