AIS Commutative Algebra (2012)

In this AIS in Commutative Algebra, we aim to cover certain interesting topics in classical and combinatorial commutative algebra. We will begin the programme by revising certain basic commutative algebra results. We then develope some amount of homological algebra which itself is interesting and also is a tool for proving more interesting results. After developing enough of background material in commutative and homological algebra, we prove Quillen-Suslin theorem and Stanley's upper bound theorem in the final week.

In this AIS in Commutative Algebra, we aim to cover certain interesting topics in classical and combinatorial commutative algebra. We will begin the programme by revising certain basic commutative algebra results. We then develope some amount of homological algebra which itself is interesting and also is a tool for proving more interesting results. After developing enough of background material in commutative and homological algebra, we  prove Quillen-Suslin theorem and Stanley's upper bound theorem in the final week.

Syllabus: Nakayama's lemma, localization of rings and modules, Noetherian rings and modules, spectrum of a ring, support of a module, associated primes of rings and modules, primary decomposition of ideals and modules. Filtered rings and modules, topology defined by a filtration, completions of filtered modules, flatness of completion,
graded rings and modules, Artin­-Rees lemma, Krull intersection theorem.

Integer valued polynomials and polynomial functions, Hilbert polynomial of a graded module, Hilbert­Samuel polynomial of a filtration. Integral extensions, Cohen­Seidenberg theorems, systems of parameters,
dimension theorem, normal rings, Krull's theorem. Dimension theory for Affine algebras of polynomial rings, normalization lemma, transcendence degree and dimension, Hilbert's Nullstellensatz, integral closure of a
finitely generated algebra.

Chain complexes, long exact sequence of homology. Projective and injective modules, Derived functors, Ext, Tor. Tensor product of complexes, Koszul complex, depth and Koszul homology. Regular sequences, Ext and grade, depth and projective dimension, Auslander­ Buchsbaum formula. preservation of Cohen­Macaulay property, unmixed
theorems of Macaulay and Cohen, CM rings are universally catenary.

Projective modules, Free resolutions, Quillen-Suslin Theorem

Simplicial complexes, Stanley-Reisner rings, polytopes, Local cohomology of Stanley-Reisner rings, Stanley's upper bound theorem.

Speakers: Prof. Balwant Singh (IIT Bombay)
Prof. Clare D'Cruz (CMI)
Prof. J. K. Verma (IIT Bombay)
Prof. A. V. Jayanthan (IIT Madras)
Prof. T. E. Venkata Balaji (IIT Madras)
Prof. Krishna Hanumanthu (CMI)

The IIT campus is located on the Sardar Patel Road, midway between the Raj Bhavan and Adyar bus terminus and is around 12 km from Chennai Central Railway Station. Other landmarks include the Guindy Snake Park near the IIT main gate and the Central Leather Reasearch Institute (CLRI) campus opposite the IIT campus.

If you have light luggage (a bag or an attache) you could consider taking a bus to IIT. The bus stop is just across the road from the Central Railway Station. You could use the subway to cross the road which is perpetually busy. There are direct buses to IIT, which start at the Central Station bus stop. If you take any of them, get down at Gandhi Mandapam, which is at walking distance from the IIT main gate. Otherwise, take any bus to Saidapet and from there, any bus heading for Adyar or Besant Nagar which will drop you right in front of the IIT gate. You could also take any bus to Adyar; there are frequent buses from Adyar Signal to the campus. Once you reach the In gate, there are
buses to Gajendra Circle (near the administrative block and Central Library), the academic zone and the hostels (they are around 3.8 km from the main gate!) almost every 30 minutes.

If you have luggage, an auto-rickshaw or taxi would it would be the most comfortable option. The fare will be about Rs. 150 for an auto-rickshaw and Rs. 300 for a taxi. If you arrive at Egmore Station, you could take buses to IIT, Gandhi Mandapam or Adyar from where there are number of buses to the IIT. Current auto-rickshaw and taxi fares
from Egmore are approximately Rs.130 and Rs.250 respectively.

From the airport, probably the only convenient way is to take a prepaid taxi which is available at the exit point of domestic terminal.

A strong warning regarding the autorikshaws in Chennai: As soon as you get down at the railway station (Egmore or Central), there would be many autorikshaws trying to offer you to take you to IIT for a "cheap" price or a price which is "same as the prepaid ones". Do not give ear to them. Just come out and look for the prepaid autorikshaw counter and get auto from there.