Description
Commutative Algebra seminar (Please note that this talk will be via Skype).
Speaker: Soumi Tikader.
Affiliation: ISI Kolkata.
Date and Time: Thursday 31 October, 11:45 am - 12:30 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Orbit spaces of unimodular rows over smooth real affine algebras.
Abstract: In this talk we will discuss about the group structure on orbit
spaces of unimodular rows over smooth real affine algebras. With a few
definition and some results to start, we will prove a structure theorem of
elementary orbit spaces of unimodular rows over aforementioned ring with
the help of similar kind results on Euler class group. As a consequences,
we will prove that :
Let $X=Spec(R)$ be a smooth real affine variety of even dimension $d > 1$,
whose real points $X(R)$ constitute an orientable manifold. Then the set
of isomorphism classes of (oriented) stably free $R$ of rank $d > 1$ is a
free abelian group of rank equal to the number of compact connected
components of $X(R)$.
In contrast, if $d > 2$ is odd, then the set of isomorphism classes of
stably free $R$-modules of rank $d$ is a $Z/2Z$-vector space (possibly
trivial). We will end this talk by giving a structure theorem of Mennicke
symbols.
PS: Soumi Tikader is a post doctoral candidate.