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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.
13J10 Complete rings, completion
13B10 Morphisms of commutative rings
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