Ordinary and symbolic powers are Golod

Let S$S$ be a positively graded polynomial ring over a field of characteristic 0$0$, and I⊂S$I\subset S$ a proper graded ideal. In this note it is shown that S/I$S/I$ is Golod if ∂(I)²⊂I$\partial {\left(I\right)}^2\subset I$. Here ∂(I)$\partial \left(I\right)$ denotes the ideal generated by all the partial derivatives of elements of I$I$. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals.     

The absolutely Koszul property of Veronese subrings and Segre products

Absolutely Koszul algebras are a class of rings over which any finite graded module has a rational Poincaré series. We provide a criterion to detect non-absolutely Koszul rings. Combining the criterion with Macaulay2 computations, we identify large families of Veronese subrings and Segre products of polynomial rings which are not absolutely Koszul. In particular, we classify completely the absolutely Koszul algebras among Segre products of polynomial rings, when the base field has characteristic 0.     

A remark on regularity of powers and products of ideals

We give a simple proof for the fact that the Castelnuovo–Mumford regularity and related invariants of products of powers of ideals are asymptotically linear in the exponents, provided that each ideal is generated by elements of constant degree. We provide examples showing that the asymptotic linearity is false in general. On the other hand, the regularity is always given by the maximum of finitely many linear functions whose coefficients belong to the set of the degrees of generators of the ideals.     

Principal Borel ideals and Gotzmann ideals

In this paper we characterize all principal Borel ideals with Borel generator up to degree 4 which are Gotzmann. We also classify principal Borel ideals with a Borel generator of degree d which are lexsegment and we describe the shadows of principal Borel ideals. Finally, we discuss the corresponding results for squarefree monomial ideals.Received: 10 May 2002     

Computing residue currents of monomial ideals using comparison formulas

Given a free resolution of an ideal a$\mathfrak{a}$ of holomorphic functions, one can construct a vector-valued residue current R  , which coincides with the classical Coleff–Herrera product if a$\mathfrak{a}$ is a complete intersection ideal and whose annihilator ideal is precisely a$\mathfrak{a}$.     

Taylor complexes for Koszul boundaries

For all boundary modules of the Koszul complex of a monomial sequence we construct complexes, which we call Taylor complexes. For a monomial d-sequences these complexes provide free resolutions of the boundary modules. Let M be the ideal generated by a monomial d-sequence. We use the Taylor complexes to construct minimal free resolutions of the Rees algebra and the associated graded ring of M. Received: 13 November 1997 / Revised version: 6 March 1998     

Binomial Cycle Bases on Koszul Homology Modules

Let I be a principal p-Borel ideal of the polynomial ring S in variables x over a field. Then the Koszul homology module H ₃(x;S/I) has binomial cycle bases.     

A monomial cycle basis on Koszul homology modules

We give a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have a monomial cyclic basis.     

The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1

Let S be a polynomial ring and I be the Stanley-Reisner ideal of a simplicial complex Δ. The purpose of this paper is investigating the Buchsbaum property of S/I^(r) when Δ is pure dimension 1. We shall characterize the Buchsbaumness of S/I^(r) in terms of the graphical property of Δ. That is closely related to the characterization of the Cohen-Macaulayness of S/I^(r) due to the first author and N.V. Trung.     

A study of the lex plus powers conjecture

An {a₁,…,a_n}-lex plus powers ideal is a monomial ideal in I⊂k[x₁,…,x_n] which minimally contains the regular sequence x₁^a₁,…,x_n^a_n and such that whenever m∈R_t is a minimal generator of I and m′∈R_t is greater than m in lex order, then m′∈I. Conjectures of Eisenbud et al. and Charalambous and Evans predict that after restricting to ideals containing a regular sequence in degrees {a₁,…,a_n}, then {a₁,…,a_n}-lex plus powers ideals have extremal properties similar to those of the lex ideal. That is, it is proposed that a lex plus powers ideal should give maximum possible Hilbert function growth (Eisenbud et al.), and, after fixing a Hilbert function, that the Betti numbers of a lex plus powers ideal should be uniquely largest (Charalambous, Evans). The first of these assertions would extend Macaulay's theorem on Hilbert function growth, while the second improves the Bigatti, Hulett, Pardue theorem that lex ideals have largest graded Betti numbers. In this paper we explore these two conjectures. First we give several equivalent forms of each statement. For example, we demonstrate that the conjecture for Hilbert functions is equivalent to the statement that for a given Hilbert function, lex plus powers ideals have the most minimal generators in each degree. We use this result to prove that it is enough to show that lex plus powers ideals have the most minimal generators in the highest possible degree. A similar result holds for the stronger conjecture. In this paper we also prove that if the weaker conjecture holds, then lex plus powers ideals are guaranteed to have largest socles. This suffices to show that the two conjectures are equivalent in dimension ?3, which proves the monomial case of the conjecture for Betti numbers in those degrees. In dimension 2, we prove both conjectures outright.     

Borel-plus-powers monomial ideals

Let S=K[x₁,…,x_n] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing , where p is a prime number.     

Characteristic-independence of Betti numbers of graph ideals

In this paper, we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first 6 Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n=11, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion.In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field.     

Projective bundle ideals and Poincaré duality algebras

The study of maximal-primary irreducible ideals in a commutative graded connected Noetherian algebra over a field is in principle equivalent to the study of the corresponding quotient algebras. Such algebras are Poincaré duality algebras. A prototype for such an algebra is the cohomology with field coefficients of a closed oriented manifold. Topological constructions on closed manifolds often lead to algebraic constructions on Poincaré duality algebras and therefore also on maximal-primary irreducible ideals. It is the purpose of this note to examine several of these and develop some of their basic properties.     

Stabilization of Boij–Söderberg decompositions of ideal powers

Given an ideal I we investigate the decompositions of Betti diagrams of the graded family of ideals ${\{I^k\}}_k$ formed by taking powers of I. We prove conjectures of Engström from [5] and show that there is a stabilization in the Boij–Söderberg decompositions of $I^k$ for $k>>0$ when I is a homogeneous ideal with generators in a single degree. In particular, the number of terms in the decompositions with positive coefficients remains constant for $k>>0$, the pure diagrams appearing in each decomposition have the same shape, and the coefficients of these diagrams are given by polynomials in k. We also show that a similar result holds for decompositions with arbitrary coefficients arising from other chains of pure diagrams.     

Cohen-Macaulay monomial ideals with given radical

We study the set of Cohen-Macaulay monomial ideals with a given radical. Among this set of ideals are the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. It is shown that if there exists one such modification, then there exist indeed infinitely many.     

The Lex-Plus-Powers Conjecture holds for pure powers

We prove Evans' Lex-Plus-Powers Conjecture for ideals containing a monomial regular sequence.     

The rationality of the Hilbert-Kunz multiplicity in graded dimension two

We show that the Hilbert-Kunz multiplicity is a rational number for an R₊−primary homogeneous ideal I=(f₁, . . . , f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. More specific, we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f₁, . . . , f_n) on the projective curve Y=ProjR.