ATMW Singularity Categories in Algebraic Geometry and Commutative Algebra (2013)

Schedule:

Workshop:

Participants will arrive on 1st January, 2013 a Tuesday. The workshop runs from 2nd January through the afternoon of 10th January, the following Thursday.  There will be no lectures on 6th January.

A tentative plan for the lectures is as follows:

Wednesday, 2nd January.

1. Dimension and depth for modules over commutative rings; Gorenstein rings.
References: Bruns and Herzog[1], Matsumura [12]

2. Sheaf theory and affine schemes
References: [5]

3. Triangulated categories: definitions and basic properties
References: Neeman [14],
Exercise session: Krull-Remak-Schmidt

Thursday, 3rd January.

4. Coherent sheaves on projective schemes
References: Serre [5, 19]

5. Thick subcategories and quotient categories (abelian and triangulated version); examples (homotopy categories and derived categories)
References: [10, 14]

6. Derived category of a ring and of a scheme
References: [5, 7]

Friday, 4th January.

7. Stable category of MCM modules over Gorenstein rings
References: Buchweitz [2], Herzog [6]

8. MCM modules over hypersurfaces and matrix factorizations
References: Eisenbud [4]

9. Cech cohomology, computations for Pⁿ ; Serre duality
References: Serre [19]

Saturday, 5th January.

10. Hypersurfaces of finite representation type
References: Buchweitz, Greuel, and Schreyer [3]

11. Kn ̈rrer periodicity - I

12. Kn ̈rrer periodicity - II
References: Kn ̈rrer [8]

Exercise session: ADE singularities

Monday, 7th January.

13. Derived functors in geometry; Grothendieck duality
References: Huybrechts [7]

14. Singularity categories in geometry; Zariski localization
References: Orlov [15]

15. Geometric version of Kn ̈rrer periodicity
References: Orlov [15]

Tuesday, 8th January.

16. Graded matrix factorizations and graded singularity categories.
References: Orlov [16]

17. Semi-orthogonal decompositions; the derived category of projective space bundles.
References: Huybrechts [7]

18. The derived category of a projective hypersurface and the singularity category of its affine cone - I
Exercise session: the quotient of the category of graded modules by torsion modules, and coherent sheaves on projective space.

Wednesday, 9th January.

19. Duality for singularity categories - I
References: Murfet [13]

20. The derived category of a projective hypersurface and the singularity category of its affine cone - II
References: Orlov [16]

21. Riemann-Roch for singularity categories - I

Thursday, 10th January.

22. Duality for singularity categories - II
References: Murfet [13]

23. Riemann-Roch for singularity categories - II

24. Riemann-Roch for singularity categories - III

Conference Schedule (11th and 12th January 2013)

References
[1] W. Bruns, J. Herzog, Cohen-Macaulay rings, Revised edition, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1998.

[2] R.-O. Buchweitz, Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Ring, http://hdl.handle.net/1807/16682

[3] R.-O. Buchweitz, G.-M. Greuel, , F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II. Invent. Math. 88 (1987), 165–182.

[4] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.

[5] R. Hartshorne, Algebraic Geometry, Graduate Texts Math. 52, Springer-Verlag, New York, 1977.

[6] J. Herzog, Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen–Macaulay–Moduln, Math. Ann. 233 (1978), 21–34.

[7] D. Huybrechts, Fourier-Mukai transforms in Algebraic Geometry, Oxford Mathematical Monographs (2006).

[8] H. Kn ̈rrer, Cohen-Macaulay modules on hypersurface singularities. I. 88 (1987), 153–164.

[9] H. Krause, The stable derived category of a Noetherian scheme, Compositio Math. 141 (2005), 1128–1162.

[10] H. Krause, Derived categories, resolutions, and Brown representability, Interactions between homotopy theory and algebra (Chicago, 2004), Contemp. Math. 436, Amer. Math.
Soc., Providence, RI 2007; 101–139.
http://front.math.ucdavis.edu/0511.5047

[11] G. Leuschke, R. Wiegand, Cohen-Macaulay Representations, Mathematical Surveys and Monographs 181, Amer. Math. Soc., Providence, RI 2012.

[12] H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, 1986.

[13] D. Murfet, Residues and duality for singularity categories of isolated Gorenstein singularities, preprint 2010.  front.math.ucdavis.edu/0912.1629

[14] A. Neeman, Triangulated categories, Annals of Math. Studies 148, Princeton Univ. Press, Princeton, NJ, 2001.

[15] D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 246 (2004), 227–248.
http://front.math.ucdavis.edu/0302.5304

[16] D. Orlov, Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities; http://front.math.ucdavis.edu/0503.5632

[17] D. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sb. Math. 197 (2006), 1827–1840.
http://front.math.ucdavis.edu/0503630

[18] D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226 (2011), 206–217.
http://front.math.ucdavis.edu/0901.1859

[19] J.-P. Serre, Faisceaux alg ́briques coh ́rents, Ann. of Math. 61 (1955), 197–278.

[20] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990.