Algebraic Geometry Seminar

Travis SchedlerImperial
Tate—Hochschild cohomology, matrix factorisations, and singularities of deformations

Friday, December 18, 2020 - 11:30am
Virtual (note the different time)

Tate—Hochschild cohomology is an analogue for associative algebras of Tate cohomology of groups. Keller proved it is isomorphic to the Hochschild cohomology of singularity categories. For a hypersurface this is also known as the matrix factorisation category. I will explain an explicit approach to compute it with its algebraic structure in the hypersurface case. It turns out that its deformations correspond to (commutative) deformations of the original singularity. As a result, every noncommutative deformation (aka quantisation) has “equivalent singularities” to a commutative one. I will sketch conjectural extensions to symplectic singularities. Finally if time allows I will compare this Z-differential graded story to the often more well studied Z/2-dg version (appearing eg in work of Dyckerhoff and others).