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).