Periodic Free Resolutions from Twisted Matrix Factorizations
School of Science
Journal of Algebra
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.
Matrix factorization, Zhang twist, Singularity category, Minimal free resolution, Maximal Cohen–Macaulay
SMC Affiliated Work
Andrew Conner (Mathematics and Computer Science): "Periodic free resolutions from twisted matrix factorizations," by T. Cassidy, A. Conner, E. Kirkman, W.F. Moore, in the Journal of Algebra, Volume 455, June 2016, pp. 137-163. https://doi.org/10.1016/j.jalgebra.2016.01.037
Cassidy, Thomas; Conner, Andrew; Kirkman, Ellen; and Moor, W. Frank. Periodic Free Resolutions from Twisted Matrix Factorizations (2016). Journal of Algebra. 455, 137-163. 10.1016/j.jalgebra.2016.01.037 [article]. https://digitalcommons.stmarys-ca.edu/school-science-faculty-works/11