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Exact categories of topological vector spaces with linear topology

Leonid Positselski

Abstract

We explain why the naive definition of a natural exact category structure on complete, separated topological vector spaces with linear topology fails. In particular, contrary to arXiv:0711.2527, the category of such topological vector spaces is not quasi-abelian. We present a corrected definition of exact category structure which works OK. Then we explain that the corrected definition still has a shortcoming in that a natural tensor product functor is not exact in it, and discuss ways to refine the exact category structure so as to make the tensor product functors exact.

Exact categories of topological vector spaces with linear topology

Abstract

We explain why the naive definition of a natural exact category structure on complete, separated topological vector spaces with linear topology fails. In particular, contrary to arXiv:0711.2527, the category of such topological vector spaces is not quasi-abelian. We present a corrected definition of exact category structure which works OK. Then we explain that the corrected definition still has a shortcoming in that a natural tensor product functor is not exact in it, and discuss ways to refine the exact category structure so as to make the tensor product functors exact.

Paper Structure

This paper contains 13 sections, 62 theorems, 29 equations.

Table of Contents

  1. Abelian Groups with Linear Topology
  2. The Construction of Counterexamples
  3. Vector Spaces with Linear Topology
  4. Exact Categories
  5. Quasi-Abelian Categories
  6. Maximal Exact Structures
  7. Categories of Incomplete VSLTs are Quasi-Abelian
  8. The Maximal Exact Category Structure on VSLTs
  9. Pro-Vector Spaces
  10. Supplemented Pro-Vector Spaces
  11. The Strong Exact Category Structure on VSLTs
  12. Beilinson's Yoga of Tensor Product Operations
  13. Refined Exact Category Structures on VSLTs

Key Result

Lemma 1.1

Let $\mathfrak B$ be a complete, separated topological abelian group, and let $A\subset\mathfrak B$ be a subgroup, endowed with the induced topology. Then the morphism of completions $A\sphat\,\longrightarrow\mathfrak B\sphat=\mathfrak B$ induced by the inclusion $A\longrightarrow\mathfrak B$ identi

Theorems & Definitions (139)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 129 more