PL - Commutative algebra and modular representation theory (2025)

Speakers and Syllabus

 

Speaker and affiliation Number of Lectures Detailed Syllabus
Srikanth Iyengar
University of Utah
USA		 5
  1. Perfect complexes and the theorem of Hopkins and Neeman;
  2. Complete intersections and the BGG correspondence;
  3. Support varieties for modules over complete intersections;
  4. Stratification for the bounded derived category of a complete intersection;
  5. The stable module category of a finite group.
Manoj Kummini
CMI		 1

Review of derived categories and DG algebras
Sarang Sane
IIT Madras		 2

Stable module categories. The stable module category of matrix factorizations.
Upendra Kulkarni
CMI		 2

Group algebras and group cohomology

 References

  1.  L. L. Avramov. Infinite free resolutions. In J. Elías, J. M. Giral, R. M. Miró-Roig, and S. Zarzuela, editors, Six lectures on commutative algebra, Papers from the Summer School (Ballaterra, 1996), Mod. Birkhäuser Class., pages 1–118. Birkhäuser/Springer,Basel, 1998.
  2.  L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71-101.
  3.  L. L. Avramov, R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000),285-318.
  4.  L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar and C. Miller, Homology of perfect complexes, Advances in Math. 223 (2010),1731-1781. Corrigendum: Advances in Math. 225 (2010), 3576-3578.
  5.  L. L. Avramov and S. B. Iyengar, Constructing modules with prescribed cohomological support, Ill. Jour. Math. 51 (2007), 1-20.
  6.  L. L. Avramov and S. B. Iyengar, Restricting homology to hypersurfaces, Geometric and topological aspects of the representation theory of finite groups, 1–23, Springer Proc. Math. Stat., 242, Springer, Cham, ; MR3901154
  7.  D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, 1991, reprinted in paperback, 1998.
  8.  D. J. Benson, S. B. Iyengar, and H. Krause, Stratifying modular representations of finite groups, Annals of Math 174 (2011),1643-1684.
  9.  R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology, Mathematical Surveys and Monographs, 262,Amer. Math. Soc., Providence, RI, [2021] ©2021; MR4390795
  10.  J. F. Carlson, Cohomology and induction from elementary abelian subgroups, Q. J. Math. 51 (2000), no. 2, 169–181;MR1765788
  11.  J. F. Carlson and S. B. Iyengar, Thick subcategories of the bounded derived category of a finite group, Trans. Amer. Math. Soc.367 (2015), no. 4, 2703–2717; MR3301878
  12.  D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math.Soc. 260 (1980), 35-64.
  13.  M. J. Hopkins, Global methods in homotopy theory, Homotopy Theory, Durham 1985, Lecture Notes in Mathematics, vol.117, Cambridge University Press, 1987.
  14.  S. Iyengar, Modules and Cohomology over Group Algebras: One Commutative Algebraist’s Perspective, Trends in Commutative Algebra, MSRI Publications, Volume 51, 2004, pp. 51–86
  15.  J. Tate. Homology of Noetherian rings and local rings. Illinois J. Math., 1:14–27, 1957.

Secondary references

  1. L. L. Avramov and S. B. Iyengar, Cohomology over complete intersections via exterior algebras, Triangulated categories (Leeds,2006), London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press, Cambridge, 2010, 52-75.
  2. D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, 1991, reprinted in paperback, 1998.
  3. D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59–80; MR1450996
  4. T. H. Gulliksen and G. Levin. Homology of local rings. Number 20 in Queen’s Papers in Pure and Applied Mathematics. Queen’s University, Kingston, Ont., 1969.
  5. H. Krause, Homological theory of representations, Cambridge Studies in Advanced Mathematics, 195, Cambridge Univ. Press, Cambridge, 2022; MR4327095
  6. M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Mem. AMS, vol. 128, American Math.Society, 1997.

Time Table

 

Day Date Lecture 1
9.30–11.00 		  Lecture 2
11.30–1.00 		  Tutorial
2.30–3.30 		  Tutorial
4.00-5.00 		 
Mon 15/12/2025 MK

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 SI L
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MK, T1, T2 T
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 SI, T1, T3

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Tue 16/12/2025 SS SI SS, T2, T3 SI, T1, T2
Wed 17/12/2025 SS SI SS, T1, T3 SI, T2, T3
Thu 18/12/2025 UK SI UK, T1, T2 SI, T1, T3
Fri 19/12/2025 UK SI UK, T2, T3 SI, T1, T2

 

SI : Srikanth Iyengar
UK : Upendra Kulkarni
MK : Manoj Kummini
SS : Sarang Sane

 

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