Elsevier

Journal of Algebra

Volume 586, 15 November 2021, Pages 140-153
Journal of Algebra

Resolution and tor algebra structures of grade 3 ideals defining compressed rings

https://doi.org/10.1016/j.jalgebra.2021.06.027Get rights and content

Abstract

Let R=k[x,y,z] be a standard graded 3-variable polynomial ring, where k denotes any field. We study grade 3 homogeneous ideals IR defining compressed rings with socle k(s)k(2s+1), where s3 and 1 are integers. The case for =1 was previously studied in [8]; a generically minimal resolution was constructed for all such ideals. The paper [7] generalizes this resolution in the guise of (iterated) trimming complexes. In this paper, we show that all ideals of the above form are resolved by an iterated trimming complex. Moreover, we apply this machinery to construct ideals I such that R/I is a ring of Tor algebra class G(r) for some fixed r2, and R/I may be chosen to have arbitrarily large type. In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman in [5].

Introduction

Let (R,m,k) be a regular local ring with maximal ideal m and residue field k. A result of Buchsbaum and Eisenbud (see [3]) established that any quotient R/I of R with projective dimension 3 admits the structure of an associative commutative differential graded (DG) algebra. Later, a complete classification of the multiplicative structure of the Tor algebra TorR(R/I,k) for such quotients was established by Weyman in [9] and Avramov, Kustin, and Miller in [2].

Given an m-primary ideal I=(ϕ1,,ϕn)R, one can “trim” the ideal I by, for instance, forming the ideal (ϕ1,,ϕn1)+mϕn. This process is used by Christensen, Veliche, and Weyman (see [5]) in the case that R/I is a Gorenstein ring to produce ideals defining rings with certain Tor algebra classification, negatively answering a question of Avramov in [1].

This trimming procedure also arises in classifying certain type 2 ideals defining compressed rings. More precisely, it is shown in [8] that every homogeneous grade 3 ideal Ik[x,y,z] defining a compressed ring with socle Soc(R/I)=k(s)k(2s+1) is obtained by trimming a Gorenstein ideal. A complex is produced that resolves all such ideals; it is generically minimal. This resolution is then used to bound the minimal number of generators and, consequently, parameters arising in the Tor algebra classification.

In the current paper, we continue with this theme. In particular, much of the work done in [8] can be generalized to the case that I is a homogeneous grade 3 ideal Ik[x,y,z] defining a compressed ring with Soc(R/I)=k(s)k(2s+1) for some 1. The values s and 2s1 are interesting because they are extremal; more precisely, it is not possible to have a quotient ring as above with socle minimally generated in degrees s1 and s2, with s22s1 (see [4, Proposition 1.6] for a proof of this fact). We show that all such ideals are then obtained as iterated trimmings of a Gorenstein ideal (see Proposition 3.14), and the Tor algebra structure may be computed in similar fashion.

We employ a piece of machinery from [7], namely an iterated trimming complex, in order to resolve all homogeneous grade 3 ideals I with Soc(R/I)=k(s)k(2s+1) for some 1. This complex arises as a natural generalization of the complex constructed in [8] and has applications to the resolution of a variety of other classes of ideals, explored in [7].

The paper is organized as follows. In Section 2, we recall the construction of trimming complexes as given in [7]. In Sections 3 and 4, we consider grade 3 homogeneous ideals Ik[x,y,z] (a standard graded polynomial ring over a field k) defining a compressed ring with Soc(R/I)=k(s)k(2s+1). We first show that all such ideals define a ring with tipping point (see Definition 3.4) s and type s+2. Using this information we deduce the previously mentioned fact that I is obtained as the iterated trimming of a grade 3 Gorenstein ideal. In particular, I is resolved by the complex of Section 2. Moreover, we show precisely when such rings have Tor algebra class G(r), and r is computed in terms of the minimal number of generators of I and the variable .

In Section 5, we show that there are ideals of arbitrarily large type of class G(r), for any r2. The construction of these ideals is remarkably simple and is a generalization of the ideals constructed in [8]. One sees that the machinery of iterated trimming complexes allows for a quick proof that these ideals satisfy all of the required hypotheses of Proposition 4.4. In particular, these ideals provide a class of counterexamples to the conjecture of Avramov that is not already contained in [5].

Section snippets

Iterated trimming complexes

In this section, we recall the construction of trimming complexes. All proofs of the following results may be found in Section 2 and 3 of [7]; the purpose here is to give a concise and efficient introduction to the machinery, as it will be used in later sections.

Setup 2.1

Let R=k[x1,,xn] be a standard graded polynomial ring over a field k. Let IR be a homogeneous ideal and (F,d) denote a homogeneous free resolution of R/I.

Write F1=F1(i=1mRe0i), where each e0i generates a free direct summand of F1.

Compressed rings of higher type

In this section and Section 4 we generalize some of the work done in [8]. In particular, Corollary 3.15 says that the ideals introduced in Setup 3.6 are resolved by the iterated trimming complex of Theorem 2.4. In Proposition 4.4, we show that under suitable hypotheses, all ideals introduced in Setup 3.6 define rings of Tor algebra class G(r) for some r (see Definition 4.1). This information will be used to produce rings of Tor algebra class G with arbitrarily large type in Section 5.

Definition 3.1

Let A be a

Tor algebra structures for higher type ideals

In this section, we generalize the results of [8] to the case of higher type. Proposition 4.4 is the main result of this section and will be used to construct interesting examples of rings with Tor algebra class G(r) in Section 5, but most of the work done for this result is contained in the proof of Theorem 4.2. To begin the section, we recall the definition of the Tor algebra class G(r):

Definition 4.1

[2], Theorem 2.1

Let (R,m,k) be a regular local ring with Im2 and ideal such that pdR(R/I)=3. Let T:=TorR(R/I,k). Then R/I

Realizability in the higher type case

In this section, we construct ideals with Tor algebra class G(r) and arbitrarily large type. To begin with, we recall a collection of matrices introduced in Section 7 of [8] (which were inspired by matrices considered in [5]). It will be particularly easy to apply the construction of Theorem 2.4 to the submaximal pfaffians of these matrices to produce interesting classes of ideals attaining Tor algebra class G(r) but with arbitrarily large type. This is stated more precisely in Theorem 5.4,

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