The lower semicontinuity of the Frobenius splitting numbers

We show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.Comment: 13 pages, revised version, to appear in Mathematical Proceedings of the Cambridge Philosophical Societ

On the upper semi-continuity of the Hilbert-Kunz multiplicity

We show that the Hilbert-Kunz multiplicity of a $d$-dimensional nonregular complete intersection over the algebraic closure of $F_p$, $p>2$ prime, is bounded by below by the Hilbert-Kunz multiplicity of the hypersurface $\sum _{i=0}^{d} x_i^2=0$, answering positively a conjecture of Watanabe and Yoshida in the case of complete intersections.Comment: 14 pages, preprint version, final version to appear in Journal of Algebra, 285, no 1, 222-23

The Frobenius Structure of Local Cohomology

Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an anti-nilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition.Comment: 35 pages. Section 3 was revised to emphasize Theorem 3.1, and some minor corrections/changes were performed. To appear in Algebra and Number Theor

The Frobenius Complexity of a Local Ring of Prime Characteristic

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 $2 \times 2$ minors of a $2 \times 3$ 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 $\mathbb{Q}$-Gorenstein ring.Comment: 20 pages, minor revision