**Title**

\nLocal
and Global Surjective Capacities

**Speaker**

\nR
obin Nanda Baidya\, Department of Mathematics and Statistics\, Georgia Sta
te University

**Abstract**

\nThis talk will pres
ent work that will be included in the speaker’s PhD dissertation. Let R be
a commutative Noetherian ring\, and let M and N be finitely generated R-m
odules. The global surjective capacity of M with respect to N over R is th
e supremum of the nonnegative integers d such that there exists a surjecti
ve R-module homomorphism from 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 supre
mum of the nonnegative integers d such that there exists a surjective (R_p
)-module homomorphism from (M_p) to (N_p)^d. Our main result states that l
arge local surjective capacities collectively guarantee a large global sur
jective capacity. In the case that M is projective and N is R\, our main r
esult fully recovers\, and in fact still properly generalizes\, Serre’s Sp
litting Theorem from algebraic K-theory. In the case that R is a direct pr
oduct of a combination of semilocal rings and Dedekind domains\, we can su
pplement our main result with conditions equivalent to a given global surj
ective capacity. To close\, we will mention several conjectures that natur
ally arise from this work\, and we will briefly discuss some related resul
ts that will serve as the basis for future seminar talks.