F-signature of pairs and the asymptotic behavior of Frobenius splittings

We generalize F-signature to pairs (R, D) where D is a Cartier subalgebra on R as defined by the first two authors. In particular, we show the existence and positivity of the F-signature for any strongly F-regular pair. In one application, we answer an open question of Aberbach and Enescu by showing that the F-splitting ratio of an arbitrary F-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the F-signature and the F-splitting ratio in the spirit of the work of R. Fedder. (C) 2012 Elsevier Inc. All rights reserved.