Seminar zur Motiven WS 2025/26

Introduction to maximal Cohen-Macaulay modules, Matrix Factorizations and Singularity Categories

The motives seminar this term will be giving an introduction to matrix factorizations, maximal Cohen-Macaulay modules and singularity categories. We hope this will form a foundation for further work in developing versions of these theories that incorporate a duality in the underlying categories.

Preliminary program details

You can find a preliminary program as pdf here. We meet in WSC-N-U-4.04, 16-18 Uhr (c.t.). The meeting will also be accessible via Zoom (for the coordinates, please see the weekly seminar announcements).

Detailed Schedule

Lecture 1, (October 28, Marc Levine) Gorenstein rings and maximal Cohen-Macaulay modules.
Lecture 1 Video

Lecture 2, (November 4, Marc Levine) Stable categories and equivalences between them.
Introduce the matrix factorisation category $MF(A, x)$ (as a dg category) for $(A,\mathfrak{m})$ a regular local ring and $x$ a non-zero element of $\mathfrak{m}$. State and prove Eisenbud's periodicity theorem in the case of a local hypersurface. Define the stable category of $S$-modules, the stable category of maximal Cohen-Macaulay modules $\underline{MCM}(S)$ (for $S$ Gorenstein) as a triangulated category, the stabilized derived category (singularity category) $\underline{D^b}(S)$ and the homotopy category of acyclic complexes of projective modules $\underline{APC}(S)$. Prove Theorem 4.4.1 of [Buchweitz]: For $S$ Gorenstein, there are equivalences $\Omega_0:\underline{APC}(S)\to \underline{MCM}(S)$ and $\iota_S:\underline{MCM}(S)\to \underline{D^b}(S)$. Show that the homotopy category $\underline{MF}(A,x)$ is a triangulated category and describe an equivalence of triangulated categories $\underline{MCM}(A/(x))\cong \underline{MF}(A,x)$ (see [Dyckerhoff], §2).

Lecture 3, (Nov. 11, Louisa Bröring) Cohen-Macaulay modules on quadrics. Present the paper [BEH], concentrating on Theorems 2.1 and 2.2. Sketch the material in §1 without giving proofs.

Lecture 4 (Nov. 18, N.N.) Knörrer periodicity. Present Knörrer's paper [Knoerrer], §1-3. You can ignore the statements on the Auslander-Reiten quiver.

Lectures 5,6,7,8 Present Dyckerhoff's paper [Dyckerhoff], sections 2-6.
Lecture 5 (Nov 25, N.N.) Give an overview of Toën's paper [Toen], concentrating on the definitions and statements of main results as needed for Dyckerhoff's paper, without proofs.
Lecture 6 (Dec. 2, N.N.): [Dyckerhoff] § 2,3, just the local case. § 2 should just be recollections from Lectures 1 and 2.
Lecture 7 (Dec. 9, N.N.): [Dyckerhoff] § 4.
Lecture 8 (Dec. 16, N.N.):[Dyckerhoff] § 5,6.

Lecture 9 (Jan. 6, N.N.) Real Knörrer periodicity. Present the paper by Brown [Brown]

Lecture 10 (Jan. 13, N.N.) Real Knörrer periodicity. Present the paper by Spellmann-Young [SpellmannYoung].

Lecture ??? Further directions, time permitting. These could include looking at more modern work on singularity categories, such as the papers of Preygel [Preygel], Toën-Vezzosi [ToenVezzosi], Blanc-Robalo-Toën-Vezzosi [BRTV], or the two papers of Hennion-Holstein-Robalo [HHRI, II].

Bibliography

[BRTV] A. Blanc, M. Robalo, B. Toën, G. Vezzosi,
Motivic realizations of singularity categories and vanishing cycles. J. Éc. polytech. Math. 5 (2018), 651--747.

[Brown] M.K. Brown,
Knörrer periodicity and Bott periodicity, Doc. Math. 21 (2016), 1459--1501.

[Buchweitz] R.-O. Buchweitz,
Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorentstein rings. Preprint 1987 Buchweitz Notes

[BuchweitzAMS] R.-O. Buchweitz,
Maximal Cohen-Macaulay modules and Tate cohomology. With appendices and an introduction by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar and Janina C. Letz Math. Surveys Monogr., 262 American Mathematical Society, Providence, RI, [2021]. xii+175 pp.

[BEH] R.-O. Buchweitz, D. Eisenbud, J. Herzog,
Cohen-Macaulay modules on quadrics. Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 58--116. Lecture Notes in Math., 1273.

[Dyckerhoff] T. Dyckerhoff,
Compact generators in categories of matrix factorizations. Duke Math. J. 159 (2011), no. 2, 223--274.

[Eisenbud] D. Eisenbud,
Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260 (1980), no. 1, 35--64.

[HHRI] B. Hennion, J. Holstein, M. Robalo,
Gluing invariants of Donaldson--Thomas type -- Part I: the Darboux stack. arXiv:2407.08471.

[HHRII] B. Hennion, J. Holstein, M. Robalo,
Gluing invariants of Donaldson--Thomas type -- Part II: Matrix factorizations. arXiv:2503.15198.

[Knoerrer] H. Knörrer,
Cohen-Macaulay modules on hypersurface singularities. I Invent. Math. 88 (1987), no. 1, 153--164.

[Preygel] A. Preygel,
Thom-Sebastiani & duality for matrix factorizations. preprint (2011) arxiv 1101.5834

[SpellmannYoung] J.-L. Spellmann, M. Young,
Matrix factorizations, reality and Knörrer periodicity. J. Lond. Math. Soc. (2) 108 (2023), no. 6, 2297--2332.

[Toen] B. Toën,
The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615--667.

[ToenVezzosi] B. Toën, G. Vezzosi,
The $\ell$-adic trace formula for dg-categories and Bloch's conductor conjecture. Boll. Unione Mat. Ital. 12 (2019), no. 1-2, 3--17.