Periodic free resolutions from twisted matrix factorizations

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) ma...

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Bibliographic Details
Published in:Journal of algebra 2016-06, Vol.455, p.137-163
Main Authors: Cassidy, Thomas, Conner, Andrew, Kirkman, Ellen, Moore, W. Frank
Format: Article
Language:English
Subjects:
Matrix factorization
Maximal Cohen–Macaulay
Minimal free resolution
Singularity category
Zhang twist
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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 Orlov.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2016.01.037