# Algebra seminar

**Current seminar:**

**Matt Booth (Univ. Lancaster): Global Koszul duality**

**Abstract**: Koszul duality is the name given to various duality phenomena between dg algebras and dg coalgebras involving the bar and cobar constructions. These results are usually expressed as the existence of model structures on the above categories for which the bar-cobar adjunction is a Quillen equivalence. Koszul duality can be found across many different parts of mathematics - for example, the fact that noncommutative formal moduli problems are controlled by associative algebras is the Koszul duality between dg algebras and conilpotent dg coalgebras. More familiar may be the related correspondence between formal moduli problems and dg Lie algebras, which is commutative-Lie Koszul duality.

I'll survey the above ideas and results, before turning to the global setting, where one wants to work with coalgebras that are not necessarily conilpotent - geometrically, this corresponds to using all finite-dimensional algebras as bases over which to deform, not just the Artinian local ones. I'll talk about the existence of model structures for global Koszul duality, for which the (extended) bar-cobar adjunction is a Quillen equivalence. New here is that one has to use a new notion of weak equivalences for both algebras and coalgebras, defined in a more symmetric manner than those for usual conilpotent Koszul duality.

This is joint work with Andrey Lazarev, hopefully to appear on the ArXiv soon.

**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.