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Noncommutative Kn\"orrer type equivalences via noncommutative resolutions of singularities

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arxiv 1707.02836 v1 pith:KAI6FOOY submitted 2017-07-10 math.AG math.ACmath.RT

classification math.AGmath.ACmath.RT
keywords algebrassingularitiescategoriesequivalencesnoncommutativeorrersingularitytype
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abstract

We construct Kn\"orrer type equivalences outside of the hypersurface case, namely, between singularity categories of cyclic quotient surface singularities and certain finite dimensional local algebras. This generalises Kn\"orrer'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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Categorical absorptions of cone singularities

    math.AG 2026-07 conditional novelty 7.0 of 10

    For anticanonical cones over many Fano varieties, the derived category decomposes as a finite-dimensional algebra component together with two line bundles, and the algebra is explicitly a truncation of a Calabi-Yau co...

  2. The singularity category as a stable module category

    math.RT 2025-09 conditional novelty 7.0 of 10

    For left artinian rings with a suitable semisimple subring, the singularity category is triangle equivalent to the stable module category of the zeroth component of the Leavitt ring, which is an FC ring.

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