Global Frobenius Betti numbers and F-splitting ratio

Abstract

We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair $(R,\mathcal{D})$, where $\mathcal{D}$ is a Cartier algebra, is equivalent to the positivity of the global F-signature $\mathrm{s}(R,\mathcal{D})$ of the pair. This extends a result previously proved by these authors, by removing an extra assumption on the Cartier algebra.

31 pages