Sampat Kumar Sharma :TIFR Mumbai

Title: On a question of Suslin about completion of unimodular rows


R.G. Swan and J. Towber showed that if (a2, b, c) is a unimodular row
over any commutative ring R then it can be completed to an invertible
matrix over R. This was strikingly generalised by A.A. Suslin who showed
that if (a

0 , a1, . . . , ar) is a unimodular row over R then it can be com-
pleted to an invertible matrix. As a consequence A.A. Suslin proceeds to

conclude that if 1

r! ∈ R, then a unimodular row v(X) ∈ Umr+1(R[X])
of degree one, with v(0) = (1, 0, . . . , 0), is completable to an invertible
matrix. Then he asked
(Sr(R)): Let R be a local ring such that r! ∈ GL1(R), and let p =
(f0(X), . . . , fr(X)) ∈ Umr+1(R[X]) with p(0) = e1(= (1, 0, . . . , 0)). Is it
possible to embed the row p in an invertible matrix?
Due to Suslin, one knows answer to this question when r = d + 1,
without the assumption r! ∈ GL1(R). In 1988, Ravi Rao answered this
question in the case when r = d.
In this talk we will discuss about the Suslin’s question Sr(R) when r =
d − 1. We will also discuss about two important ingredients; “homotopy
and commutativity principle” and “absence of torsion in Umd+1(R[X])
Ed+1(R[X]) ”,
to answer Suslin’s question in the case when r = d − 1, where d is the
dimension of the ring.
Room No. 216 Department of Mathematics
Fri, July 26, 2019
Start Time
4:00pm-5:00pm IST
1 hour
Created by
Thu, July 25, 2019 6:24pm IST