Mon, August 26, 2019
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August 2019
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3:00pm [3:30pm] R.V. Gurjar
Commutative Algebra seminar. Speaker: R V Gurjar. Affiliation: IIT Bombay. Date and Time: Monday 26 August, 3:30 pm - 5:00 pm. Venue: Room 215, Department of Mathematics. Title: Lecture series on Ramification in Commutative Algebra and Algebraic Geometry. Abstract: We will consider mainly the following situation. Let R,S be complete normal local domains over an alg. closed field k of char. 0 such that S is integral over R. Our aim is to describe three ideals in S; I_N, I_D, I_K (Noether, Dedekind, Kahler differents resp.) each of which capture the ramified prime ideals in S over R. In general these three ideals are not equal. An important special case when all are equal is when S is flat over R. We will prove many of these statements. The case when there is a finite group G of k-automorphisms of S such that R is the ring of invariants is already very interesting. Then many nice results are proved. These include works of Auslander-Buchsbaum, Chevalley-Shephard-Todd, Balwant Singh, L. Avramov, P. Roberts, P. Griffith, P. Samuel.... I will try to discuss all these results. I believe that these results and ideas involved in them will be very valuable to students and faculty both. Prerequisites. Basic knowledge of Commutative Algebra and language of Algebraic Geometry (no sheaf theory!). I will "throw in" topological proofs from time to time to make the results intuitively more clear.