Cohomology of finite modules over local rings

Liana M Sega, Purdue University

Abstract

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 > 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's delta invariants, by showing that [special characters omitted] = 0 for all i ≥ 0 and all n > 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.

Degree

Ph.D.

Advisors

Avramov, Purdue University.

Subject Area

Mathematics

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