Excellence in prime characteristic

Abstract

Fix any field K of characteristic p such that [K : K p ] is finite. We discuss excellence for domains whose fraction field is K, showing for example, that R is excellent if and only if the Frobenius map is finite on R. Furthermore, we show R is excellent if and only if it admits some nonzero p −e -linear map for R (in the language of [3]), or equivalently, that the Frobenius map R → F * R defines a solid R-algebra structure on F * R (in the language of [11]). In particular, this means that generically Ffinite, Frobenius split Noetherian domains are always excellent. We also show that non-excellent rings are abundant and easy to construct in prime characteristic, even within the world of regular local rings of dimension one inside function fields.

Towards Vorst's conjecture in positive characteristic

“…Note that a reduced -algebra which satisfies for all maximal points is excellent if and only if it is Frobenius finite, as shown by Kunz and Datta–Smith [DS18, Corollary 2.6].…”

Essential finite generation of extensions of valuation rings

“…The defect of 𝜈∕𝜈 𝑝 controls interesting properties of 𝜈. For example, it is shown in [7, Proof of Theorem 5.1] (see also [6,Corollary IV.23]) that if 𝐾 is a function field of a variety over a ground field 𝑘, and 𝜈 is a valuation of 𝐾∕𝑘, then 𝜈∕𝜈 𝑝 is defectless if and only if 𝜈 is an Abhyankar valuation of 𝐾∕𝑘.…”