Lattices over finite group schemes and stratification

TL;DR

This paper studies the structure of lattices over finite flat group schemes, showing their stable categories are stratified and costratified by cohomology, with applications to support, cosupport, and ideal classification.

Contribution

It proves the stable category of lattices over finite group schemes is stratified and costratified by cohomology, providing new tools for understanding their module categories.

Findings

Stable category is rigidly-compactly generated and tensor triangulated.

Supports and cosupports can be computed explicitly.

Classification of thick ideals in the stable category.

Abstract

This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full subcategory of all $G$-modules projective over $R$ generated by the lattices. The stable category of such $G$-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of $G$. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.