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This is a Quantitative Study study.

October 5, 2025Open Access

Closed exact categories of modules over generalized adic rings. Part 1: The bounded case

Key Points

Developing foundations in topological algebra yields significant insights into modules over generalized adic rings.
Quasi-abelian categories are defined, focusing on complete linearly topologized k-modules with closed symmetric monoidal structures.
Examples of bounded modules demonstrate the project's depth and applicability to both formal schemes and analytic adic spaces.
The findings suggest potential for establishing more categories with enough projectives or injectives linked to these structures.

Abstract

We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining quasi-abelian categories of complete linearly topologized k-modules, which are also closed symmetric monoidal for a suitable choice of tensor product and internal Hom, and have enough projectives or injectives. For k a suitably generalized adic ring, we describe here a few examples of such categories consisting of bounded modules.

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Francesco Baldassarri (Wed,) studied this question.

synapsesocial.com/papers/68e2537cd6d66a53c24743cb https://doi.org/https://doi.org/10.48550/arxiv.2510.00848