Localization algebras and deformations of Koszul algebras

Abstract

We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular, the center of A acts by characters on the deformed standard modules, providing a "localization map". We construct a universal graded deformation of A and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. 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 O for gln is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category O" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.