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Periodic free resolutions from twisted matrix factorizations. (English) Zbl 1378.16003

Summary: The notion of a matrix factorization was introduced by Eisenbud in the commutative case in his study of bounded (periodic) free resolutions over complete intersections. Since then, matrix factorizations have appeared in a number of applications. In this work, we extend the notion of (homogeneous) matrix factorizations to regular normal elements of connected graded algebras over a field.
Next, we relate the category of twisted matrix factorizations to an element over a ring and certain Zhang twists. We also show that in the setting of a quotient of a ring of finite global dimension by a normal regular element, every sufficiently high syzygy module is the cokernel of some twisted matrix factorization. Furthermore, we show that in the noetherian AS-regular setting, there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface, following work of D. Orlov [Prog. Math. 270, 503–531 (2009; Zbl 1200.18007)].

MSC:

16E05 Syzygies, resolutions, complexes in associative algebras
16E35 Derived categories and associative algebras
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)

Citations:

Zbl 1200.18007

Software:

BERGMAN; Macaulay2

References:

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