IPM

“ Schools ”

“ Centers and Units ”

“ Research Groups ”

“ Useful Links ”

“ E-Services ”

“ Bulletin Board ”

School of Mathematics

Short Course

Projective modules: some old and new results
S. M. Bhatwadekar
The Tata Institute of Fundamental Research
Mumbai, India

Abstract

Let A be a commutative ring. A finitely generated A-module P is said to be projective if there exists a finitely generated A-module Q such that P إQ @ Aⁿ where Aⁿ denotes the free A-module of rank n.

The study of projective modules, apart from its intrinsic interest, also has motivation from topology. For example, the famous conjecture of Serre viz. all projective modules over a polynomial algebra over a field are free has genesis in the fact that Euclidean space over reals being contractible, all topological vector bundles over Rⁿ are trivial. Therefore, one can ask whether there exist purely algebraic analogues of known results on topological vector bundles.

Motivated from topology, we have following results which have profound effect on the subsequent developments of the study of projective modules :
�

Theorem 1. (Quillen-Suslin) Let k be a field and let A = k[X₁, �, X_n] be a polynomial algebra. Then all finitely generated projective modules over A are free.

Theorem 2. (Serre) Let A be a commutative Noetherian ring of (Krull) dimension d. Let P be a projective A-module of rank > d. Then P splits off a free direct summand of rank 1 i.e. P @ A إQ.

Theorem 3. (Bass) Let A be a commutative Noetherian ring of dimension d. Let P and P₁ be projective A-modules of rank > d. If A إP @ A إP₁ then P @ P₁.

In series of 10/12 lectures I am going to present proofs of above results with all details. Subsequently I will also give a proof of a theorem of Mohan Kumar and Murthy ( assuming some results) which gave impetus to active research in the area in last 10 years. The lectures will be presented in a conversational style. Technical terms encountered in my lectures will be explained in simple possible manner. First couple of lectures will be devoted to give definitions, set up some notations and prove some simple results.
�

Prerequisites: Familiarity with following topics in Algebra : Noetherian commutative rings, Krull dimension, Primary decomposition, Regular rings.

Time:	Sep. 4-21, 10:00-12:00 every Mondays, Tuesdays, Wednesdays , and Thursdays
Place:	Lecture Hall, Niavaran Bldg., Niavaran Sqr., Tehran, Iran

See photos

See photos