# Algebra seminar

**Current seminar:**

**Leonid Positselski (IM CAS): Exact category structures and tensor products of topological vector spaces with linear topology**

**Abstract: **Abelian categories are rare in functional analysis or topological algebra, and the classical approach to developing homological algebra in topological settings has been to construct and use exact category structures. The quasi-abelian categories form a class of additive categories closest to the abelian ones; they have very natural exact category structures. The categories of all topological vector spaces or Hausdorff topological vector spaces are quasi-abelian, but the category of complete topological vector spaces is not, because the quotient space of a complete topological vector space by a closed subspace need not be complete. I will explain that the category of complete Hausdorff topological vector spaces is right, but not left quasi-abelian, contrary to a claim on the first page of a 2008 paper of Beilinson. Furthermore, Beilinson defined three important tensor product operations on topological vector spaces with linear topology and claimed that they are exact functors in the (nonexistent) quasi-abelian exact category structure. I will explain that at least two of the three tensor products are not exact in the maximal exact structure on topological vector spaces.

**Forthcomming:**

The Algebra Seminar was founded by Vladimir Korinek in the early 1950's and continued by Karel Drbohlav until 1981. The seminar resumed its activities in 1990 under the guidance of Jaroslav Jezek and Tomas Kepka. Since 1994, the seminar is headed by Jan Trlifaj.

Presently, the seminar is supported by GACR. It serves primarily as a platform for presentation of recent research of the visitors to the Department of Algebra as well as members of the Department and their students.