Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module $M$ over an Artin algebra $\Lambda$, there exists a positive integer $n_M$ such that for all finitely generated $\Lambda$-modules $N$, if \Ext_{\Lambda}^i(M,N)=0 for all $i\gg 0$, then \Ext_{\Lambda}^i(M,N)=0 for all $i\geq n_M$. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.Comment: 16 page

Free resolutions over short local rings

The structure of minimal free resolutions of finite modules M over commutative local rings (R,m,k) with m^3=0 and rank_k(m^2) < rank_k(m/m^2)is studied. It is proved that over generic R every M has a Koszul syzygy module. Explicit families of Koszul modules are identified. When R is Gorenstein the non-Koszul modules are classified. Structure theorems are established for the graded k-algebra Ext_R(k,k) and its graded module Ext_R(M,k).Comment: 17 pages; number of minor changes. This article will appear in the Journal of the London Math. So

Simplicial Resolutions of Powers of Square-free Monomial Ideals

The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.Comment: 32 pages, 3 figures, 1 tabl

Cohomology of finite modules over local rings

It is known that the powers [special characters omitted] of the maximal ideal [special characters omitted] of a local Noetherian ring R share certain homological properties for all sufficiently large integers n. When M is a finite R-module, Levin proved that the induced maps [special characters omitted] are zero for all large n and all i. In Chapter 1 we show that these maps are zero for all n \u3e pol reg M, where pol reg M denotes the Castelnuovo-Mumford regularity of the associated graded module [special characters omitted] over the symmetric algebra Symk([special characters omitted]). We also give a new application to the theory of Auslander\u27s delta invariants, by showing that [special characters omitted] = 0 for all i ≥ 0 and all n \u3e pol reg M; this extends and gives an effective version of a theorem of Yoshino. In Chapter 2 we deal with the base change in (co)homology induced by the natural ring homomorphisms R → R/[special characters omitted]. These maps are known to be Golod, respectively, small, for all large n. We determine bounds on the values of n for which these properties begin to hold. When R is a complete intersection, Avramov and Buchweitz proved that the asymptotic vanishing of [special characters omitted](−, −) is symmetric in the module variables and raised the question whether this property holds for all Gorenstein rings. Recently, Huneke and Jorgensen gave a positive answer for Gorenstein rings of minimal multiplicity. In Chapter 3 we answer the question positively for all Gorenstein rings of codimension at most 4