**Title**

\nLocal and Global Surjective Capac
ities

**Speaker**

\nRobin Nanda Baidya\, Departm
ent of Mathematics and Statistics\, Georgia State University

\nThis talk will present work that will be inclu
ded in the speaker’s PhD dissertation. Let R be a commutative Noetherian r
ing\, and let M and N be finitely generated R-modules. The global surjecti
ve capacity of M with respect to N over R is the supremum of the nonnegati
ve integers d such that there exists a surjective R-module homomorphism fr
om M to N^d. For every prime ideal p of R\, the local surjective capacity
of M with respect to N over R at p is the supremum of the nonnegative inte
gers d such that there exists a surjective (R_p)-module homomorphism from
(M_p) to (N_p)^d. Our main result states that large local surjective capac
ities collectively guarantee a large global surjective capacity. In the ca
se that M is projective and N is R\, our main result fully recovers\, and
in fact still properly generalizes\, Serre’s Splitting Theorem from algebr
aic K-theory. In the case that R is a direct product of a combination of s
emilocal rings and Dedekind domains\, we can supplement our main result wi
th conditions equivalent to a given global surjective capacity. To close\,
we will mention several conjectures that naturally arise from this work\,
and we will briefly discuss some related results that will serve as the b
asis for future seminar talks.