Compact generators of the contraderived category of contramodules

TL;DR

This paper studies the contraderived category of contramodules over a topological ring, showing it is compactly generated and characterizing its compact objects, with applications to flat and projective periodicity.

Contribution

It establishes conditions under which the contraderived category of contramodules is compactly generated and describes its compact objects, extending the understanding of contramodule homotopy categories.

Findings

Contraderived category is compactly generated under certain conditions.

Full subcategory of compact objects is described as the opposite of a derived category.

Proves flat and projective periodicity theorems for contramodules.

Abstract

We consider the contraderived category of left contramodules over a right linear topological ring $\mathfrak R$ with a countable base of neighborhoods of zero. Equivalently, this is the homotopy category of unbounded complexes of projective left $\mathfrak R$-contramodules. Assuming that the abelian category of discrete right $\mathfrak R$-modules is locally coherent, we show that the contraderived category of left $\mathfrak R$-contramodules is compactly generated, and describe its full subcategory of compact objects as the opposite category to the bounded derived category of finitely presentable discrete right $\mathfrak R$-modules. Under the same assumptions, we also prove the flat and projective periodicity theorem for $\mathfrak R$-contramodules.