# Localization algebras and deformations of Koszul algebras - Mathematics > Rings and Algebras

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

Autor: Tom Braden, Anthony Licata, Christopher Phan, Nicholas Proudfoot, Ben Webster

Fuente: https://arxiv.org/