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I have a "well known question" for which I do not find a reference.

Let $A$ and $B$ be a commutative rings and $A\rightarrow B$ be a faithfully flat morphism. Let $C$ be a ring over $A$. Is it true that $$C_{red}\otimes B\cong (C\otimes B)_{red}?$$ In other words does faithful flatness commute with taking the reduced structure?

liao
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  • Let me make sure I understand your notation. First, I assume you mean $C$ is a module over $A$. By $C_{red}$ do you mean $C$ as a module over the reduced ring $A_{red}?$ –  Dec 17 '13 at 19:38
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    I think he means $C$ is an $A$-algebra and $C_{red}$ is $C/\operatorname{Nil}(C)$ as an $A$-algebra. – Dori Bejleri Dec 17 '13 at 19:56
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    There is no reason for $C_{red}\otimes B$ to be reduced, so the isomorphism does not hold in general. – Cantlog Dec 17 '13 at 21:01

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It's not hard to find a field extension $k\subset K$ such that $K\otimes_kK$ is not reduced.

Community
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