Description

Speaker: Sampat Kumar Sharma.

Affiliation: ISI, Kolkata.

Date and Time: Friday 16 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

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

Abstract:

R.G. Swan and J. Towber showed that if (a

2

, 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

r!

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.

Affiliation: ISI, Kolkata.

Date and Time: Friday 16 August, 4:00 pm - 5:00 pm.

Venue: Ramanujan Hall, Department of Mathematics.

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

Abstract:

R.G. Swan and J. Towber showed that if (a

2

, 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

r!

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.

Description

Ramanujan Hall, Department of Mathematics

Date

Fri, August 16, 2019

Start Time

4:00pm-5:00pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

DEFAULT ADMINISTRATOR

Updated

Tue, August 13, 2019 2:45pm IST