Idempotent modules, locus of compactness and local supports

TL;DR

This paper explores the structure of modules over group algebras of finite group schemes in characteristic p, introducing local support concepts based on idempotent modules and their properties, with implications for module classification.

Contribution

It develops a new framework for local supports of modules using idempotent modules and their restrictions, extending the understanding of module categories over group schemes.

Findings

Idempotent modules have restrictions that are compact objects along flat maps.

The stable Hom from an idempotent module to any module is finitely generated under certain conditions.

A realization theorem for local supports is established.

Abstract

Let $kG$ be the group algebra of a finite group scheme defined over a field $k$ of characteristic $p>0$. Associated to any closed subset $V$ of the projectivized prime ideal spectrum $\operatorname{Proj} \operatorname{H}^*(G,k)$ is a thick tensor ideal subcategory of the stable category of finitely generated $kG$-module, whose closure under arbitrary direct sums is a localizing tensor ideal in the stable category of all $kG$-modules. The colocalizing functor from the big stable category to this localizing subcategory is given by tensoring with an idempotent module $\mathcal{E}$. A property of the idempotent module is that its restriction along any flat map $\alpha:k[t]/(t^p) \to kG$ is a compact object. For any $kG$-module $M$, we define its locus of compactness in terms of such restrictions. With some added hypothesis, in the case that $V$ is a closed point, for a $kG$-module $M$, we show that in the stable category $\operatorname{Hom}(\mathcal{E}, M)$ is finitely generated over the endomorphism ring of $\mathcal{E}$, provided the restriction along an associated flat map is a compact object. This leads to a notion of local supports. We prove some of its properties and give a realization theorem.