Algebraický seminář

Leonid Positselski (IM CAS): The pseudopullback diagram of contramodule forgetful functors

Abstract: Let f: R --> S be a homomorphism of complete, separated two-sided linear topological rings with countable bases of neighborhoods of zero. Consider the square diagram of forgetful functors between the contramodule and module categories S-Contra, R-Contra, S-Mod, and R-Mod. Under certain flatness assumption on f, together with the assumption that the forgetful functor R-Contra --> R-Mod is fully faithful (which happens more often than one might think), we prove that this is a pseudopullback diagram of categories. In other words, any datum of an R-contramodule and an S-module structures on the same set for which the underlying R-module structures agree extends uniquely and functorially to an S-contramodule structure. If one is only interested in separated contramodules, then the flatness assumption can be relaxed, but a counterexample shows that for nonseparated contramodules the relaxed assumption is not sufficient. The proof uses a construction of a tower of epimorphisms in the pseudopullback category by transfinite induction up to the successor cardinal of the cardinality of the (contra)module. The nonseparatedness kernel gets shrunk in the tower. This work is motivated by the theory of contraherent cosheaves of contramodules on formal schemes.