Fri, August 16, 2019
Public Access Category: All |
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- Time:
- 2:00pm-3:30pm
- Location:
- Room No. 215 Department of Mathematics
- Description:
- Commutative Algebra Seminars
Speaker: Dilip Patil.
Affiliation: IISc, Bangalore.
Seminar II - Date and Time: Friday 16 August, 2:00 pm - 3:30 pm.
Venue: Room 215, Department of Mathematics.
Title: The Cohen-Structure Theorems.
Abstract: The purpose of these two lectures is to provide the proof of
Cohen’s structure theorem for complete local rings (which Cohen proved in
his PhD thesis 1942, Johns Hopkins University under the guidance of Oscar
Zariski). In these lecture we deal with the equicharacteristic case. We
give a modern and concise treatment by using the notion of formal
smoothness which was introduced by Grothendieck in 1964 in EGA Chapter IV.
It is closely connected with the differentials and throws new light to the
theory of regular local rings and used in proving Cohen’s structure
theorem of complete local rings.
- Time:
- 4:00pm-5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- 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.