Surjective capacity and splitting capacity

Abstract

Let M and N be finitely generated modules over a d-dimensional commutative Noetherian ring R. We show that, if M maps onto N⊕(t+d) locally, then M maps onto N⊕t globally. In a similar manner, we demonstrate that, if M locally admits N⊕(t+d) as a direct summand, then M globally admits N⊕t as a direct summand. We also generalize splitting theorems due to Serre, De Stefani–Polstra–Yao, and Bass by exploiting the j-spectrum of R and by allowing M and N to be modules over a possibly noncommutative module-finite R-algebra.