**Date & Time:** Wedenesday, July 21, 2010, 14:30-15:30.

**Venue:** Ramanujan Hall

**Title:** Splitting of Projective Modules

**Speaker:** Sarang Sane, TIFR Mumbai

**Abstract:** We investigate when a projective module P over a Noetherian
ring A can split off a free summand of rank 1, i.e. when P is isomorphic
to A \oplus Q. A result of Serre allows us to restrict to the case where
rank (P) \leq dim (A). We initially consider results in the situation when
A is a Dedekind domain where there is a natural obstruction in the ideal
class group.
For higher dimensional rings, we have the notion of the Euler
class group where obstructions to splitting take values. We will describe
a bit of this theory and then consider results when A is a smooth affine
domain over a real closed field. In this case, one can calculate the Euler
class group. This allows us to get a complete classification of when the
phenomenon P is isomorphic to A\oplus Q occurs when rank (P) = dim (A).