The Auslander-Reiten translate on monomial quotient rings

TL;DR

This paper studies the Auslander-Reiten translate on positively t-determined modules over polynomial rings, computing cohomology and Betti spaces for iterates, extending classical results on local cohomology of Stanley-Reisner rings.

Contribution

It provides explicit calculations of cohomology and Betti spaces of the Auslander-Reiten translate iterates, generalizing Hochster and Gr"abe's results.

Findings

Computed multigraded cohomology and Betti spaces for Na_t^k(S/I)

Determined the S-module structure of these cohomology modules

Extended classical local cohomology results to a broader setting

Abstract

For a multidegree t in N^n, E.Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander-Reiten translate, Na_t, on the (derived) category of such modules. A monomial ideal I is positively t-determined if every generator x^a has a \leq t. We compute the multigraded cohomology- and betti spaces of Na_t^k(S/I) for every iterate k, and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gr\"abe on local cohomology of Stanley-Reisner rings.