Singularity Category, MCM modules and Matrix Factorization Seminar

Schedule

Program and schedule (pdf)

Notes

Notes (pdf) typed by Rory Potter

References:

Algebraic perspective

D. Eisenbud.: Homological Algebra on a Complete Intersection with an Application to Group Representations (1980). The main result is that for hypersurface singularities modules have periodic free resolutions. Matrix factorizations were introduced in this paper.

R.-O. Buchweitz: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings (1987). In the first 4 chapters the singularity category of a ring is introduced and studied. Singularity category is proved to be equivalent to the category of MCM modules.

Y. Yoshino: Maximal Cohen-Macaulay Modules Over Cohen-Macaulay Rings (1990).

M. Auslander, I. Reiten, S. Smalo: Representation Theory of Artin Algebras (1995).

G. Leuschke, R. Wiegand: Cohen-Macaulay Representations (2011)

Geometric perspective

D.Orlov: Triangulated categories of singularities and D-branes in Landau-Ginzburg models (2004). In this paper Orlov introduces Singularity Categories (presumably unaware of work of Buchweitz below). Topcis covered: Homological Algebra of Gorenstein schemes, Singularity category, Knorrer periodicity, Matrix factorizations, A_n example.

D.Orlov: Landau-Ginzburg Models, D-branes, and Mirror Symmetry (2011). Singularity categories and Matrix Factorizations discussed in the context of Mirror Symmetry.

McKay correspondence perspective

M. Auslander: Rational Singularities and Almost Split Exact Sequences (1986)

M. Wemyss: Lectures on Noncommutative Resolutions (2014)

Graded singularity category

J. Bernstein, I. Gelfand S. Gelfand: Algebraic Bundles on P^n and Problems of Linear Algebra (1978). BGG correspondence: Derived category of coherent sheaves on the projective space is proved to be a equivalent to the graded singularity category of the exterior algebra. This is also explained in Gelfand-Manin, "Methods of Homological Algebra", IV.3. Proof of the BGG is explained by D. Eisenbud, G. Floystad and F.-O. Schreyer in Sheaf cohomology and free resolutions over the exterior algebra" (2001).

D.Orlov: Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities (2009). Category of singularities in the graded case is considered. The category of singularities of a cone CX over a projective variety X is related to the category of sheaves on X.