Description

Commutative Algebra seminar I.

Speaker: Soumi Tikader.

Affiliation: ISI Kolkata.

Date and Time: Monday 21 October, 11:30 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.

Speaker: Soumi Tikader.

Affiliation: ISI Kolkata.

Date and Time: Monday 21 October, 11:30 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.

Description

Ramanujan Hall, Department of Mathematics

Date

Mon, October 21, 2019

Start Time

11:30am-12:30pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

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Updated

Mon, October 21, 2019 11:39am IST