Commutative Algebra seminar
Speaker: Sourjya Banerjee (IMSc)
Host: Manoj Keshari
Title: From Unimodular Rows to Zero Cycles over Real Varieties
Time, day and date: 4:00:00 PM - 5:00:00 PM, Monday, September 1
Venue: Ramanujan Hall
Abstract: A unimodular row of length $n$ over a commutative Noetherian ring $R$ (with $1 \neq 0$) is a row vector $(v_1,\ldots,v_n) \in R^n$ such that the ideal generated by $v_1,\ldots,v_n$ is the whole ring $R$. We discuss unimodular rows and their connections with projective modules, specifically addressing when a unimodular row of length $n$ can be completed to a row of an invertible matrix and how this question arises naturally in the study of projective modules. We present a classical example of a unimodular row of length $(d+1)$ over a $d$-dimensional smooth real variety that cannot be completed to such a row. We then describe a class of $d$-dimensional real varieties where every unimodular row of length $d+1$ is completable to a row of an invertible matrix. We discuss some applications to the study of the $d$-th Euler class group $\mathrm{E}^d(R)$ defined by Bhatwadekar and Raja Sridharan, the Levine--Weibel Chow group of zero cycles $\mathrm{CH}_0(\mathrm{Spec}(R))$, and the natural maps between them. If time permits, we discuss unimodular rows of length $d$ over $d$-dimensional smooth real varieties. The final part is based on ongoing joint work with Jean Fasel.