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When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic geometry, except sometimes as a word for an immersion of varieties. And the notion of an "immersion" of schemes, especially an "open immersion," seems much more similar to the topologists' "embedding" than their "immersion." [Closed immersions at least have the somewhat flimsy rationale that the scheme structure does not depend solely on the choice of subset.]

Does anyone know of a good reason, other than cultural momentum, to use the word "immersion" rather than "embedding"?

[Note: this has come up in Ravi Vakil's blog on his Algebraic Geometry notes.]

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    $\begingroup$ +1: I was wondering about this very question myself recently, but I was a little embarrassed to ask. $\endgroup$ Dec 7, 2010 at 3:34
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    $\begingroup$ What do you call a morphism $f:X \rightarrow Y$ such that $f$ carries $X$ homeomorphically onto an open subset $U$ but the induced map $X \rightarrow U$ is not necessarily a scheme isom? I like to call it an open embedding. That is, "embedding" encodes the topological aspect, and "immersion" means one keeps appropriate track of the structure sheaves too. The distinction is invisible in differential geometry since an immersion between connected manifolds has the manifold structure on the source uniquely determined by the topology of the situation and the manifold structure on the target. $\endgroup$
    – BCnrd
    Dec 7, 2010 at 4:08
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    $\begingroup$ It's a little confusing that things are the other way round in differential geometry -- an immersion need not be an embedding. (That's a distinction that's missing in algebraic geometry, at least for "closed immersions"!) $\endgroup$ Dec 7, 2010 at 7:21
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    $\begingroup$ Vivek--In that case, you should also be irritated that we have 'covering space' in our vocabulary. $\endgroup$ Dec 7, 2010 at 16:30
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    $\begingroup$ Dear Dave: In EGA IV$_4$, sections 16.9 and 19, such maps are called "regular immersions"! Dear Vivek: etale maps that are injective on geometric points are open immersions, so Keerthi's comment seems quite apt and you may be amused to know that Grothendieck's original attempt at defining the etale topology was via finite etale covers of Zariski opens (before Artin convinced him to switch to etale surjections). $\endgroup$
    – BCnrd
    Dec 7, 2010 at 17:56

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I am wondering whether this has to do with language. Algebraic geometry's current foundations were established in French, a language where "immersion" translates both immersion and embedding.

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    $\begingroup$ Well, not really -- the standard french translation of embedding (used by topologists, among others) is plongement. I think the use of immersion in algebraic geometry is due to Grothendieck, who was probably not aware of the standard usage (or did not care). $\endgroup$
    – abx
    Nov 27, 2017 at 17:24
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    $\begingroup$ I learned differential geometry in French (in France in case this is a country-dependent issue), and I always heard "plongement" being used for "embedding" and "immersion" restricted to the same meaning as in English. $\endgroup$
    – Gro-Tsen
    Nov 27, 2017 at 17:29

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