Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

The triangular spectrum of matrix factorizations is the singular locus


Author: Xuan Yu
Journal: Proc. Amer. Math. Soc. 144 (2016), 3283-3290
MSC (2010): Primary 18E30; Secondary 13D02
DOI: https://doi.org/10.1090/proc/13001
Published electronically: February 2, 2016
MathSciNet review: 3503696
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

The singularity category of a ring/scheme is a triangulated category defined to capture the singularities of the ring/scheme. In the case of a hypersurface $ R/f$, it is given by the homotopy category of matrix factorizations $ [MF(R,f)]$. In this paper, we apply Balmer's theory of tensor triangular geometry to matrix factorizations by taking into consideration their tensor product. We show that the underlying topological space of the triangular spectrum of $ [MF(R,f)]$ is the singular locus of the hypersurface by using a support theory developed by M. Walker.


References [Enhancements On Off] (What's this?)

  • [1] Paul Balmer, Presheaves of triangulated categories and reconstruction of schemes, Math. Ann. 324 (2002), no. 3, 557-580. MR 1938458 (2003j:18016), https://doi.org/10.1007/s00208-002-0353-1
  • [2] Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149-168. MR 2196732 (2007b:18012), https://doi.org/10.1515/crll.2005.2005.588.149
  • [3] Paul Balmer, Tensor triangular geometry, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 85-112. MR 2827786 (2012j:18016)
  • [4] Ragnar-Olaf Buchweitz,
    Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings,
    Unpublished manuscript, page 155, 1987.
  • [5] Tobias Dyckerhoff and Daniel Murfet, The Kapustin-Li formula revisited, Adv. Math. 231 (2012), no. 3-4, 1858-1885. MR 2964627, https://doi.org/10.1016/j.aim.2012.07.021
  • [6] T. Dyckerhoff and D. Murfet,
    Pushing forward matrix factorisations,
    arXiv preprint arXiv:1102.2957, 2011.
  • [7] David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. MR 570778 (82d:13013), https://doi.org/10.2307/1999875
  • [8] D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240-262 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3 (246) (2004), 227-248. MR 2101296 (2006i:81173)
  • [9] Greg Stevenson, Subcategories of singularity categories via tensor actions, Compos. Math. 150 (2014), no. 2, 229-272. MR 3177268, https://doi.org/10.1112/S0010437X1300746X
  • [10] Ryo Takahashi, Classifying thick subcategories of the stable category of Cohen-Macaulay modules, Adv. Math. 225 (2010), no. 4, 2076-2116. MR 2680200 (2011h:13014), https://doi.org/10.1016/j.aim.2010.04.009
  • [11] Mark E Walker,
    Notes for Walker's matrix factorization seminar,
    Spring 2012.
  • [12] Xuan Yu, Geometric study of the category of matrix factorizations, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)-The University of Nebraska - Lincoln. MR 3187316
  • [13] Xuan Yu, Chern character for matrix factorizations via Chern-Weil, J. Algebra 424 (2015), 416-447. MR 3293227, https://doi.org/10.1016/j.jalgebra.2014.09.024

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 18E30, 13D02

Retrieve articles in all journals with MSC (2010): 18E30, 13D02


Additional Information

Xuan Yu
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Email: xuanyumath@gmail.com

DOI: https://doi.org/10.1090/proc/13001
Received by editor(s): November 5, 2014
Received by editor(s) in revised form: October 9, 2015
Published electronically: February 2, 2016
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society