Fri, August 16, 2019
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August 2019
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2:00pm [2:00pm] Dilip P Patil, IISc Bangalore
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.

4:00pm [4:00pm] Sampat Kumar Sharma : ISI, Kolkata
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.