Cohomology of Commutative Algebras (2024)

Convener(s)

 
Name: Prof. Sarang Sane Prof. Manoj Kummini
Mailing Address: Assistant Professor
Indian Institute of Technology Madras		 Professor
Chennai Mathematical Institute
Email: sarangsanemath at gmail.com mkummini at cmi.ac.in

This is a proposal for an Advanced Instructional School on the topic of André-Quillen and Hochschild cohomology functors on commutative algebras. The lecture series will explore a family of results and conjectures in homological commutative algebra, each making predic- tions about the complexity of certain cohomological objects.
This school will introduce to early career researchers the above cohomology theories and highlight important connections to algebraic topology, and finally present state of the art arguments to uncover the structure of these cohomology theories.

Dates: 

Monday, December 2, 2024 - 09:00 to Saturday, December 14, 2024 - 21:00

Venue: 

Venue Address: 

 NAC
644, Department of Mathematics
IIT Madras, Adyar
Chennai, Tamilnadu
600 036.

Venue State: 

Venue City: 

PIN: 

600036

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

 

 

Selected Applicants: 

Selected outstaion applicants should confirm their participation by Monday 2024-Oct-14.o the following mail id mkummini@cmi.ac.in

Sr.n SID Full name Gender Affiliation Position in College/University University/Institute M.Sc./M.A. Year of Passing M.Sc./M.A Ph.D. Degree Date
1 59514 Mr. Suraj Kumar Male IIT, DELHI PhD IIT MADRAS 2019  
2 59521 Mr. Siddhartha Pramanik Male IIT Kharagpur PhD student IIT Madras 2022  
3 59525 Mr. Sai Krishna P M S Male IIT Bombay PhD Indian Statistical Institute Kolkata 2020  
4 59530 Ms. Mansi Mittal Female IIT PATNA PhD M.Sc. 2019  
5 59534 Mr. Rohit Pokhrel Male Harish Chandra Research Institute, Prayagraj PhD Sikkim University 2019  
6 59592 Ms Shruti Priya Female Indian Institute of Technology Kharagpur PhD NIT Rourkela 2020  
7 59594 Mr. Anoot Kumar Yadav Male Indian Institute of Technology Patna PhD Student Ewing Christian College (University of Allahabad) 2018  
8 59600 Mr. Aniruddha Saha Male IIIT Hyderabad (IIT H) Phd University of Hyderabad 2018  
9 59623 Mr. Aman Singh Male IISER Mohali PhD Kerala School of Mathematics 2022  
10 59626 Ms Ananya Pal Female Indian Statistical Institute, Kolkata PhD Student M.Sc from Calcutta University 2018  
11 59647 Ms. Mouma Samanta Female IIT Kharagpur Ph.D. IIT Kharagpur 2022  
12 59702 Ms. Anusuiya Baishya Female IIT Kharagpur PhD Tezpur University 2021  
13 59703 Mr. Sudip Manna Male IIT Kharagpur PhD University of Hyderabad 2022  
14 59730 Mr. Samarendra Sahoo Male IIT Bombay PhD Ravenshaw University, Cuttack 2017  
15 59732 Mr. Debojyoti Saha Male Indian Statistical Institute PhD 3rd year student M.Math from Indian Statistical Institute Kolkata 2022  
16 59760 Ms Deepanshi Saraf Female Indian Institute of Science Education and Research (IISER) Mohali PhD M.S. 2021  
17 59779 Mr. Subhadip Bhowmick Male Indian Institute of Technology Kharagpur PhD Indian Institute of Technology Kharagpur 2023  
18 59878 Mr. Himanshu Rewri Male Indian Institute of Technology Delhi PhD Indian Institute of Technology Jodhpur 2019  
19 59894 Ms. Siddhi Balu Ambhore Female Indian Institute of Technology Gandhinagar PhD Indian Institute of Technology Gandhinagar 2020  
20 59917 Mr. Sayed Sadiqul Islam Male IIT Bombay PhD Jadavpur University 2021  
21 59973 Mr. Om Prakash Male Indian Institute of Technology Bombay Institute Post Doctoral Fellow Maharshi Dayanand university Rohtak 2016 29/07/2023
22 59977 Ms Divya Ahuja Female Department of Mathematics, Indian Institute of Technology Delhi PhD M.Sc. Mathematics, Indian Institute of Technology Madras 2019  
23 60017 Mr Anirban Sarkar Male Tata Institute of Fundamental Research IPhD Student Tata Institute of Fundamental Research Awaiting Result  
24 60046 Mr Jebasingh R Male IISER Kolkata PhD IIT Madras 2019  
25 60048 Mr Omkar Deepak Javadekar Male IIT Bombay MSc & PhD Dual Degree Student Indian Institute of Technology, Bombay 2020  
26 60077 Mr. Manav Sanjay Batavia Male Purdue University PhD Indian Institute of Technology Bombay 2022  
27 60093 Mr. Shashi Ranjan Sinha Male Indian Institute of Technology Hyderabad PhD Patna University 2020  
28 60142 Mr. Shrikant Shekhar Male IISER BHOPAL Ph.D. Scholar IIT Gandhinagar 2019  
29 60332 Ms. Sutapa Dey Female IIT  Hyderabad PhD Birla Institute of Technology, Mesra 2017  
30 60371 Mr. Subham Jaiswal Male Indian Statistical Institute Msc. Student Indian Statistical Institute 2022  
 Local participants      
1 59537 Mr. Abhiram Subramanian Male Chennai Mathematical Institute PhD Chennai Mathematical Institute, M.Sc. 2022  
2 59544 Mr. Sourjya Banerjee Male IMSc PDF University of Calcutta 2016 20/01/2023
3 59556 Mr. Ganapathy K Male Indian Institute of Technology Madras PhD student IIT Madras    
4 59673 Mr. Kanoy Kumar Das Male Chennai Mathematical Institute Postdoctoral Fellow Ramkrishna Mission Vidyamandira 2018 06/10/2023
5 59699 Mr Kesavan Mohana Sundaram Male University of Nebraska-Lincoln PhD Indian Institute of Technology Madras 2023  
6 59785 Mr. Suraj Rajendra Kulkarni Male The Institute of Mathematical Sciences Integrated PhD The Institute of Mathematical Sciences 2024  
7 60013 Mr. Soham Mondal Male IIT MADRAS PhD VISVA-BHARATI 2017  
8 60211 Mr. Buddhadev Hajra Male Chennai Mathematical Institute Postdoc IIT Bombay, Mumbai 2016 20/08/2022
9 60263 Mr. Cyril J Jacob Male Chennai Mathematical Institute PhD St. Berchmans College, Kerala 2019  
Waiting list
1 59542 Mr. Manohar Kumar Male Indian Institute of Technology Kharagpur PhD Indian Institute of Technology Kharagpur 2020  
2 59939 Mr. Tapas Kumar Roy Male Indian Institute of Technology Kharagpur PhD Jadavpur University 2019  
3 59593 Mr. Vivek Bhabani Lama Male Indian Institute of Technology Kharagpur PhD St. Joseph's College (Autonomous), Bangalore City University 2021  
4 59642 Mr. Balaji Subramoniam P Male Indian Statistical Institute (Bangalore Centre) Junior Research Fellow in Mathematics Chennai Mathematical Institute 2024  
5 60172 Ma. Aditya Dwivedi Male IIT Bombay BS-MSc Student IIT Bombay Appeared  
6 60097 Mr. Om Sai Male Indian Statistical Institute M-Math Student Indian Statistical Institute Kolkata    
7 60016 Mr. Swapnajit Das Male TIFR Mumbai Integrated Phd TIFR Mumbai Appeared  
8 59515 Mr. Biplab Dawn Male IIT PATNA PhD Student IIT PATNA 2023  
9 59601 Mr. Ajay P Joseph Male NIT Karnataka Ph.D Cochin University of Science and Technology 2021  
10 59618 Ms Paromita Bordoloi Female IIT Jammu PhD IIT Guwahati 2021  
11 59765 Mr. Shashank Gupta Male Indian Institute of Science Education and Research, Mohali Integrated PhD Student Indian Institute of Science Education and Research, Mohali Appeared  

 

 

How to Reach: 

TBA

School Short Name: 

cca

Last Date Application: 

Sunday, May 14, 2023

School Type: 

AIS

Separate faculty form: 

0