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, D), where D is a Cartier algebra, is equivalent to the positivity of the global F-signature s(R, D) of the pair. This extends a result proved in [DSPY16], by removing an extra assumption on the Cartier algebra. is part of a free resolution of F e * (M s ) over the ring R s . In particular, by localizing at any prime Q ∈ Spec(R) not containing s, the complex is still exact, and it becomes a free resolution of F e * (M Q ) over R Q . However, it may not be minimal. That is,∈ Q}, we therefore have that β i (e, Q, M) β i (e, P, M) for all Q ∈ D(s). This shows dense upper semi-continuity of the function P → β i (e, P, M). We now focus on the function P → χ i (e, P, M). Let P, ϕ j , ψ j and s ∈ R P be as above. Let Q ∈ D(s), and denote Ω j = ker((ψ j−1 ) Q ), for all j = 0, . . . , i.

Equimultiplicity Theory of Strongly $F$-regular rings

“…We refer the reader to [Li08, AL08, DSHNnB17, DPY18] for basics on Frobenius Betti numbers and [DPY18] for basics on Frobenius Euler characteristic. Our study begins with a simple application of the Auslander-Buchsbaum formula.…”

“…It has been known for some time that Hilbert-Kunz multiplicity, F -signature, and Frobenius Betti numbers serve as measurements of singularities, see [WY00,HY02], [HL02,AL03], and [AL08] respectively. Frobenius Euler characteristic was introduced in [DPY18] as a tool to prove that the functions β F i : Spec(R) → R sending P → β F i (R P ) are upper semicontinuous and it was unclear from those techniques whether or not Frobenius Euler characteristic could be used to detect regular rings. In the present article, we show Frobenius Euler characteristic does indeed serve as a measurement of singularities under the strongly F -regular hypothesis.…”