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)].


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.)


Zbl 1200.18007


BERGMAN; Macaulay2
Full Text: DOI arXiv


[1] Anick, David J., Noncommutative graded algebras and their Hilbert series, J. Algebra, 78, 1, 120-140 (1982), MR 677714 (84g:16001) · Zbl 0502.16002
[2] Backelin, Jørgen; Cojocaru, Svetlana; Ufnarovski, Victor, Bergman, a system for computations in commutative and non-commutative algebra, Available at: · Zbl 1084.13503
[3] Ballard, Matthew; Deliu, Dragas; Favero, David; Umut Isik, M.; Katzarkov, Ludmil, Resolutions in factorization categories · Zbl 1353.13016
[4] Buchweitz, Ragnar-Olaf, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, Available at: · Zbl 1062.16012
[5] Eisenbud, David, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc., 260, 1, 35-64 (1980), MR 570778 (82d:13013) · Zbl 0444.13006
[6] Etingof, Pavel; Ginzburg, Victor, Noncommutative complete intersections and matrix integrals, Special Issue: In Honor of Robert D. MacPherson. Part 3. Special Issue: In Honor of Robert D. MacPherson. Part 3, Pure Appl. Math. Q., 3, 1, 107-151 (2007), MR 2330156 (2008b:16044) · Zbl 1151.14006
[7] Gelfand, Sergei I.; Manin, Yuri I., Methods of Homological Algebra, Springer Monographs in Mathematics (2003), Springer-Verlag: Springer-Verlag Berlin, MR 1950475 (2003m:18001)
[8] Golod, E. S., Homology of the Shafarevich complex, and noncommutative complete intersections, Fundam. Prikl. Mat., 5, 1, 85-95 (1999), MR 1800120 (2001k:16012) · Zbl 0962.16006
[9] Golod, E. S.; Šafarevič, I. R., On the class field tower, Izv. Ross. Akad. Nauk Ser. Mat., 28, 261-272 (1964), MR 0161852 (28 #5056) · Zbl 0136.02602
[10] Grayson, Daniel R.; Stillman, Michael E., Macaulay2, a software system for research in algebraic geometry, Available at:
[11] Jørgensen, Peter, Non-commutative graded homological identities, J. Lond. Math. Soc. (2), 57, 2, 336-350 (1998), MR 1644217 (99h:16010) · Zbl 0922.16025
[12] Jørgensen, Peter, Ext vanishing and infinite Auslander-Buchsbaum, Proc. Amer. Math. Soc., 133, 5, 1335-1341 (2005), (electronic), MR 2111939 (2005k:16018) · Zbl 1076.16005
[13] Kirkman, E.; Kuzmanovich, J.; Zhang, J. J., Invariant theory of finite group actions on down-up algebras, Transform. Groups, 20, 1, 113-165 (2015), MR 3317798 · Zbl 1356.16021
[14] Kirkman, E.; Kuzmanovich, J.; Zhang, J. J., Noncommutative complete intersections, J. Algebra, 429, 253-286 (2015), MR 3320624 · Zbl 1360.16012
[15] Lam, T. Y., Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189 (1999), Springer-Verlag: Springer-Verlag New York, MR 1653294 (99i:16001) · Zbl 0911.16001
[16] Leuschke, Graham J.; Wiegand, Roger, Cohen-Macaulay Representations, Mathematical Surveys and Monographs, vol. 181 (2012), American Mathematical Society: American Mathematical Society Providence, RI, MR 2919145 · Zbl 1252.13001
[17] Orlov, Dmitri, Derived categories of coherent sheaves and triangulated categories of singularities, (Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. II. Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. II, Progr. Math., vol. 270 (2009), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA), 503-531, MR 2641200 (2011c:14050) · Zbl 1200.18007
[18] Polishchuk, Alexander; Positselski, Leonid, Quadratic Algebras, University Lecture Series, vol. 37 (2005), American Mathematical Society: American Mathematical Society Providence, RI, MR MR2177131 · Zbl 1145.16009
[19] Zhang, J. J., Twisted graded algebras and equivalences of graded categories, Proc. Lond. Math. Soc. (3), 72, 2, 281-311 (1996), MR 1367080 (96k:16078) · Zbl 0852.16005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.