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

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

2011-07-06

arxiv.org/abs/1107.1082v1

Mathematics

Commutative Algebra

26 pages, comments welcome

Scientific paper

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 I. Aberbach and F. 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.

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