Commutative algebraic groups up to isogeny. II

Abstract

This paper develops a representation-theoretic approach to the isogeny category $\underline{\mathcal{C}}$ of commutative group schemes of finite type over a field $k$, studied in arXiv:1602:00222. We construct a ring $R$ such that $\underline{\mathcal{C}}$ is equivalent to the category $R$-mod of all left $R$-modules of finite length. We also construct an abelian category of $R$-modules, $R$-$\widetilde{\rm mod}$, which is hereditary, has enough projectives, and contains $R$-mod as a Serre subcategory; this yields a more conceptual proof of the main result of [loc. cit.], asserting that $\underline{\mathcal{C}}$ is hereditary. We show that $R$-$\widetilde{\rm mod}$ is equivalent to the isogeny category of commutative quasi-compact $k$-group schemes.