Selecta Mathematica

, 17:533 | Cite as

Localization algebras and deformations of Koszul algebras

  • Tom Braden
  • Anthony Licata
  • Christopher Phan
  • Nicholas ProudfootEmail author
  • Ben Webster


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 \({\mathcal{O}}\) 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.

Mathematics Subject Classification (2000)

Primary 16S37 Secondary 55N91 16S80 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Tom Braden
    • 1
  • Anthony Licata
    • 2
  • Christopher Phan
    • 3
  • Nicholas Proudfoot
    • 4
    Email author
  • Ben Webster
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Department of MathematicsStanford UniversityPalo AltoUSA
  3. 3.Department of MathematicsBucknell UniversityLewisburgUSA
  4. 4.Department of MathematicsUniversity of OregonEugeneUSA

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