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We construct Kn\"orrer type equivalences outside of the hypersurface case,namely, between singularity categories of cyclic quotient surface singularitiesand certain finite dimensional local algebras. This generalises Kn\"orrer'sequivalence for singularities of Dynkin type A (between Krull dimensions $2$and $0$) and yields many new equivalences between singularity categories offinite dimensional algebras.  Our construction uses noncommutative resolutions of singularities, relativesingularity categories, and an idea of Hille & Ploog yielding stronglyquasi-hereditary algebras which we describe explicitly by building on Wemyss'swork on reconstruction algebras. Moreover, K-theory gives obstructions togeneralisations of our main result.

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