AIS - Cohomology of Commutative Algebras (2024)

Speakers and Syllabus

Speaker and affiliation Number of Lectures Detailed Syllabus
Sukhendu Mehrotra CMI 4 Kähler derivations and differentials, the fun- damental exact sequence, Regular rings and complete intersection rings, Properties of maps: Smoothness and locally complete in- tersection; only for maps essentially of finite type.
Anjan Gupta IISER Bhopal 4 Commutative differential graded algebras and modules, Koszul complex, as a DG al- gebra, Tate resolutions and semi-free resolu- tions, Characterising complete intersections rings in terms of Koszul homology; the re- sults of Tate, Assmus, Gulliksen, Wiebe.
ManojCMI Kummini 4 Gorenstein rings and Koszul homology, the result of Avramov, and Golod, Minimality of acyclic closures, the result of Gulliksen and Schöller, Obstructions to multiplicative struc- tures on free resolutions.
Anand Sawant TIFR Mumbai 4 Simplicial algebraic geometry
Benjamin Briggs Imperial College London, UK. and Srikanth Iyengar University of Utah USA. 4 + 4 Cotangent complex of a map and basic prop- erties. This is the one place we need simplicial methods. André-Quillen functors, the Jacobi– Zariski exact sequence, and links to smooth- ness and locally complete intersection prop- erty. Hochschild homology and cohomology, the Hochschild-Kostant-Rosenberg Theorem, and rigidity properties. Hochschild and its re- lation to the cotangent complex; the Atiyah character. Ext as a graded Hopf algebra and the homotopy Lie algebra of a map. Rad- ical of the homotopy Lie algebra, and the Avramov-Halperin result. Relating the cotan- gent complex to its dg avatar. Rigidity of the cotangent complex, the Quillen conjectures, and future directions.

References

  • [AG71] L. L. Avramov and E. S. Golod. The homology of algebra of the Koszul complex of a local Gorenstein ring. Mat. Zametki, 9:53–58, 1971.
  • [And74] M. André. Homologie des algèbres commutatives, volume Band 206 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin-New York, 1974.
  • [Avr98] 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.
  • [BI23] B. Briggs and S. Iyengar. Rigidity properties of the cotangent complex. J. Amer. Math. Soc., 36(1):291–310, 2023. 1
  • [GL69] 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.
  • [Gul68] T. H. Gulliksen. A proof of the existence of minimal R-algebra resolutions. Acta Math., 120:53–58, 1968. [MR10] J. Majadas and A. G. Rodicio. Smoothness, regularity and complete intersection, volume 373 of London Math. Soc. Lecture Note Ser. Cambridge University Press, Cambridge, 2010.
  • [Qui70] D. Quillen. On the (co-) homology of commutative rings. In A. Heller, editor, Applications of categorical alge- bra (New York, 1968), volume XVII of Proc. Sympos. Pure Math., pages 65–87. Amer. Math. Soc., Providence, R.I., 1970.
  • [Sch67] C. Schoeller. Homologie des anneaux locaux noethériens. C. R. Acad. Sci. Paris Sér. A-B, 265:A768–A771, 1967.
  • [Tat57] J. Tate. Homology of Noetherian rings and local rings. Illinois J. Math., 1:14–27, 1957. [Wie69] H. Wiebe. Über homologische Invarianten lokaler Ringe. Math. Ann., 179:257–274, 1969.

 

Time Table

 

Day Date Lecture 1   Lecture 2   Tutorial   Tutorial  
    9.30–11.00

T
E
A

 11.30–1.00

L
U
N
C
H

 2.30–3.30

 
T
E
A

4.00-5.00

S
N
A
C
K
S 
Mon 2/12/2024 MK AG MK, T1, T2 AG, T1, T3
Tue 3/12/2024 AG SM AG, T2, T3 SM, T1, T2
Wed 4/12/2024 SM MK SM, T1, T3 MK, T2, T3
Thu 5/12/2024 MK AG MK, T1, T2 AG, T1, T3
Fri 6/12/2024 AG SM AG, T2, T3 SM, T1, T2
Sat 7/12/2024 SM MK SM, T1, T3 MK, T2, T3
Mon 9/12/2024 AS BB AS, T1, T2 BB, T1, T3
Tue 10/12/2024 SI AS SI, T2, T3 AS, T1, T2
Wed 11/12/2024 BB AS BB, T1, T3 AS, T2, T3
Thu 12/12/2024 AS SI AS, T1, T2 SI, T1, T3
Fri 13/12/2024 SI BB SI, T2, T3 BB, T1, T2
Sat 14/12/2024 BB SI BB, T1, T3 SI, T2, T3

BB : Benjamin Briggs
AG : Anjan Gupta
SI : Srikanth Iyengar
MK: Manoj Kummini
SM :Sukhendu Mehrotra
AS : Anand Sawant

 

 

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