Etaleness and Normality

Let A be an excellent normal semilocal ring with strict Henselization Ahs. If A D Ahs is a normal semilocal intermediate ring semilocally dominated by Ahs, then D is a direct limit of semilocal-etale extensions of A. If A is an excellent normal local domain then the integral closure of A in a finite separable field extension L of the quotient field Q(A) is etale over A if and only if there is a Q(A) morphism L → Q(Ahs) and Q(Ahs) contains a normal closure of L. We apply these results to the normalizations of an arbitrary excellent reduced local ring A and its strict Henselization Ahs to obtain a characterization for local-etaleness of a local intermediate ring A D Ahs whose total quotient ring L is finitely generated over Q(A). Specifically, such an intermediate ring D is local-etale over A if and only if the normalization of D is semilocal and has the proper residual field structure relative to à and to D. We generalize this result to characterize local-etaleness of arbitrary injective reduced local morphisms A → D, where A is excellent, finding that such a morphism is local-etale if and only if the morphism à → is semilocal-etale and there is an isomorphism D A Ã.