Title: Excellence 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.