Localization algebras and deformations of Koszul algebras

Abstract

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 gl n 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.

On the Spectrum of the Equivariant Cohomology Ring

“…Note added in proof: In the intervening years since this article was submitted for publication, a general setting for this duality has been developed by Braden, Licata, Phan, Proudfoot, and Webster [5].…”

“…It was implicitly considered in [2,3], and explicitly in [6], [16, p. 27], [7, p. 83], and [8,Theorem 2]. We also wish to draw attention to the related articles [5,11,19,[22][23][24][25][26]32] and to the unrelated article [27].…”

Cohomology of Spaltenstein varieties

“…The precise form of the relations (1.4)-(1.5) was originally worked out in [B], which is concerned with the centers of integral blocks of parabolic category O for the general linear Lie algebra. These centers provide another natural occurrence of the algebras C λ µ ∼ = H * (X λ µ , C); see also [St,Theorem 4.1.1] (which treats regular µ only) and [BLPPW,Remark 9.10] (for all µ).…”