IST Commutative Algebra (2017)

Venue: St. Joseph's College,  Irinjalakuda, Kerala
Date:  24th, Apr 2017 to 6th, May 2017

 

School Convener(s)
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 and Syllabus 

 Speakers

sr. Speaker Affiliation Topics covered
1 Manoj Kummini CMI, Chennai Ring and morphisms-There 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 straight-forward.
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 well-defined 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 non-Noetherian 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 x-axis; hyperbola over
the x-axis (i.e., xy − 1) and the union of two axes over x-axis (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.
Going-up 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 Atiyah-Macdonald during the tutorials.
The tutors Mr. Suhas B, Ms. Mitra Koley and Mr. Praveenkumar Roy were very
much helpful during the tutorials.

 

Tutors

  1.  ManojKummini (CMI, Chennai)
  2.  Clare D’Cruz (CMI, Chennai)
  3.  Krishna Hanumanthu (CMI, Chennai)
  4.  Sarang Sane (IIT Madras)
  5.  Suhas M B (IIT Madras)
  6.  MitraKoley (CMI, Chennai)
  7.  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 14:30-15:30 15:30 - 16:00 Tutorial 2 16:00-17:00 17:00- 17:30
Mon 24/4/17 MK

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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
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Mon 1/5/17 CD

 

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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, Ulhasnagar-03. 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 Visva-Bharati PhD Visva-Bharati 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

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