Categories of comodules and chain complexes of modules

TL;DR

This paper establishes an equivalence between the category of comodules over certain bialgebroids and chain complexes of modules, generalizing known results from vector spaces to modules over rings using advanced non-commutative techniques.

Contribution

It introduces a new approach using non-commutative Tannaka reconstruction to relate comodules over bialgebroids to chain complexes of modules, extending previous field-based results.

Findings

Category of comodules over epimorphic images of $L(A)$ is equivalent to chain complexes of $R$-modules.

Equivalence is monoidal when $R$ is commutative and $A$ is an $R$-algebra.

Generalizes results by Pareigis and Tambara beyond vector spaces over fields.

Abstract

Let $\lL(A)$ denote the coendomorphism left $R$-bialgebroid associated to a left finitely generated and projective extension of rings $R \to A$ with identities. We show that the category of left comodules over an epimorphic image of $\lL(A)$ is equivalent to the category of chain complexes of left $R$-modules. This equivalence is monoidal whenever $R$ is commutative and $A$ is an $R$-algebra. This is a generalization, using entirely new tools, of results by B. Pareigis and D. Tambara for chain complexes of vector spaces over fields. Our approach relies heavily on the non commutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.