F-signature exists

Details F-signature exists F-signature exists

2011-03-21

arxiv.org/abs/1103.4173v1

Mathematics

Commutative Algebra

19 pages. Substantially updated from a version which was circulated on a limited basis in the fall of 2010

Scientific paper

Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings of the iterates of the Frobenius endomorphism of R. This invariant was first formally defined by C. Huneke and G. Leuschke and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of finite quotient singularities.

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