IST Commutative Algebra (2017)
Venue:  St. Joseph's College, Irinjalakuda, Kerala 
Date:  24th, Apr 2017 to 6th, May 2017 
Name  Clare D'Cruz  Manoj Kummini  Mangalambal N. R. 
Mailing Address  Chennai Mathematical Institute  Chennai Mathematical Institute  Department of Mathematics St. Joseph’s College (Autonomous), Irinjalakuda 
Speakers
sr.  Speaker  Affiliation  Topics covered 
1  Manoj Kummini  CMI, Chennai  Ring and morphismsThere was some confusion about why we define ring morphisms the way we do (with 1 7→ 1). ManojKummini gave examples from rings of continuous functions and later interpreted polynomial rings as rings of functions (from R or C, so they can think of these as continuous functions).He gave many concrete examples of ring morphisms (over R and C) and what they mean geometrically, Hilbert Basis Theorem. It was stated without proof; the proof in AtiyahMacdonald is quite straightforward. Nullstellensatz. It was used (without proof) to give many examples of geometric interest. Spectrum. Manoj defined prime spectrum Spec, and showed that contraction gives a continuous map in the Zariski topology and sketched that in artinian rings, all prime ideals are maximal and that there are only finitely many maximal ideals. 
2  Sarang Sane  IIT Madras  Sarang Sanetalked about tensor products, localization and exact sequences after some preliminary discussion and examples of modules. In particular, he defined and covered basic notions of modules, submodules, operations on them, free modules, generating sets for modules, basis for a free module, sums, direct sums, intersections, etc.Some time was spent on furthering the example of continuous functions by talking about continuous functions from X to R n and proving that it is a (free) module over the ring of continuous functions from X to R. He then looked at the case of X a sphere and considered those submodules of those functions which give orthogonal vectors at each point (tangent bundle) and parallel (normal bundle). Made them prove that they are submodules and that their direct sum is the free one. For the circle, they checked that both these submodules are themselves free and the parallel About tensor products, Sarang stated the construction and the universal property, and the participants did several exercises where they had to define module homomorphisms, and after initially struggling, it may have sunk in that one has to use the universal property or the construction to define these. He stated several times that M ⊗ R/I ' M/IM and that tensor is right exact. About exact sequences, he defined exactness, short exact sequences and the notion of a free resolution. The participants haven’t solved any exercises of note on this. About localization, he defined it, and the participantsgot a feel about how to view it (solved several problems).He also proved the property that localization captures when a module is 0 (and hence injectivity and surjectivity of maps). We also saw that localization preserves exact sequences and that only ideals disjoint from the multiplicative set are preserved. In particular that prime ideals in the localization are in bijective correspondence with the prime ideals not intersecting the multiplicative set. Using this,he also proved that rank is welldefined and that a generating set for any finite rank free module with the same size as the rank must be a basis. Finally, Sarang also proved prime avoidance. 
3  Clare D’Cruz  CMI, Chennai  Definition of primary ideals, Primary ideals and localization, First Uniqueness Theorem for Primary Decompositin with examples, Associated Primes, Second Uniqueness Theorem for Primary Decomposition, Primary Decomposition in Noetherian Rings, Examples of ideals in Noetherian rings where the minimal primary components were computed using Second Uniqueness theorem, Symbolic Powers, Examples of Artin rings, The tutorial sessions consisted of: Computing Primary components corresponding to minimal primes, Assoicated primes, Primary Decomosiiton in certain Examples, Symbolic powers, Example of a nonNoetherian ring, Examples of rings which satisfy acc and dcc, and Introduction to Macaulay  (Rings, Ideals, Edge Ideals, Primary Decomposion) 
4  Krishna Hanumanthu  CMI, Chennai  Integral extensions  definition, characterization in terms of finite generation as modules; examples like the cusp, node, x 2 − y 2 over the xaxis; hyperbola over the xaxis (i.e., xy − 1) and the union of two axes over xaxis (i.e., xy). First one is integral and the the last two are not. He indicated why integrality fails (empty fibre, infinite fibre) in these two examples. Goingup theorem and all its manifestations: lying over theorem, spec map is surjective, closed etc.Krishna also proved that the number of primes lying over a fixed prime is finite. Valuation rings: Proved all the results leading to showing that integral closure of A in a field K is the intersection of all valuation rings of K containing A. Using the results on valuation rings, proved the version of Nullstellensatz that Manoj mentioned: a field K which is finitely generated as an algebra over a field F is a finite extension of F. Also gave four other versions of Nullstellensatz and proved the first two of them: (1) maximal ideals in k[x1, ..., xn] are in bijection with k n; (2) every proper ideal in k[x1, ..., xn] has zeroes; (3) I(V (I)) = rad(I); all three versions above over an alg closed field k; and finally (4) radical of an ideal in a finitely generated algebra over a field is the intersection of maximal ideals. Noether normalization: Krishna stated and sketched a proof over infinite fields. But he didn’t have time to fully prove it. He did prove Nullstellensatz using Noether normalization, to give an idea of its power. We did many of the exercises in the fifth chapter of AtiyahMacdonald during the tutorials. The tutors Mr. Suhas B, Ms. Mitra Koley and Mr. Praveenkumar Roy were very much helpful during the tutorials. 
Tutors
 ManojKummini (CMI, Chennai)
 Clare D’Cruz (CMI, Chennai)
 Krishna Hanumanthu (CMI, Chennai)
 Sarang Sane (IIT Madras)
 Suhas M B (IIT Madras)
 MitraKoley (CMI, Chennai)
 Praveen Kumar Roy (CMI, Chennai)
Schedule of Lectures and Tutorials
Day  Date  Lecture 1 9:30 11:00  11:00 11:30  Lecture 2 11:30 13:00  13:00 14:30  Tutorial 1 14:3015:30  15:30  16:00  Tutorial 2 16:0017:00  17:00 17:30 
Mon  24/4/17  MK 
T E A 
SS 
L U N C H 
MK+Tut1+Tut2 
T E A 
SS+Tut1+Tut2 
S N A C K S 
Tues  25/4/17  MK  SS  MK+Tut1+Tut2  SS+Tut1+Tut2  
Wed  26/4/17  MK  SS  MK+Tut1+Tut2  SS+Tut1+Tut2  
Thu  27/4/17  MK  SS  MK+Tut1+Tut2  SS+Tut1+Tut2  
Fri  28/4/17  MK  SS  MK+Tut1+Tut2  SS+Tut1+Tut2  
Sat  29/4/17  MK  SS  MK+Tut1+Tut2  SS+Tut1+Tut2  
SUNDAY : OFF  
Mon  1/5/17  CD 
T E A 
KH 
L U N C H 
CD+Tut3+KH 
T E A 
KH+Tut3+CD 
S N A C K S 
Tue  2/5/17  CD  KH  CD+Tut3+KH  KH+Tut3+CD  
Wed  3/5/17  CD  KH  CD+Tut3+KH  KH+Tut3+CD  
Thu  4/5/17  CD  KH  CD+Tut3+KH  KH+Tut3+CD  
Fri  5/5/17  CD  KH  CD+Tut3+KH  KH+Tut3+CD  
Sat  6/5/17  CD  KH  CD+Tut3+KH  KH+Tut3+CD 
Actual Participants 
Sr  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  10161  Mr. Ravi Shankar Kapildev Yadav  Male  SSR College of Artrs Commerce and Science  Asst. Prof.  Maharaja Sayajirao University of Baroda,Vadodara  2016  
2  10174  Mr. N. Annamalai  Male  Bharathidasan University  PhD  Bharathidasan University/ M.Sc  2011  
3  10226  Mrs Viji M.  Female  St. Thomas' College, Thrissur  Asst. Prof.  Cochin University of Science and Technology  2006  2/19/2016 
4  10257  Ms Meera Sn  Female  Manonmaniam Sundaranar University  PhD Student  Manonmaniam Sundaranar University  2015  
5  10278  Mr. Amit Sharma  Male  Pratap Institute Of Technology And Science  Asst. Prof.  Maharshi Dayanand University, Rohtak  2006  
6  10339  Prof. Dipak Sandu Jadhav  Male  Smt. Chandibai Himathmal Mansukhani College  Asst. Prof.  University of Mumbai  2008  
7  10361  Mr. Harish V N  Male  Sree Kerala Varma College  Asst. Prof.  St. Thomas College, Thrissur  2000  
8  10404  Mr. V. Ramanathan  Male  Manonmaniam sundaranar University, Tirunelveli  Research Scholar  G. Venkataswamy Naidu College, Kovilpatti  2010  
9  10442  Mr. Gaurav Shantaram Jadhav  Male  V.P. Varde College of Commerce & Economics  Asst. Prof.  70  2011  
10  10565  Mrs Reshma E  Female  Payyanur College Payyanur  Asst. Prof.  Payyanur college  2008  
11  10607  Mr. Subhash Mallinath Gaded  Male  R K Talreja College of Arts, Science & Commerce, Ulhasnagar03.  Asst. Prof.  University of Mumbai  2008  
12  10616  Mr Raveesh R Varrier  Male  ST.ALOYSIUS COLLEGE  Asst. Prof.  CUSAT  2011  
13  10618  Ms Akshara R  Female  St ALoysius College  Guest Lecturer  Cochin University of Science & Technology  2013  
14  10713  Prof. Pooja Gurmukhdas Rajani  Female  Smt. Chandibai Himathmal Mansukhani College.  Asst. Prof.  Mumbai University Kalina  2008  
15  10845  Ms Athira Satheesh K  Female  Payyanur college  Guest lecturer  Payyanur college  2015  
16  10849  Ms. Chathely Briji Jacob  Female  Birla College of Arts, Science and Commerce.  Visiting Faculty  National Institute of Technology Calicut  2016  
17  11008  Mr. Sushobhan Maity  Male  VisvaBharati  PhD  VisvaBharati  2013  
18  11069  Mr. Kamalakar Ramesh Surwade  Male  University of Mumbai  Asst. Prof.  IIT Guwahati  2009  
19  11202  Mrs. Deepthi A N  Female  Sree Narayana College  Asst. Prof.  St.Joseph's College,Irinjalakuda,Thrissur, Kerala  2007  
20  Mrs. Sabna K S  KKTM College, Kodungallur  Asst. Prof.  
21  Mrs. Sherin Jose T  St.Joseph's College, Irinjalakuda  Asst. Prof.  
22  Mrs. Sinda Joy  St.Joseph's College, Irinjalakuda  Asst. Prof.  
23  Ms. Dhanya V S  St.Joseph's College, Irinjalakuda  Asst. Prof.  
24  Dr. Sr. Deeni C J  St.Joseph's College, Irinjalakuda  Guest Faculty  
25  Sonadas P  St.Joseph's College, Irinjalakuda  Guest Faculty  
26  Shajila K Y  St.Joseph's College, Irinjalakuda  Guest Faculty 
How to reach
Web Address: http://www.stjosephs.edu.in/
St. Joseph’s college is located in Irinjalakuda town, 22 km away towards south via kodungallor state highway from Thrissur Railway Station and 38 km away from Cochin International Airport.
Accommodation – Hostel of Sahrdaya College of engineering and technology, Kodakara ,Thrissur, district, Kerala 680684
Web address: http://www.sahrdayacas.ac.in/hostel1.html
Sahrdaya College Hostel is located at kodakara, 27 km away from Thrissur railway station via National Highway, 34 km away from Cochin International Airport and 11 km away from venue.

To Sahrdaya Hostel
From
Mode of
Transportation
Expected
Charge
Thrissur Railway
Station
By car
Rs. 600 min
By Bus and Auto( Get Kodakara route bus from KSRTC bus station which is near to railway station, Get Auto From Kodakara bus stop to sahrdaya hostel)
Rs 25 +Rs 100
=Rs. 125 min
Cochin
International
Airport
By Car
Rs. 700 min
link for maps
https://goo.gl/maps/bvHwWj4y4cq
Please Contact for further Assistance to
Ms. Sherin Jose T. ( 9725474495), Asst. Prof. St. Joseph’s College irinjalakuda
Ms. Sinda Joy (9495465285), Asst. Prof. St. Joseph’s College irinjalakuda