## Unibranch and etaleness

Is it true that if A->B is an etale map of local rings, then A is unibranch if and only B is?

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 You surely meant to require the map to be local and "essentially etale", not "etale" (since local maps are rarely even finite type). Anyway, so then the condition you ask for is actually what characterizes "geometrically unibranched"; see 18.6.12, 18.8.15, and 18.8.16 in EGA IV$_4$. The distinction comes about from residue field extensions in the etale extension of rings. In down to earth terms, a nodal singularity on an irreducible curve could have tangent directions not rational over the base field. – BCnrd Apr 16 2010 at 23:10