1) Buchweitz, Eisenbud, etc.

2) Orlov’s singularity category and relation to Buchweitz

3) exceptional sequences; relation to tilting and representation theory of quivers; see Bondal, “Helices, representations of quivers, and Koszul algebras” in the volume “Helices and vector bundles” (edited by Rudakov; search the Benson archive for Rudakov to find it).

4) Orlov’s theorem on graded matrix factorizations and their relation to the singularity category

5) Standard resolution of modules over a hypersurface ring - i.e.  if R = S/(f) is a hypersurface, this is a functorial way of taking a finite resolution of an R-module M over S and producing a resolution over R. See Eisenbud section 8, and Avramov, Buchweitz, “Homological algebra modulo a regular sequence…” section 2

6) Buchweitz’s Herband difference, Hochster’s theta function: relation to each other, connection to rigidity of Tor vanishing


1) Homological perturbation lemma

2) Grothendieck’s residue symbol, local cohomology

3) Dyckerhoff/Murfet I or II

Connections to representation theory

1) Kajiura, Saito, Takahashi - “Matrix factorizations and representations of quivers II”

2) Burban, Iyama, Keller, Reiten

3) Lenzing, de la Pena

Cluster-tilting, non-commutative crepant resolutions

1) basics on cluster tilting

2) Van Den Bergh - “Noncommutative crepant resolutions”

3) Iyama-Wemyss

4) Buchweitz, Leuschke, Van Den Bergh for hypersurfaces

Polishchuk-Vaintrob, Hirzebruch-Riemann-Roch Formula

1) Toen’s homotopy theory for dg-categories

2) Dyckerhoff’s calculation of HH(MF) for an isolated singularity

3) Polishchuk-Vaintrob - HRR paper

Other ideas

1) Aspinwall, Morrsion - “Quivers from matrix factorizations” (on the arxiv)