Idempotent modules, locus of compactness and local supports

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.