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Noncommutative Knörrer periodicity and noncommutative Kleinian singularities. (English) Zbl 1490.16063

Summary: We establish a version of Knörrer’s Periodicity Theorem in the context of noncommutative invariant theory. Namely, let \(A\) be a left noetherian AS-regular algebra, let \(f\) be a normal and regular element of \(A\) of positive degree, and take \(B = A /(f)\). Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over \(B\) and those over (a noncommutative analog of) its second double branched cover \((B^{\#})^{\#}\). Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.

MSC:

16S38 Rings arising from noncommutative algebraic geometry
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16G50 Cohen-Macaulay modules in associative algebras
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
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References:

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