Resolution and tor algebra structures of grade 3 ideals defining compressed rings
Introduction
Let be a regular local ring with maximal ideal and residue field k. A result of Buchsbaum and Eisenbud (see [3]) established that any quotient 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 for such quotients was established by Weyman in [9] and Avramov, Kustin, and Miller in [2].
Given an -primary ideal , one can “trim” the ideal I by, for instance, forming the ideal . This process is used by Christensen, Veliche, and Weyman (see [5]) in the case that 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 defining a compressed ring with socle 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 defining a compressed ring with for some . The values s and 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 and , with (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 for some . 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 (a standard graded polynomial ring over a field k) defining a compressed ring with . We first show that all such ideals define a ring with tipping point (see Definition 3.4) s and type . 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 , and r is computed in terms of the minimal number of generators of I and the variable ℓ.
In Section 5, we show that there are ideals of arbitrarily large type of class , for any . 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 be a standard graded polynomial ring over a field k. Let be a homogeneous ideal and denote a homogeneous free resolution of . Write , where each generates a free direct summand of .
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 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 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 :
Definition 4.1 Let be a regular local ring with and ideal such that . Let . Then [2], Theorem 2.1
Realizability in the higher type case
In this section, we construct ideals with Tor algebra class 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 but with arbitrarily large type. This is stated more precisely in Theorem 5.4,
References (9)
A cohomological study of local rings of embedding codepth 3
J. Pure Appl. Algebra
(2012)- et al.
Poincaré series of modules over local rings of small embedding codepth or small linking number
J. Algebra
(1988) - et al.
Free resolutions of artinian compressed algebras
J. Algebra
(2018) On the structure of free resolutions of length 3
J. Algebra
(1989)
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