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.