Comodules, contramodules and Pontryagin duality

TL;DR

This paper explores dualities among categories of comodules, contramodules, and modules over coalgebras, completing a square of dualities and extending a known isomorphism in the context of pseudocompact and discrete modules.

Contribution

It introduces the category of pseudocompact right contramodules and completes the duality square, extending Takeuchi's isomorphism.

Findings

Established dualities between pseudocompact modules, comodules, and contramodules.

Completed the duality square by including pseudocompact right contramodules.

Extended Takeuchi's natural isomorphism within this duality framework.

Abstract

Let $C$ be a $k$-coalgebra, where $k$ is a field. The category of pseudocompact left $C^*$-modules is dual to both the category of discrete right $C^*$-modules and to the category of left $C$-comodules. We obtain this way two sides of a square of dualities. In this note we complete the square by introducing as fourth corner the category of pseudocompact right $C$-contramodules. We use these dualities to extend a natural isomorphism of Takeuchi.