The Stacks project

Remark 41.19.6. The result on “reducedness” does not hold with a weaker definition of étale local ring maps $A \to B$ where one drops the assumption that $B$ is essentially of finite type over $A$. Namely, it can happen that a Noetherian local domain $A$ has nonreduced completion $A^\wedge $, see Examples, Section 110.16. But the ring map $A \to A^\wedge $ is flat, and $\mathfrak m_ AA^\wedge $ is the maximal ideal of $A^\wedge $ and of course $A$ and $A^\wedge $ have the same residue fields. This is why it is important to consider this notion only for ring extensions which are essentially of finite type (or essentially of finite presentation if $A$ is not Noetherian).

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