Title: The Steinberg formula for orbit spaces

Speaker: Dr. Anuradha Garge

IIT-Bombay

Time: 4 to 5 pm

Venue: Room no: 113, Maths Department

Abstract:

The orbit space of unimodular rows of size $n$

(denoted by $\Um_n$) modulo elementary action has

been an object of study for both topologists and algebraists.

L. N. Vaserstein and W. van der Kallen showed that in certain cases,

depending on the dimension of the ring,

the orbit space admits a group structure and this can be explained

using the Vaserstein and universal weak Mennicke symbols. These will

be detailed in the talk.

Throughout this talk, for us $R$ will be a commutative ring with unity.

For a map $\Um_n (R) \stackrel{\varphi}{\longrightarrow} A$,

$A$ an abelian group, we say that the Steinberg formula holds if

for $1 \leq i \neq j \leq n, \lambda \in R$,

\begin{itemize}

\item $\varphi (a_1, \ldots, a_n) =

\varphi (a_1, \ldots, a_i+ \lambda a_j, a_{i+1}, \ldots, a_n)$.

\item $\varphi (a_1, \ldots, a_i, \ldots, a_n) +

\varphi (a_1, \ldots, (1 - a_i), \ldots, a_n) =

~~~~~~~~$

$~~~~~~~~~~~~~~~~~~ \varphi (a_1, \ldots, a_i(1- a_i), \ldots, a_n).$

\end{itemize}

The aim of this talk is to show that the Steinberg formula holds

for the Vaserstein symbol and the weak Mennicke symbol. The main

feature is that this formula holds for the above symbols

independent of the dimension assumptions on the ring.

Tony J. Puthenpurakal

Convener

SCC