Description
Date: February 25 (Friday), 2.30 - 3.30 pm
Link: meet.google.com/pbb-odky-xvs
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.