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

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