Purity, ascent, and periodicity for Gorenstein flat cotorsion modules

TL;DR

This paper explores the pure structure and properties of Gorenstein flat cotorsion modules within Frobenius categories, establishing new conditions for scalar extension and revealing a triangulated equivalence related to hypersurface singularities.

Contribution

It introduces a detailed analysis of purity in Gorenstein flat cotorsion modules and extends Knörrer periodicity to certain singularity categories, providing new insights into their structure.

Findings

Pure structure of the stable category relates to Gorenstein flat modules.

Extension of scalars can preserve Gorenstein flat cotorsion modules under certain conditions.

Triangulated equivalence between singularity categories of hypersurfaces and their double covers.

Abstract

We investigate purity within the Frobenius category of Gorenstein flat cotorsion modules, which can be seen as an infinitely generated analogue of the Frobenius category of Gorenstein projective objects. As such, the associated stable category can be viewed as an alternative approach to a big singularity category, which is equivalent to Krause's when the ring is Gorenstein. We study the pure structure of the stable category, and show it is fundamentally related to the pure structure of the Gorenstein flat modules. Following that, we give conditions for extension of scalars to preserve Gorenstein flat cotorsion modules. In this case, one obtains an induced triangulated functor on the stable categories. We show that under mild conditions that these functors preserve the pure structure, both on the triangulated and module category level. Along the way, we consider particular phenomena over commutative rings, the cumulation of which is an extension of Kn\"{o}rrer periodicity, giving a triangulated equivalence between Krause's big singularity categories for a complete hypersurface singularity and its twofold double-branched cover.