Commutative algebraic groups up to isogeny. II

TL;DR

This paper introduces a representation-theoretic framework for understanding the category of commutative algebraic groups up to isogeny, establishing an equivalence with module categories over a constructed ring and providing new insights into their hereditary properties.

Contribution

It constructs a ring R such that the isogeny category of commutative group schemes is equivalent to R-modules, and develops an abelian category of R-modules that clarifies the hereditary nature of this category.

Findings

Established an equivalence between the isogeny category and R-modules.

Constructed a hereditary abelian category of R-modules with enough projectives.

Provided a conceptual proof of the hereditary property of the isogeny category.

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.