# Excellence in prime characteristic

## TL;DR

This paper explores the conditions under which Noetherian domains in prime characteristic are excellent, linking excellence to properties of the Frobenius map and $p^{-e}$-linear maps, and discusses the abundance of non-excellent rings.

## Contribution

It establishes new equivalences for excellence in prime characteristic rings, connecting Frobenius properties with $p^{-e}$-linear maps and solid algebra concepts.

## Key findings

- Frobenius finiteness characterizes excellence in these rings.
- Existence of non-zero $p^{-e}$-linear maps implies excellence.
- Non-excellent rings are common and easily constructed in prime characteristic.

## Abstract

Fix any field $K$ of characteristic $p$ such that $[K:K^p]$ is finite. We discuss excellence for Noetherian 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 non-zero $p^{-e}$-linear map for $R$ or equivalently, that $R$ is a solid $R$-algebra under Frobenius. In particular, this means that Frobenius split Noetherian domains that are generically $F$-finite 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 in function fields. This paper is mostly expository in nature.