Split embedding problems over complete domains

Abstract

We prove that every finite split embedding problem is solvable over the field $K(\mskip-1.5mu(X_1,\ldots,X_n)\mskip-1.5mu)$ of formal power series in $n \geq 2$ variables over an arbitrary field $K$, as well as over the field $\operatorname{Quot}(A[\mskip-2mu[X_1,\ldots,X_n]\mskip-2mu])$ of formal power series in $n \geq 1$ variables over a Noetherian integrally closed domain $A$. This generalizes a theorem of Harbater and Stevenson, who settled the case $K(\mskip-1.5mu(X_1,X_2)\mskip-1.5mu)$.