


Commutative algebra seminar Tuesday, 27 September 2022 @3.30 pm. Venue: Ramanujan Hall Speaker: Tony Puthenpurakal, IIT Bombay Title: On coefficient idealsII Abstract: Let (A, m) be a CohenMacaulay local ring of dimension d ≥ 2 with infinite residue field and let I be an mprimary ideal. Let For 0 ≤ i ≤ d let Ii be the i thcoefficient ideal of I. Also let Ie = Id denote the RatliffRush closure of A. Let G = GI (A) be the associated graded ring of I. We show that if dim H j G+ (G) ∨ ≤ j−1 for 1 ≤ j ≤ i ≤ d−1 then (I n)d−i = Ifn for all n ≥ 1. In particular if G is generalized CohenMacaulay then (I n)1 = Ifn for all n ≥ 1. As a consequence we get that if A is an analytically unramified domain with G generalized CohenMacaulay, then the S2ification of the Rees algebra A[It] is L n≥0 Ifn.
Dear all, Prof. Nitin Nitsure will give a series of lectures on 'Algebraic Stacks and Moduli Theory' starting next week. The draft announcement is attached. The first talk will be introductory and will be accessible to everyone with very minimal knowledge of algebraic geometry. For the benefit of students, this is a very central and important area of algebraic geometry and Prof. Nitsure is a very good lecturer. Do try and attend the first lecture in Ramanujan Hall on Tuesday 27th at 5:10 pm. The first talk will be of 1 hour.
Date 28 September 2022
Time 45 pm
Venue: Ramanujan Hall
Speaker: Prof. Eknath Ghate, TIFR, Mumbai
Title: Semistable representations as limits of crystalline representations
Abstract: We construct an explicit sequence of crystalline representations
converging to a given irreducible twodimensional semistable
representation of the Galois group of Q_p. The convergence takes place in
the blowup space of twodimensional trianguline representations studied
by Colmez and Chenevier. It is connected to a classical formula going back
to Greenberg and Stevens expressing the Linvariant as a logarithmic
derivative.
Our convergence result can be used to compute the reductions of any
irreducible twodimensional semistable representation in terms of the
reductions of certain nearby crystalline representations of exceptional
weight. For instance, using our zigzag conjecture on the reductions of
crystalline representations of exceptional weights, we recover completely
the work of BreuilMezard and GuerberoffPark on the reductions of
irreducible semistable representations of weights at most p+1, at least
on the inertia subgroup. As new cases of the zigzag conjecture are
proved, we further obtain some new information about the reductions for
small odd weights.
Finally, we use the above ideas to explain away some apparent violations
to local constancy in the weight of the reductions of crystalline
representations of small weight that were noted in our earlier work and
which provided the initial impetus for this work.
This is joint work with Anand Chitrao and Seidai Yasuda.