The \'etale open topology over the fraction field of a henselian local domain

TL;DR

This paper compares the étale open topology and the R-adic topology over the fraction field of a henselian local domain, establishing conditions under which they refine each other and coincide.

Contribution

It characterizes the relationship between the étale open topology and the R-adic topology over fraction fields of henselian local domains, especially in regular cases.

Findings

If R is henselian, the R-adic topology refines the étale open topology.

If R is regular, the étale open topology refines the R-adic topology.

Over fields like L((t_1,...,t_n)), the étale open topology coincides with the L[[t_1,...,t_n]]-adic topology.

Abstract

Suppose that $R$ is a local domain with fraction field $K$. If $R$ is Henselian then the $R$-adic topology over $K$ refines the \'etale open topology. If $R$ is regular then the \'etale open topology over $K$ refines the $R$-adic topology. In particular the \'etale open topology over $L((t_1,\ldots,t_n))$ agrees with the $L[[t_1,\ldots,t_n]]$-adic topology for any field $L$ and $n \ge 1$.