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A property of non excellent rings. (English) Zbl 0878.13016
From the introduction: We consider the following situation: (*) $$f:A\to A'$$ is a flat homomorphism of commutative noetherian rings, $$I\subset A$$ an ideal such that $$A/I\simeq A'/IA'$$. Under (*) $$\widehat{A}\simeq \widehat{A}'$$ where $$\widehat{\phantom{A}}$$ denotes $$I$$-adic completion. It is known that $$A'$$ is a filtered direct limit of smooth $$A$$-algebras of finite type if and only if the fibers of $$\text{Spec}(f)$$ are geometrically regular. We prove that in characteristic 0 the weaker condition that $$A'$$ is a filtered direct limit of flat $$A$$-algebras of finite type implies that the fibers of $$\text{Spec}(f)$$ are reduced.
##### MSC:
 13J10 Complete rings, completion 13B10 Morphisms of commutative rings
##### Keywords:
flat homomorphism; completion
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##### References:
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