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

Manuel Blickle, Karl Schwede, Kevin Tucker

Number | 10 |
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Author | Prof. Dr. Manuel Blickle |

Year | 2011 |

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.