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In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local structure plays a big role. For instance it is used to prove that fibers of $C^p$ maps locally of constant rank are embedded submanifolds (of known codimension).

Where is the local structure of étale morphisms needed further on in the theory? Which important theorems and proofs crucially require actually knowing an explicit local form?

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  • $\begingroup$ As you say, the local structure condition in the smooth category has to do with finite codimension--not just codimension 1. That seems analogous to condition (1) in your link to Stacks. That theorem on étale maps is used constantly. But standard étale maps correspond to embeddings with codimension precisely 1. So I would like to see an answer by someone more expert, but probably the local existence of standard étale forms is less practically useful than condition (1) just as local exist of codimension 1 embeddings in the smooth case is less useful than the implicit function theorem. $\endgroup$
    – Colin McLarty
    Commented Nov 22, 2018 at 15:00
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    $\begingroup$ The beauty of the stacks project is that you can see exactly where this lemma is used (and where those results are used, etc.): stacks.math.columbia.edu/tag/02GT/statistics#dependencies. so you can decide for yourself if you think it's necessary/useful $\endgroup$
    – Dylan Wilson
    Commented Nov 24, 2018 at 16:18

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