Leonid Positselski, Czech Academy of Science, Prague

Title: Countably generated endo-Sigma-coperfect modules have perfect decomposition

Abstract: In a 2006 paper of Angeleri Hugel and Saorin, it was shown that any module with a perfect decomposition is Sigma-coperfect over its endomorphism ring, and a question was asked whether the converse implication holds. In this talk, I will present a topological algebra approach to this problem. The ring of endomorphisms of any module is endowed with the so-called finite topology, making it a complete, separated topological ring with a base of neighborhoods of zero formed by open right ideals. Both the existence of a perfect decomposition and the endo-Sigma-coperfectness properties of a module are shown to be equivalent to certain properties of the topological ring of endomorphisms. The equivalence of these two properties of a topological ring is an assertion extending Bass' famous Theorem P to the realm of topological rings. One can prove it for commutative topological rings and for topological rings with a countable base of neighborhoods of zero. It follows that the answer to the question of Angeleri Hugel and Saorin is positive for modules with a commutative endomorphism ring and for countably generated modules. This talk is based on a joint work of Jan Stovicek and the speaker.