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

Francesco Baldassarri

TL;DR

This work develops a foundations-level framework for topological algebra over a linearly topologized base ring $k$, focusing on bounded $k$-modules and a finiteness system (op, clop, fop, ω-admissible) to enable exact quasi-abelian categories with closed monoidal structures. It introduces canonical $k$-modules $ ext{LM}^{ m can}_k$ as quotients of small direct sums of copies of $k$, proves they form a bicomplete quasi-abelian category with a compact projective generator, and constructs tensor products and internal Homs to obtain a closed symmetric monoidal structure on this canonical subcategory. The paper also develops a rich theory of limits and colimits for topological modules, including box products, boxed direct sums, and strict inductive limits, and analyzes separations, completions, barrelled/pseudocanonical structures, and naive canonical topologies, tying these to both formal schemes and non-archimedean analytic settings. Collectively, these results lay groundwork for quasi-coherent sheaves on formal schemes and adic spaces, bridging EGA-style bounded modules with analytic analogues and providing a robust platform for future unbounded generalizations. The constructions pave the way for a deeper understanding of tensorial operations, exactness properties, and dualities in topological module categories, with potential connections to condensed mathematics and analytic rings.

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.

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

TL;DR

This work develops a foundations-level framework for topological algebra over a linearly topologized base ring , focusing on bounded -modules and a finiteness system (op, clop, fop, ω-admissible) to enable exact quasi-abelian categories with closed monoidal structures. It introduces canonical -modules as quotients of small direct sums of copies of , proves they form a bicomplete quasi-abelian category with a compact projective generator, and constructs tensor products and internal Homs to obtain a closed symmetric monoidal structure on this canonical subcategory. The paper also develops a rich theory of limits and colimits for topological modules, including box products, boxed direct sums, and strict inductive limits, and analyzes separations, completions, barrelled/pseudocanonical structures, and naive canonical topologies, tying these to both formal schemes and non-archimedean analytic settings. Collectively, these results lay groundwork for quasi-coherent sheaves on formal schemes and adic spaces, bridging EGA-style bounded modules with analytic analogues and providing a robust platform for future unbounded generalizations. The constructions pave the way for a deeper understanding of tensorial operations, exactness properties, and dualities in topological module categories, with potential connections to condensed mathematics and analytic rings.

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.

Paper Structure

This paper contains 34 sections, 92 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Exact categories
  4. Basic definitions
  5. Projective, injective, flat objects
  6. Topological groups, rings, and modules
  7. Topological groups
  8. Topological rings and modules
  9. Boundedness
  10. Separation and separated completion
  11. Uniform and clop rings
  12. Limits of topological modules
  13. Limits and left-adjoints
  14. Box products
  15. Clop, barrelled and pseudocanonical modules.
  16. ...and 19 more sections

Key Result

Proposition 1.1.5

Let ${\mathcal{C}}$ be an additive category and let ${\mathcal{E}}$ be the family of all sequences exseq0 in ${\mathcal{C}}$ where $(u,v)$ is a kernel-cokernel pair. Then ${\mathcal{E}}$ is an exact structure on ${\mathcal{C}}$ iff

Theorems & Definitions (277)

  • Remark 1.1.2
  • Definition 1.1.3
  • Proposition 1.1.5
  • proof
  • Definition 1.1.6
  • Proposition 1.1.7
  • Remark 1.1.8
  • Definition 1.1.9
  • Remark 1.1.10
  • Definition 1.2.1
  • ...and 267 more