A contramodule generalization of Neeman's flat and projective module theorem

TL;DR

This paper extends classical theorems relating projective and flat modules to the setting of contramodules over a topological ring, establishing equivalences between various derived and homotopy categories.

Contribution

It introduces a contramodule generalization of Neeman's flat and projective module theorems, connecting homotopy and derived categories in this new context.

Findings

Homotopy category of projective contramodules is equivalent to the derived category of flat contramodules.

A complex of flat contramodules is contraacyclic iff it is acyclic with flat cocycles.

Establishes contramodule analogues of classical theorems by Neeman and others.

Abstract

This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left $\mathfrak R$-contramodules is equivalent to the derived category of the exact category of flat left $\mathfrak R$-contramodules, and also to the homotopy category of flat cotorsion left $\mathfrak R$-contramodules. In other words, a complex of flat $\mathfrak R$-contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat $\mathfrak R$-contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat $\mathfrak R$-contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortes-Izurdiaga, and Estrada.