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BIRS Workshop Lecture Videos

Non-Commutative Resolution of Singularities for Toric varieties Smith, Karen


Consider a finitely generated normal commutative algebra R over a field K. A non-commutative resolution of singularities of Spec R is a (non-commutative) R-algebra A with finite global dimension of the form End(M) where M is some finitely generated reflexive R-module. The existence of a non-commutative resolution for a commutative ring R places strong conditions on R, such as rational singularities. In this talk, we discuss how in prime characteristic, the Frobenius can be used to construct non-commutative resolutions of nice enough rings. We conjecture that for a strongly F-regular ring R, End(F_*R) is a non-commutative resolution of R, where F_*R denotes R viewed as an R-module via restriction of scalars from Frobenius. We prove this conjecture when R is the coordinate ring of an affine toric variety. We also show that for toric rings, the ring of differential operators D(R) has finite global dimension (joint with Eleonore Faber and Greg Muller).

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