Tate modules as condensed modules

TL;DR

This paper establishes a fully faithful embedding of countable Tate modules into condensed modules over arbitrary rings, characterizes their essential image for finite type rings, and extends results to the $ abla$-categorical setting with boundedness conditions.

Contribution

It introduces a new embedding of Tate modules into condensed modules and characterizes their image for rings of finite type, extending to the $ abla$-categorical context with boundedness constraints.

Findings

Embedding of countable Tate modules into condensed modules is fully faithful.

Characterization of the essential image for finite type base rings.

Boundedness condition is necessary for fullness, demonstrated by a counterexample.

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.