Alladi Ramakrishnan Hall

Zero cycles, Mennicke symbols and $\mathrm{K}_1$-stability of real affine algebras

Sourjya Banerjee

IMSc, Chennai

Let $R$ be an affine algebra of (Krull) dimension $d \geq 2$ over the
base field $\mathbb{R}$. In this talk, we shall discuss the following:
First, we define the $d$-th weak Euler class group
$\mathrm{E_0^d}(R)$, as defined by Bhatwadekar-R. Sridharan, the
Levine-Weibel Chow group of zero cycles
$\mathrm{CH}_0(\text{Spec}(R))$ modulo rational equivalence, and a
canonical map between them. Then we show that for a certain class of
real algebras, this map is an isomorphism. In the second part, we
define the weak universal Mennicke symbol and the universal Mennicke
symbol of length $d+1$ over $R$. Further, we show that these coincide
for the same class of real algebras. If time permits, then we shall
discuss some stability results for the groups $\mathrm{K_1}(R)$ and
$\mathrm{K_1Sp}(R)$.