Multigraded minimal free resolutions of simplicial subclutters

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Abstract

This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter C and its simplicial subclutter D, we compare some algebraic properties and invariants of the ideals I,J associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of J in terms of those of I and some combinatorial data about D. As a result, we see that if C admits a simplicial subclutter, then there exists a monomial uI such that the (multi)graded Betti numbers of I+(u) can be computed through those of I. It is proved that the Betti sequence of any graded ideal with linear resolution is the Betti sequence of an ideal associated to a simplicial subclutter of the complete clutter. These ideals turn out to have linear quotients. However, they do not form all the equigenerated square-free monomial ideals with linear quotients. If C admits ∅ as a simplicial subclutter, then I has linear resolution over all fields. Examples show that the converse is not true.

Introduction

Free resolutions have been a central topic in commutative algebra since the work of David Hilbert and his celebrated theorem “Hilbert Syzygy Theorem”. They are a very important tool to study the properties of a graded module over finitely generated graded K-algebras. They, in particular, are used to compute the Castelnuovo–Mumford regularity, projective dimension, depth, Hilbert function, etc. It is, in general very difficult to determine the whole resolution of a module, even using some computer programming systems. However, algebraists try to find some tools to investigate the general shape of the resolution and to compute some numerical data explaining them. An important invariant, regarding the graded minimal free resolutions, which carries most of the numerical data about them, is the (multi)graded Betti numbers. For aZn and iZ, the (i,a)-th Betti number of a finitely generated multigraded module M over a multigraded standard K-algebra R is given byβi,a(M)=dimKToriR(K;M)a. While it is again too hard to determine the exact value of the (multi)graded Betti numbers, one may try to explain them in combinatorial terms for monomial ideals over the polynomial rings. Indeed, given a monomial ideal I, passing through polarization, one can obtain a square-free monomial ideal J whereas lots of algebraic properties are preserved via this operation. Among all other properties, I and J share same graded Betti numbers. Hence, in order to study the graded free resolution of monomial ideals, it is fair enough to concentrate on the square-free ones. On the other hand, there is a bijection between the class of square-free monomial ideals over a polynomial ring and some classes of combinatorial or topological objects. Given a square-free monomial ideal I over the polynomial ring S=K[x1,,xn] one can associate a simplicial complex ΔI to I such that F[n]={1,,n} does not belong to ΔI if and only if xF:=iFxi belongs to I. A more combinatorial approach associates a collection CI of subsets of [n], called a clutter, to the minimal generating set of I as follows: A subset F[n] belongs to CI if and only if xF is an element in the minimal generating set of I. In case I is equigenerated, say in degree d, the complement of CI, denoted by C¯I, is defined to be the collection of all d-subsets of [n] not belonging to CI. Unless otherwise stated, throughout the paper, by the ideal attached to a clutter C, we mean the ideal whose generators are in correspondence with the elements of the complement of C.

In this paper, we study a class of clutters, called simplicial subclutters. In order to define this class, we first introduce, in Section 2, a reduction operation on the uniform clutters – the clutters whose elements have the same cardinality. This operation on a clutter C is based on removing an element F which contains a particular subset, called simplicial element of C. The definition of simplicial element and all other preliminaries is given in Section 1. A clutter has a simplicial subclutter if and only if it admits a non-trivial simplicial element. Not all clutters have such elements, see e.g. Example 4.5. In Section 2, we deal with the ideals associated to uniform clutters and their simplicial subclutters. Given a uniform clutter C and its simplicial subclutter D, we investigate some algebraic properties of the ideal J attached to D through those of I attached to C. By the structure of D, we see that J is an extension of I to which shares some properties with I. In case C admits a simplicial subclutter, it is guaranteed that there exists uI such that adding u to I does not change the nonlinear (multi)graded Betti numbers. Note that for any monomial ideal I generated in degree d and any monomial u with deg(u)=d, βi,a(I+(u))=βi,a(I), where aZn with |a|>d+i+r, and where r is the Castelnuovo-Mumford regularity of the ideal I:u, (cf. Theorem 1.1). But one can not expect more for a general case: Example 1.2 shows that the equality does not always hold in some multidegrees a with |a|=d+i+r even in case r=1 (i.e. I:u is generated by variables). However, in Theorem 2.4, we present a formula for the multigraded Betti numbers of the ideal I+(u), with u=xF, in terms of those of I and some numerical data about F, where I and I+(u) are given respectively by a clutter C and its simplicial subclutter obtained by removing F. As a result, one can see that the two ideals share the same regularity. In particular, I has a linear resolution if and only if so does I+(xF).

The proof of Theorem 2.4 is a direct proof having topological and combinatorial flavor. However, after presenting this result at the CMS meeting 2017, Huy Tài Hà mentioned to the first author that the ideal I+(xF) is splittable, [12]. This fact can be observed by the choice of F. On the other hand, since splittable ideals are Betti splittable, [16], one gets the Betti formula of Theorem 2.4 from the Betti splitting formula. This short proof of Theorem 2.4 is presented as well. The authors would like to thank Huy Tài Hà for this remark.

The final point in this section is that the ideal I satisfies the subadditivity condition if and only if so does I+(xF).

In Section 3, we consider the class of simplicial subclutters of the complete clutter Cn,d={F[n]:|F|=d}. We call it the class of simplicial clutters. This class is the dual of a class called chordal clutters [8]. Roughly speaking, a chordal clutter, is a clutter which can be reduced to ∅ by some reduction operations, while a simplicial clutter is obtained from Cn,d by some reduction operations. As a consequence of Theorem 2.4, it is seen that the ideals attached to the class of chordal clutters have linear resolution over all fields. This class has been studied in [8]. Later, in [5] a modification of this variation of chordality was studied for simplicial complexes. Trying to compare the properties of chordal clutters and simplicial clutters, it turns out that, like chordal clutters, the ideal associated to a simplicial clutter has linear resolution over all fields and all of its graded Betti numbers can be identified explicitly (Corollary 3.1). Moreover, these ideals have linear quotients, while not all ideals of chordal clutters have this property, [8, Example 3.14].

In Theorem 3.2 we see that the square-free stable ideals are examples of ideals associated to simplicial subclutters. In [7], it is proved that any Betti sequence of an ideal with linear resolution is the Betti sequence of an ideal associated to a chordal clutter. One of the main results of Section 3 improves this result stating that the Betti sequence of any ideal with linear resolution coincides with the Betti sequence of an ideal associated to a simplicial clutter (see Theorem 3.4). Hence all the Betti sequences of ideals with linear resolution are the Betti sequences of ideals with linear quotients.

One may ask if all the square-free equigenerated monomial ideals with linear quotients are attached to a simplicial clutter. In the last section, Section 4, we give a negative answer to this question by presenting a simple example. On the other hand, some known classes of ideals with linear resolution over all fields, such as square-free stable ideals, matroidal ideals and Alexander dual of vertex decomposable ideals, are associated to chordal clutters. Hence one may ask if all the square-free monomial ideals with linear resolution over all fields are attached to chordal clutters, [8, Question 1]. We close Section 4 showing by some examples that, this is not the case. However, to the best of our knowledge, this class is one of the largest known classes of clutters whose associated ideal has linear resolution over every field.

Section snippets

Preliminaries

Throughout this paper, S=K[x1,,xn] denotes the polynomial ring over the field K endowed with multigraded standard grading; that is deg(xi)=ei, where eis are the standard basis of Rn.

Simplicial subclutters and their multigraded resolution

In this section, we consider a class of subclutters, called simplicial subclutters, which have the property that all the (multi)graded Betti numbers of their associated ideals can be determined via those of the ideal of the original clutter.

Definition 2.1

Let C be a d-uniform clutter on the vertex set [n] and let DC. We say that D is a simplicial subclutter of C if there exists a sequence of (d1)-subsets of [n] (not necessarily distinct), say e=e1,,et, and Ai{FCA1Ai1:eiF}, i=1,,t, such that

  • (a)

    e1 is

Simplicial clutters

In this section, we consider the simplicial subclutters of a complete clutter. A simplicial subclutter of a complete clutter is called a simplicial clutter. Since I(Cn,d)=0, one would expect that the Betti table of the ideals of the simplicial clutters can be explicitly determined through the data obtained from the simplicial sequences. The following result which is a direct conclusion of Corollary 2.9, Corollary 2.7, gives rise to computing Betti numbers of simplicial subclutters of the

Contractible complexes vs. chordal clutters

In Corollary 3.1, it is stated that I(C¯) has linear quotients if C is a simplicial clutter. Simple examples show that the class of simplicial clutters are not equivalent to the class of equigenerated square-free monomial ideals with linear quotients. It follows that they are not equivalent to the class of square-free ideals with linear resolution over all fields.

Example 4.1

Let G be the graph in Fig. 4.

We haveI(G¯)=(x1x2,x1x3,x2x3,x1x2,x3x4,x3x5,x4x5) which has linear quotients, and hence linear

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    Mina Bigdeli was supported by a grant from IPM.

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