Maria Ann Mathew, IIT Bombay

Date: February 25 (Friday), 2.30 - 3.30 pm Link: Title: Generalization of Serre Splitting to monoid algebras R[M] Abstract: In the search for an answer to his conjecture, Serre gave a splitting theorem which states that for a commutative noetherian ring R and a projective R-module P of rank r, if r > dim(R), then P admits a splitting with a free direct summand. This result, often aptly referred to as Serre splitting theorem, shrinks the class of projective R-modules one needs to study to projective R-modules of rank < dim(R) + 1. One may thus ask if a similar splitting exists for projective R[M]-modules of rank > dim(R), when M is a submonoid of Z^n. This problem will be addressed in two parts. In the first part, when rank(P) coincides with dim(R[M]), the said splitting will be demonstrated. The second part will tackle the problem when rank(P) dips even further, i.e., dim(R) < rank(P) < dim(R[M])-1. For n > 0, we define classes of monoids M_n such that if M in M_n is seminormal and rank(P) > dim(R[M]) – n, then P admits a splitting. As a consequence, it can be shown that for a projective module P over Segre extensions S_mn over R, splitting is possible when rank(P) > dim(S_mn)-[(m+n-1)/min(m,n)]. We will also discuss the possibilities of splitting under monic inversion.
Fri, February 25, 2022
Start Time
2:30pm-3:30pm IST
1 hour
Created by
Thu, February 24, 2022 11:51am IST