  Let R be a commutative ring, S a module-finite R-algebra, M a right S-module,
and N a finitely generated right S-module such that the intersection of Max(R)
and Supp(N) is finite-dimensional and Noetherian. Working under various
combinations of additional hypotheses on R, M, and N, we give lower bounds on
the global surjective capacity of M with respect to N over S, that is, the
supremum of the nonnegative integers t such that there is an S-linear
surjection from M onto the direct sum of t copies of N. We express our lower
bounds in terms of local analogues of global surjective capacity and
topological properties of Spec(R). Assuming that N is finitely presented over
S, we also give lower bounds on the global splitting capacity of M with respect
to N over S, that is, the supremum of the nonnegative integers t such that
there is an S-linear split surjection from M onto the direct sum of t copies of
N. In the process, we generalize Serre's Splitting Theorem from algebraic
K-theory and a theorem from stable algebra due to Bass. We also generalize a
theorem on Noetherian modules by De Stefani, Polstra, and Yao that serves as an
analogue of an older result by Stafford. To close, we consider the case of
finitely generated modules over a Dedekind domain; in this case, we show that
we can provide conditions equivalent to having a given global surjective or
splitting capacity.
