Basic information
The seminar has already finished.
The seminar is scheduled on Thursdays, from 10.40am, in the seminar room of the Dept. of Algebra and currently runing in a hybrid mode (contact me if you do not have Zoom coordinates). The topis is matrix factorizations and related areas of mathematics.
The initial idea is easy: given a commutative ring S and an element w∈S, one studies pairs of square matrices A, B of the same size over S such that A·B=w·I and B·A=w·I. Even if w is an irreducible element, it may have a non-trivial matrix factorization with matrices of size greater than one.
Matrix factorization were discovered by Eisenbud as a means of description of maximal Cohen-Macaulay modules over hypersurface singularities. However, there are connections to many other areas of mathematics and physics, see the literature section below.
Program
A brief overview of what has been told in the seminar will be updated below.
- March 17, 2022
- The relation between matrix factorzations of an element f in a regular local ring and maximal Cohen-Macaulay modules over S/(f). Knörrer's periodicity and Orlov's more general statement in the form of equivalence of singularity categories. Matrix factorizations after Kontsevich (D-branes of type B).
- March 24, 2022
- Generalization of matrix factorizations to higher codimension: introduction and relations to maximal Cohen-Macaulay modules, singularity categories and minimal free resolutions. Orlov's singular equivalence for complete intersection rings and matrix factorizations over projective bundles. A version of the two-periodicity result for complete intersections (the talk was mostly based on [B86], [BW15], [EP16] and [O06])
- April 7, 2022
- Derived categories of matrix factorizations - the absolute derived and coderived categories of locally free and (quasi-)coherent matrix factorizations and their main properties, following [EP15].
- April 14, 2022
- Derived categories of matrix factorizations and their connection with relative singularity categories, following [EP15].
- April 28, 2022
- Introduction to abelian model categories. Extending the singularity category of a nice enough ring to a compactly generated triangulated category (using [K05], [J05], [N08]) - there are two types of such extensions, one based on injective modules (the coderived case) and one on projective modules (the contraderived case). Model structures for (absolute) singularity categories coming from the coderived and contraderived categories, as constructed in [B14].
- May 5, 2022
- Becker's relative singularity categories. A Quillen equivalence between the contraderived category of matrix factorizations and the relative contraderived singularity category of a Koszul DG algebra [B14, §3.2].
- May 19, 2022
- A DG model for the singularity category of a finite dimensional algebra following (the appendix of) [CW21]: Leavitt path DG algebras and their construction using the Koszul duality and universal localization.
Literature
It seems hopeless to provide a reasonably complete list of literature which relates to matrix factorizations. There are simply too many sources and relations. As an illustration, here is a link to references collected for a
Of course, there have also been new and interesting developments in the last decade. Therefore, we only list selected literature suggested by participats of the seminar. The story started with this paper by Eisenbud:
[E80] | D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. [link] |
Generalization of matrix factorizations to a sequence of elements of a ring (which should correspond to studying singularities of higher codimension than one) can be found in
[BW15] | J. Burke, M. E. Walker, Matrix factorizations in higher codimension, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3323-3370. [link] [arXiv] |
[E21] | D. Eisenbud, Layered resolutions of Cohen–Macaulay modules, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 3, 845-867. [link] |
[EP16] | D. Eisenbud, I. Peeva, Minimal free resolutions over complete intersections, Lecture Notes in Mathematics, 2152. Springer, Cham, 2016. [link] |
Categories of matrix factorizations are also closely related to singularity categories (i.e. quotients of the bounded derived category by the triangulated subcategory of perfect complexes); see for example
[B86] | R. O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, manuscript, 1986. [link] |
[O06] | D. Orlov, Triangulated categories of singularities, and equivalences between Landau-Ginzburg models, Sb. Math. 197 (2006), no. 11-12, 1827-1840. [link] [arXiv] |
Sometimes it is useful to consider the usual, essentially small singularity categories as compact objects of larger, compactly generated triangulated categories. Algebraic results for this approach are developed in
[IK06] | S. Iyengar, H. Krause, Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math. 11 (2006), 207-240. [link] [arXiv] |
[J05] | P. Jørgensen, The homotopy category of complexes of projective modules, Adv. Math. 193 (2005), no. 1, 223-232. [link] [arXiv] |
[K05] | H. Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), 1128-1162. [link] [arXiv] |
[N08] | A. Neeman, The homotopy category of flat modules, and Grothendieck duality, Invent. Math. 174 (2008), no. 2, 255–308. [link] |
[Š14] | J. Šťovíček, On purity and applications to coderived and singularity categories, preprint, 2014. [arXiv] |
Matrix factorizations, where A and B are not matrices, but rather maps of quasi-coherent sheaves on a possibly non-affine scheme or modules over a non-commutative ring, and relative singularity categories were studied in
[B14] | H. Becker, Models for singularity categories. Adv. Math. 254 (2014), 187-232. [link] [arXiv] |
[EP15] | A. I. Efimov, L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories, Algebra Number Theory 9 (2015), no. 5, 1159-1292. [link] [arXiv] |
Papers concerned with matrix factorizations and their role in mathematical physics (D-branes of type B in Landau-Ginzburg models) include
[DM13] | T. Dyckerhoff, D. Murfet, Pushing forward matrix factorizations, Duke Math. J. 162 (2013), no. 7, 1249-1311. [link] [arXiv] |
[M19] | D. Murfet, Constructing A_{∞}-categories of matrix factorisations, preprint, 2019. [arXiv] |
[O12] | D. Orlov, Landau-Ginzburg Models, D-branes, and Mirror Symmetry, Mat. Contemp. 41 (2012), 75-112. [arXiv] |
Recently, there was a progress in understanding singularity categories of finite dimensional algeras (this is only loosely related to matrix factorizations, but still interesting):
[C11] | X.-W. Chen, The singularity category of an algebra with radical square zero, Doc. Math. 16 (2011), 921-936. [link] [arXiv] |
[CW21] | X.-W. Chen, Z. Wang (with an appendix by B. Keller, Y. Wang), The dg Leavitt algebra, singular Yoneda category and singularity category, preprint, 2021. [arXiv] |