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

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.