Projective modules: some old and new results
S. M. Bhatwadekar
The Tata Institute of Fundamental Research
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,
¼, Xn] 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
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.
Sep. 4-21, 10:00-12:00|
every Mondays, Tuesdays, Wednesdays , and Thursdays
|Place:||Lecture Hall, Niavaran Bldg., Niavaran Sqr., Tehran, Iran|