Hochschild cohomology and derived categories

June 20—26, 2022


The event will focus on topics related to homological aspects of the representation theory of associative algebras, the main focus being its most famous derived invariant: the Hochschild (co)homology.

Derived categories were introduced and developed by Grothendieck and Verdier in the early 1960s, at which point they were needed to state and prove the extensions of Serre’s duality theorem that Grothendieck had announced at the 1958 ICM. Since then, derived categories have received extensive attention in many different research areas of mathematics. To illustrate the ubiquity of derived categories, one needs only to mention applications in microlocal analysis (following the work of Sato and Kashiwara in the 1970s adapting the techniques of Grothendieck and Verdier to study systems of PDEs) or representation theory of Lie groups and finite Chevalley groups (following the proof of the Kazhdan–Lusztig conjecture by Brylinski and Kashiwara).

In research areas like algebraic geometry and representation theory of algebras, derived categories proved irreplaceable, offering among other things a foundation for the common generalization of the two areas: noncommutative algebraic geometry. One question of utmost importance here is to determine when two Abelian categories have equivalent derived categories. One very subtle invariant used to answer this question is given by the Hochschild (co)homology. Of course, this cohomology theory is invaluable in other research areas, most importantly, the deformation theory of algebras and modules.

Thu Nov 18 2021 14:10:51 GMT+0300 (Moscow Standard Time)