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
