**Date & Time:** Wednesday, January 11, 2012, 14:00 - 15:00

**Venue:**Room No. 216

**Title:** Projective modules over overrings of polynomial rings.

**Speaker:** Prof. Alpesh Dhorajia

**Abstract:** Some cancellation results for projective modules are discussed in
the thesis. The first result is the following which extends the cancellation
result of Ravi Rao (for polynomial extensions) and Lindel (for Laurent
polynomial extension) to overrings of polynomial rings: Let $A$ be a
commutative noetherian ring of dimension $d$ and let
$R=A[X_1,\ldots,X_n,Y_1,\ldots,Y_m,(f_1\ldots f_m)^{-1}]$, where $f_i\in
A[Y_i]$. If $P$ is a finitely generated projective $R$-module of rank $\geq
max (2,d+1)$, then $P$ is cancellative. More generally, the elementary group
$E(R \oplus P)$ acts transitively on the set of unimodular rows of $R\oplus
R$.

The second result extends a cancellation result of Gubeladze for projective modules over normal monoid algebras. When $A$ is an affine algebra of dimension $d\geq 4$ over the algebraic closure of finite field, then it is proved that projective modules over $A$ of rank $d-1$ is cancellative, thus settling a question of A.Suslin in affirmative in this case.

Note: This seminar is the speaker's Ph. D. Defence talk.