Algebraic Stacks lecture series

Tuesday, 6 February 2024, 11:30 AM

===========================

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: Bhaskaracharya Pratishthana

Title: Going up, liftings and valuative criteria.

Abstract: This lecture is in the series on separated and proper morphisms of topological spaces, schemes, algebraic spaces and algebraic stacks, We will begin with some commutative algebraic results which have the broad theme of `going up', `extensions' and `lifts'. These are of three kinds: (1) Results about lifting primes, (2) Properties of valuation rings and extensions of local domains to maximal such in their quotient fields, and (3) the formulations of valuative criteria as problems of lifting morphisms. This will be followed by a close look at specialization and generalization of points. (This will be re-visited when we go to algebraic stacks, as surprising new behaviour can occur.) With the above preliminaries done, we will give an exposition of the proof of the valuative criteria for separatedness and universal closedness of morphisms of schemes. Preparatory reading for students: Hartshorne `Algebraic Geometry' Chapter 2 section 4, and the portion on valuation rings in Atiyah-Macdonald, `Commutative Algebra' chapter 5.

Commutative Algebra Seminar

Tuesday 6 Feb, 4.00-5:30 pm

========================

Venue: Room 215

Host: Tony J. Puthenpurakal

Speaker : Samarendra Sahoo

Affiliation: IIT Bombay

Title: Eisenbud conjecture on bounded Betti number

Abstract: Eisenbud conjecture: Let Q be an NLR and I be generated by Q-regular sequence. Set A=Q/I. Let F.\to M be a minimal-free resolution of M such that the ranks of the free modules F¡ are bounded, then F is eventually periodic of period 2. He proved that it is true for the Complete intersection ring. I will continue my lecture and discuss proof of this.

Algebraic Groups seminar

Tuesday, February 6, 2024, 4 pm

========================

Venue: Ramanujan Hall

Host: Shripad M. Garge

Speaker: Shripad M. Garge

Affiliation: IIT Bombay

Title: Homogeneous Spaces - I

Abstract: We introduce the notion of homogeneous spaces for linear algebraic groups.