Surjective and splitting capacities

Abstract

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.