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Cancellation law for étale morphisms

Lemma 29.36.18. Let $f : X \to Y$ be a morphism of schemes over $S$. If $X$ and $Y$ are étale over $S$, then $f$ is étale.

Proof. See Algebra, Lemma 10.143.8. $\square$

Comments (7)

Comment #3037 by Brian Lawrence on

Suggested slogan: A map between two schemes, which are etale over a common base, is etale.

Comment #4953 by awllower on

Suggested slogan: Cancellation law for étale morphisms.

Comment #4971 by Floris Ruijter on

Suggested slogan: Morphism of étale schemes is étale.

Comment #5209 by on

OK, I went with the cancellation law slogan

Comment #8251 by DatPham on

I think it may be helpful to mention also a proof using the usual graph argument (which works because we have seen that (a) the diagonal of an unramified map is an open immersion, (b) an open immersion is étale, (c) being étale is preserved under composition and base change).

There are also:

  • 3 comment(s) on Section 29.36: Étale morphisms

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View Lemma 29.36.18 as pdf