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Tate modules as condensed modules

Valerio Melani, Hugo Pourcelot, Gabriele Vezzosi

Abstract

We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free module of infinite countable rank under direct sums, duals and retracts. In the $\infty$-categorical context, under the same assumption on the base ring, we establish a fully faithful embedding of the $\infty$-category of countable Tate objects in perfect complexes, with uniformly bounded tor-amplitude, into the derived $\infty$-category of condensed modules. The boundedness assumption is necessary to ensure fullness, as we prove via an explicit counterexample in the unbounded case.

Tate modules as condensed modules

Abstract

We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free module of infinite countable rank under direct sums, duals and retracts. In the -categorical context, under the same assumption on the base ring, we establish a fully faithful embedding of the -category of countable Tate objects in perfect complexes, with uniformly bounded tor-amplitude, into the derived -category of condensed modules. The boundedness assumption is necessary to ensure fullness, as we prove via an explicit counterexample in the unbounded case.
Paper Structure (9 sections, 15 theorems, 48 equations)

This paper contains 9 sections, 15 theorems, 48 equations.

Table of Contents

  1. Embedding of Tate modules in condensed modules
  2. The embedding using topology
  3. A non-topological description of the embedding
  4. Essential image of the embedding over a ring of finite type
  5. Duality for Tate modules
  6. Duality for solid modules
  7. Characterization of the essential image
  8. Infinity-categorical embedding in the bounded case
  9. Counter-example to fully faithfulness in the unbounded case

Key Result

Proposition 1.2

Let $R$ be a commutative ring. The fully faithful inclusion $\mathsf{Tate}^{\mathrm{Dr}}_{{\aleph}_0}(R) \to \mathsf{TopMod}_{R}$ factors through the fully faithful inclusion $\mathsf{TopMod}^{\mathrm{cgwh}}_{R} \to \mathsf{TopMod}_{R}$, i.e. the underlying topological space of an object in $\mathsf

Theorems & Definitions (38)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • Lemma 1.6
  • proof
  • ...and 28 more