Global F-splitting ratio of modules

Techniques are developed to extend the notions of F-splitting ratios to modules over rings of prime characteristic, which are not assumed to be local. We first develop the local theory for F-splitting ratio of modules over local rings, and then extend it to the global setting. We also prove that strong F-regularity of a pair (R,D), where Dis a Cartier algebra, is equivalent to the positivity of the global F-signature s(R, D) of the pair. This extends a result previously proved by these authors, by removing an extra assumption on the Cartier algebra.(c) 2022 Elsevier Inc. All rights reserved.