Resolutions of Facet Ideals

Abstract

In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determine the projective dimension and the regularity of I if the tree is 1-dimensional, show that the graded Betti numbers of I satisfy an alternating sum property if the tree is connected in codimension 1, and classify all trees whose facet ideal has a linear resolution.     

Graph representation of projective resolutions

We generalize the concept — dimension tree and the related results for monomial algebras to a more general case — relations algebras Λ by bringing Gröbner basis into play. More precisely, we will describe the minimal projective resolution of a left Λ-module M as a rooted ‘weighted’ diagraph to be called the minimal resolution graph for M. Algorithms for computing such diagraphs and applications as well will be presented.     

Lyubeznik's Resolution and Rooted Complexes

We describe a new family of free resolutions for a monomial ideal I, generalizing Lyubeznik's construction. These resolutions are cellular resolutions supported on the rooted complexes of the lcm-lattice of I. Our resolutions are minimal for the matroid ideal of a finite projective space.     

Arithmetical Rank of Squarefree Monomial Ideals Generated by Five Elements or with Arithmetic Degree Four

Let I be a squarefree monomial ideal of a polynomial ring S. In this article, we prove that the arithmetical rank of I is equal to the projective dimension of S/I when one of the following conditions is satisfied: (1) μ(I) ≤5; (2) arithdeg I ≤ 4.     

On the index of powers of edge ideals

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper, we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal I is strictly increasing if I is linearly presented. Examples show that this needs not to be the case for monomial ideals generated in degree greater than two.     

Monomial Ideals of Forest Type

As a generalization of the facet ideal of a forest, we define monomial ideal of forest type and show that monomial ideals of forest type are pretty clean. As a consequence, we show that if I is a monomial ideal of forest type in the polynomial ring S, then Stanley's decomposition conjecture holds for S/I. The other main result of this article shows that a clutter is totally balanced if and only if it has the free vertex property, and which is also equivalent to say that its edge ideal is a monomial ideal of forest type or is generated by an M sequence.     

Arithmetical rank of squarefree monomial ideals of small arithmetic degree

In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg I=reg I; (c) arithdeg I=indeg I+1. We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).     

Resolutions of monomial almost complete intersections and generalizations

Let S=K[x₁,…,x_n] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.     

Skeletons of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley–Reisner ideals we introduce the skeletons of an arbitrary monomial ideal I ? S = K [x₁, …, x_n ]. This allows us to compute the depth of S /I in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of S /I holds provided it holds whenever S /I is Cohen–Macaulay. We also discuss a conjecture of Soleyman Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Candidates for nonzero Betti numbers of monomial ideals

Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d?1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fröberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.     

Some Ideals with Large Projective Dimension

For an ideal in a polynomial ring over a field, a monomial support of is the set of monomials that appear as terms in a set of minimal generators of . Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projective dimension of grows at least exponentially with the size of a monomial support. The ideal we construct is generated by monomials and binomials.

     

Stanley depth of monomial ideals with small number of generators

For a monomial ideal I ⊂ S = K[x ₁...,x _n ], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x ₁,x ₂,x ₃] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x ₁,x ₂,x ₃]//I) +1 for any monomial ideal in I ⊂ K[x ₁,x ₂,x ₃].     

A class of square-free monomial ideals associated to two integer sequences

Given two finite sequences of positive integers α and β, we associate a square-free monomial ideal I_α,β in a ring of polynomials S, and we recursively compute the algebraic invariants of S∕I_α,β. Also, we give precise formulas in special cases.     

CW-resolutions of monomial ideals that are supported on face posets

Given a monomial ideal I with minimal free resolution ? supported in characteristic p>0 on a CW-complex X with regular 2-skeleton, in general it is not the case that the face poset of X, P(X), also supports ? in the sense of Clark and Tchernev. We construct a (not necessarily regular) CW-complex Y that also supports ? and such that the face poset P(Y) also supports ?.     

Gorenstein Injective Dimension for Complexes and Iwanaga–Gorenstein Rings

The main purpose of this article is to present some applications of the notion of Gorenstein injective dimension of complexes over an associative ring. Among the applications, we give some new characterizations of Iwanaga–Gorenstein rings. In particular, we show that an associative ring R is Iwanaga–Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero and the class of complexes of finite projective dimension are orthogonal complement of each other with respect to the ‘Ext’ functor.     

On the Denominator of the Poincaré Series for Monomial Quotient Rings

Let S = 𝕜 [x ₁,…, x _n ] be a polynomial ring over a field 𝕜 and I a monomial ideal of S. It is well known that the Poincaré series of 𝕜 over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.     

A Frobenius characterization of finite projective dimension over complete intersections

Let M be a module of finite length over a complete intersection (R,m) of characteristic . We characterize the property that M has finite projective dimension in terms of the asymptotic behavior of a certain length function defined using the Frobenius functor. This may be viewed as the converse to a theorem of S. Dutta. As a corollary we get that, in a complete intersection (R,m), an m-primary ideal I has finite projective dimension if and only if its Hilbert-Kunz multiplicity equals the length of R/I. Received June 22, 1998; in final form October 13, 1998     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.