2024/08/20 by Leonid Positselski, Positselski, Leonid
Mathematics · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2408.10928
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.