For a graph G, we construct two algebras, whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis...

We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals. INTRODUCTION Let S = K[x1, . . . ,xn] denote the polynomial ring in n variables over a field K with each degxi = 1. Let I be a monomial ideal of S and G(I) = {u1, . . . ,us} its unique minimal system of monomial generators. The Ta...

The Hilbert series of local cohomologies for monomial ideals, which are not necessarily square-free, is established. As applications, we give a sharp lower bound of the non-vanishing degree of local cohomologies and also a sharp lower bound of the positive integer k of k-Buchsbaumness for generalized CohenMacaulay monomial ideals.

We study Stanley decompositions and show that Stanley’s conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.

Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.

In the theory of monomial ideals of a polynomial ring S over a field k, it is convenient that for each such ideal I there is a standard free resolution, so called Taylor resolution, that can be canonically constructed from the minimal system of monomial generators of I (see [7], p.439 and section 2). On the other hand no construction of a minimal resolution for an arbitrary monomial ideal has b...

We show that all monomial ideals in the polynomial ring in at most 3 variables are pretty clean and that an arbitrary monomial ideal I is pretty clean if and only if its polarization I p is clean. This yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree, and it also implies that a multicomplex is shellable if and only the simplicial complex correspond...