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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.]

Charles Staats
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    +1: I was wondering about this very question myself recently, but I was a little embarrassed to ask. – Pete L. Clark Dec 07 '10 at 03:34
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    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. – BCnrd Dec 07 '10 at 04:08
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    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"!) – Dave Anderson Dec 07 '10 at 07:21
  • Also, another place the term embedding is common in AG is "regular embedding" (or "imbedding"), i.e., a subscheme locally cut out by a regular sequence. – Dave Anderson Dec 07 '10 at 07:23
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    something I find rather irritating is the fact that the thing I would think ought to be called an ''open immersion'' is called instead an ''etale morphism''... – Vivek Shende Dec 07 '10 at 15:11
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    Vivek--In that case, you should also be irritated that we have 'covering space' in our vocabulary. – Keerthi Madapusi Dec 07 '10 at 16:30
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    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). – BCnrd Dec 07 '10 at 17:56
  • Also irritating to me: AG has the class of open immersions and the class of closed immersions, but as far as I know these are not the intersection of some larger class with on the one hand the open maps and on the other hand the open maps. (This is in contrast to the situation regarding embeddings, or for that matter immersions, in topology or in differential topology.) – Tom Goodwillie Jul 17 '23 at 12:05

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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.

tiritiri
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    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). – abx Nov 27 '17 at 17:24
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    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. – Gro-Tsen Nov 27 '17 at 17:29