Localization algebras and deformations of Koszul algebras

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.

39 pages; v3: final version

References in corpus (4)

• Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology
• Hypertoric category O
• A survey of hypertoric geometry and topology
• Cohomology of Spaltenstein varieties

Cited by in corpus (6)

• Hypertoric category O
• Cohomology of Spaltenstein varieties
• Ozsvath-Szabo bordered algebras and subquotients of category O
• From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron
• Cuspidal ${\mathfrak{sl}}_n$-modules and deformations of certain Brauer tree algebras
• Strands algebras and the affine highest weight property for equivariant hypertoric categories