Purity, ascent, and periodicity for Gorenstein flat cotorsion modules
Isaac Bird
TL;DR
This paper develops purity theory for the unbounded Frobenius/triangulated setting of Gorenstein flat cotorsion modules, introducing the stable category $\underline{{\mathsf{GFlatCot}}}(R)$ as an analogue of the big singularity category. It establishes how purity and definable structures behave under change of rings via ascent of scalars, proving that extension along $R\to S$ yields triangulated functors that preserve products and, in favorable cases, coproducts, with the pure-injective and definable data transported accordingly. A key advancement is extending Knörrer periodicity to $\underline{{\mathsf{GFlatCot}}}(R)$ for hypersurface singularities, yielding a triangulated equivalence with the twofold double branched cover, and linking Ziegler spectra across these spaces. The work also analyzes kernels of extension-of-scalars, recollements arising from epimorphisms, and commutative-ring phenomena such as regular sequences, providing a cohesive framework for purity in unbounded Gorenstein settings. Overall, the results connect purity, definability, and stable/categorical dimensions to big singularity theory and broaden Knörrer-type dualities in Gorenstein homological algebra.
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 Knörrer periodicity, giving a triangulated equivalence between Krause's big singularity categories for a complete hypersurface singularity and its twofold double-branched cover.