Commutative algebra  seminar

Tuesday, 27 September 2022 @3.30 pm.

Venue: Ramanujan Hall

Speaker: Tony Puthenpurakal, IIT Bombay

Title: On coefficient ideals-II

Abstract: Let (A, m) be a Cohen-Macaulay local ring of dimension d ≥ 2 with infinite residue field and let I be an m-primary ideal. Let For 0 ≤ i ≤ d let Ii be the i th-coefficient ideal of I. Also let Ie = Id denote the Ratliff-Rush 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 Cohen-Macaulay 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 Cohen-Macaulay, then the S2-ification 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 4-5 pm

Venue: Ramanujan Hall

Speaker: Prof. Eknath Ghate, TIFR, Mumbai

Title: Semi-stable representations as limits of crystalline representations

Abstract: We construct an explicit sequence of crystalline representations
converging to a given irreducible two-dimensional semi-stable
representation of the Galois group of Q_p. The convergence takes place in
the blow-up space of two-dimensional trianguline representations studied
by Colmez and Chenevier. It is connected to a classical formula going back
to Greenberg and Stevens expressing the L-invariant as a logarithmic
derivative.

Our convergence result can be used to compute the reductions of any
irreducible two-dimensional semi-stable representation in terms of the
reductions of certain nearby crystalline representations of exceptional
weight. For instance, using our zig-zag conjecture on the reductions of
crystalline representations of exceptional weights, we recover completely
the work of Breuil-Mezard and Guerberoff-Park on the reductions of
irreducible semi-stable representations of weights at most p+1, at least
on the inertia subgroup. As new cases of the zig-zag 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.