Noncommutative Knörrer type equivalences via noncommutative resolutions of singularities

Abstract

We construct Knörrer type equivalences outside of the hypersurface case, namely, between singularity categories of cyclic quotient surface singularities and certain finite dimensional local algebras. This generalises Knörrer's equivalence for singularities of Dynkin type A (between Krull dimensions $2$ and $0$) and yields many new equivalences between singularity categories of finite dimensional algebras. Our construction uses noncommutative resolutions of singularities, relative singularity categories, and an idea of Hille & Ploog yielding strongly quasi-hereditary algebras which we describe explicitly by building on Wemyss's work on reconstruction algebras. Moreover, K-theory gives obstructions to generalisations of our main result.

arXiv commentComments welcome

References in corpus (4)

• Formal completions and idempotent completions of triangulated categories of singularities
• A Geometric Approach to Orlov's Theorem
• Relative singularity categories I: Auslander resolutions
• Relative Singularity Categories

Cited by in corpus (5)

• Singularity categories via the derived quotient
• A new equivalence between singularity categories of commutative algebras
• McKay correspondence, cohomological Hall algebras and categorification
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