We present a unified approach to the study of F -signature, Hilbert-Kunz multiplicity, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F -signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring.

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring$(R,\mathfrak{m},k)$of prime characteristic to the real numbers at reduced parameter elements with respect to the$\mathfrak{m}$-adic topology.

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We give new proofs of two results of Stafford from [Sta81], which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras. Theorem B. Let R be a commutative Noetherian ring and M a finitely generated R-module such that, for each p ∈ Spec(R), M p contains a free R p summand of rank at least dim(R/p)+ 1. For any finitely generated projective R-module Q and any finitely generatedStafford proved analogue results in [Sta81, Corollaries 5.9 and 5.11] where, instead of looking at local number of free summands, he used the notion of r-rank. Stafford's results are actually very general in that he proved analogues of Theorem A and Theorem B for rings that are not necessarily commutative.Let M be a finitely generated R-module. Observe that the conclusion of Theorem A can be viewed as a statement about the existence of a surjective homomorphism inside Hom R (M, R). For several applications, it is useful to restrict the selection of homomorphisms M → R to those belonging to a given R-submodule E of Hom R (M, R). This is the scenario arising, for instance, from the study of the F-signature of Cartier subalgebras of C R = ⊕ e Hom R (F e * R, R), where R is an F-finite local ring of prime characteristic [BST12]. We can naturally generalize the definitions in this paper and results about free-basic elements to the setup of R-submodules E of Hom R (M, R). The main achievement in this direction is a new generalization of Theorem A.Theorem C. Let R be a commutative Noetherian ring, M a finitely generated R-module, and E an R-submodule of Hom R (M, R). Assume that, for each p ∈ Spec(R), M p contains a free E p -summand of rank at least dim(R/p) + 1. Then M contains a free E -summand.As the main application of Theorem C, the authors establish in [DSPY16] the existence of a global F-signature with respect to Cartier subalgebras of C R , where R is an F-finite ring of prime characteristic, not necessarily local. Section 2, we present a version of Theorem A via rather direct and elementary techniques. However, the methods we employ are not effective enough to prove the result in its full generality. In Section 3, we introduce the notion of freebasic element, which allows us to prove Theorem A and Theorem B in their full generality. We believe that free-basic elements are worth exploring and interesting on their own, as they share many good properties both with basic elements and unimodular elements. In Section 4, we generalize free-basic elements to the setup of R-submodules of Hom R (M, R), and prove Theorem C.

Abstract. This paper establishes uniform bounds in characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the Spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the F-signature function. From this we establish that the F-signature function is lower semi-continuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally proven by Ilya Smirnov.