Frobenius categories, Gorenstein algebras and rational surface singularities

TL;DR

This paper establishes conditions under which Frobenius categories relate to Gorenstein projective modules and applies these to rational surface singularities, revealing new connections in algebraic and geometric contexts.

Contribution

It provides a framework linking Frobenius categories with Gorenstein rings and applies this to rational surface singularities, producing new Iwanaga-Gorenstein rings and results in representation theory.

Findings

Stable categories are triangle equivalent to singularity categories of partial resolutions.

Uncountably many Iwanaga-Gorenstein rings of finite GP type are constructed.

New results in Auslander-Solberg and Kong type representation theory are obtained.

Abstract

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite GP type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results.