Let (R, m, k) be a d-dimensional Noetherian reduced local ring of prime characteristic p such that R 1/p e are finite over R for all e ∈ N (i.e. R is F -finite). Consider the sequence {a e /q α(R)+d } ∞ e=0 , in which α(R) = log p [k : k p ], q = p e , and a e is the maximal rank of free R-modules appearing as direct summands of R-module R 1/q . Denote by s − (R) and s + (R) the liminf and limsup, respectively, of the above sequence as e → ∞. If s − (R) = s + (R), then the limit, denoted by s(R), is called the F -signature of R. It turns out that the F -signature can be defined in a way that is independent of the module finite property of R 1/q over R. We show that: (1) If s + (R) 1 − 1/(d!p d ), then R is regular; (2) If R is excellent such that R P is Gorenstein for every P ∈ Spec(R) \ {m}, then s(R) exists;(3) If (R, m) → (S, n) is a local flat ring homomorphism, then s ± (R) s ± (S) and, if furthermore S/mS is Gorenstein, s ± (S) s ± (R)s(S/mS).

Finitely generated modules with finite F -representation type over Noetherian (local) rings of prime characteristic p are studied. If a ring R has finite F -representation type or, more generally, if a faithful R-module has finite F -representation type, then tight closure commutes with localizations over R. F -contributors are also defined, and they are used as an effective way of characterizing tight closure. Then it is shown that lime→∞(# ( e M, M i )/(ap d ) e ) always exists under the assumption that (R, m) satisfies the Krull-Schmidt condition and M has finite F -representation type by {M 1 , M 2 , . . . , Ms}, in which all the M i are indecomposable R-modules that belong to distinct isomorphism classes and a = [R/m : (R/m) p ].

This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal a of R, there is a power Q of p, depending on a, such that the Qth Frobenius power of the Frobenius closure of a is equal to the Qth Frobenius power of a. The paper addresses the question as to whether there exists a uniform Q 0 which 'works' in this context for all parameter ideals of R simultaneously.In a recent paper, Katzman and Sharp proved that there does exists such a uniform Q 0 when R is CohenMacaulay. The purpose of this paper is to show that such a uniform Q 0 exists when R is a generalized Cohen-Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong d-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne-Speiser-Lyubeznik Theorem employed by Katzman and Sharp in the Cohen-Macaulay case. (C. Huneke), m.katzman@sheffield.ac.uk (M. Katzman), r.y.sharp@sheffield.ac.uk (R.Y. Sharp), ywyao@umich.edu (Y. Yao).

Abstract. We introduce a new invariant for local rings of prime characteristic, called Frobenius complexity, that measures the abundance of Frobenius actions on the injective hull of the residue field of a local ring. We present an important case where the Frobenius complexity is finite, and prove that complete, normal rings of dimension two or less have Frobenius complexity less than or equal to zero. Moreover, we compute the Frobenius complexity for the determinantal ring obtained by modding out the 2 × 2 minors of a 2 × 3 matrix of indeterminates, showing that this number can be positive, irrational and depends upon the characteristic. We also settle a conjecture of Katzman, Schwede, Singh and Zhang on the infinite generation of the ring of Frobenius operators of a local normal complete Q-Gorenstein ring.

It is proved that when R is a local ring of positive characteristic, φ : R → R is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through φ, then R is regular. This broad generalization of Kunz's characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If φ is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over φ of any non-zero finitely generated R-module grow at the same rate, on an exponential scale, as the Betti numbers of k over R.