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Mathematics Seminar Date: Thursday, 18 September 2025 Time: 3.30 p.m. to 4.30 p.m. Venue: Seminar Hall On unimodular rows Sourjya Banerjee Institute of Mathematical Sciences, Chennai. 18-09-25 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 $R$. In this talk, we will discuss unimodular rows and their connections with projective modules, specifically addressing when a unimodular row of length $n$ is completable to a row of an invertible matrix, and how this question arises naturally in the study of projective modules. We will first consider a classical example of a unimodular row of length $(d+1)$ over a $d$-dimensional smooth real algebraic variety that is not completable to such a row. We then describe a class of $d$-dimensional real affine algebras over which every unimodular row of length $(d+1)$ is completable to a row of an invertible matrix. Finally, we discuss the case of unimodular rows of shorter length over smooth algebras belong to the same class. The last part of the talk is based on ongoing joint work with Jean Fasel.
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