Localization algebras and deformations of Koszul algebras

TL;DR

This paper explores the structure of centers of deformed Koszul algebras, revealing their geometric and topological significance through connections to equivariant cohomology and duality principles.

Contribution

It introduces a universal deformation framework for Koszul algebras and links their centers to equivariant cohomology of geometric varieties, generalizing known dualities.

Findings

Center of universal deformation supported on hyperplane arrangements

Isomorphism between algebraic centers and equivariant cohomology rings

Generalization of Goresky-MacPherson duality in algebraic setting

Abstract

We show that the center of a flat graded deformation of a standard Koszul algebra behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming the Koszul dual algebra. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category $\mathcal{O}$ for $\mathfrak{gl}_n$ is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category $\mathcal{O}$" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.