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

Venue: IIT Madras
Dates: 2nd to 12th Jan, 2013


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name T.E.Venkata Balaji          Suresh Nayak Srikanth Iyengar
Mailing Address Dept of Maths,
ISI Bangalore, India Dept of Maths,
University of Nebraska-Lincoln,


Speakers and Syllabus 



  9:00–9:50 10:00–10:50 11:30–12:20 2:30–4:30
Wed, Jan. 2 Lecture 1


Lecture 2


Lecture 3



Krishna, Manoj, Srikanth

Thu, Jan. 3 Lecture 4


Lecture 5


Lecture 6



Sukhendu, Manoj, Srikanth

Fri, Jan. 4 Lecture 7


Lecture 8


Lecture 9



TEV, Krishna, Ajay

Sat, Jan. 5 Lecture 10


Lecture 11


Lecture 12



Krishna, Suresh, Ajay

Sun, Jan. 6 Free day
Mon, Jan. 7 Lecture 13


Lecture 14


Lecture 15



Krishna, Manoj, Sukhendu

Tue, Jan. 8 Lecture 16


Lecture 17


Lecture 18



Krishna, Krishna, Ajay

Wed, Jan. 9 Lecture 19


Lecture 20


Lecture 21



Sukhendu, Manoj, TEV

Thu, Jan. 10 Lecture 22


Lecture 23


Lecture 24



TEV, Manoj, Krishna



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)

  9:00–9:50 10:00–10:50 11:30–12:20 2:30–3:20 3:30-4:20
Fri, Jan. 11
Krause Singh Srinivas Mallick Dubey
Sat, Jan. 12
Krause Dell’Aquila Srinivas Dubey -


Umesh Dubey, TIFR

First lecture: Cohen-Macaulay representation type of singularities

In this talk we will survey some results describing a coarse classification of special classes of singularities. We will also mention Kahn correspondence between CM modules and vector bundles on curves which is used in such classifications.

Second lecture: Localisation theorems in Algebraic Geometry

Localisation theorems are important tool in computation of algebraic K-groups proved by Quillen for regular schemes. This was generalised to more general schemes by Thomason and later by Neeman in more categorical setting. Aim of this talk is to describe these results and mention some other localisation theorems proved using theorem of Neeman. If time permits we will mention our
results with V. Mallick and work in progress with A. Krishna.

Henning Krause, University of Bielefeld

First lecture: The stable derived category of a noetherian scheme

Second lecture: Cohomological length functions

Vivek Mallick, ISER, Pune.

Computing Balmer spectrum: An example

In this talk we compute the spectrum of the derived category of G-equivariant coherent sheaves on a projective variety. If time permits we shall also talk about the derived category of coherent sheaves over super-schemes. This is joint work with Dr. Umesh Dubey.

Anurag Singh, University of Utah

The F-pure threshold of a Calabi-Yau hypersurface

The F-pure threshold is a numerical invariant of prime characteristic singularities. It constitutes an analogue of a numerical invariant for complex singularities—the log canonical threshold—that measures local integrability. We will discuss the calculation of F-pure thresholds of elliptic curves, and also indicate how this calculation extends to Calabi-Yau hypersurfaces. This is work in progress with Bhargav Bhatt.

Vasudevan Srinivas, TIFR

Introduction to Grothendieck-Witt groups, after Balmer


[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,

[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.

[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.

[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.

[16] D. Orlov, Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities;

[17] D. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sb. Math. 197 (2006), 1827–1840.

[18] D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226 (2011), 206–217.

[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.


Selected Applicants


Sr Full name Affiliation Position
1 Mr Narasimha Chary B CMI, Chennai PhD student
2 Mr Saravanan Raja Periar University, Salem PhD student
3 Mr Umamaheswaran Arunachalam Periar University, Salem PhD student
4 Mr Udhayakumar Ramalingam Periar University, Salem PhD student
5 Mr. Krishanu Dan IMSc, Chennai PhD student
6 Mr. Anjan Gupta TIFR, Mumbai PhD student
7 Mr. Shuddhodan Kadattur Vasudevan TIFR, Mumbai PhD student
8 Mr. Anand Prabhakar Sawant TIFR, Mumbai PhD student
9 Mr. Gaurab Tripathi Jadavpur University PhD student
10 Mr. Susobhan Mazumdar IITM Chennai PhD student
11 Mr. Balakrishnan R IITM Chennai PhD student
12 Mr. Vivek Sadhu IIT Bombay PhD student
13 Mr. Seshadri Chintapalli IMSc Chennai PhD student
14 Mr. Umesh Vanktesh Dubey TIFR, Mumbai Post Doc
15 Dr. Vivek Mohan Mallick IISER, Pune Faculty
16 Dr. Yashonidhi Pandey IISER Mohali Faculty
17 Dr. Jishnu Biswas ISI Bangalore Faculty
18 Dr. Arvind Nair  TIFR, Mumbai Faculty