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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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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 7 '10 at 4:08
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 7 '10 at 7:21
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 7 '10 at 15:11
Vivek--In that case, you should also be irritated that we have 'covering space' in our vocabulary. – Keerthi Madapusi Pera Dec 7 '10 at 16:30
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 7 '10 at 17:56

In differential topology there is a very serious difference between the status of immersion theory that of the theory of embeddings. Saying it a bit rudely Immersion theory exists, while theory of embeddings does not. I mean that the Smale - Hirsch - Gromov theory completely reduces the theory of immersions to algebraic topology, while for embeddings such a complete solution exists only partially, in the so called metastable range (roughly when the dimension of the target is at least 3/2 times higher than that of the domain. This is due to Haefliger.) The immersions are easier to handle because being an immersion is a local property, while being an embedding is a global one. Local is easier probably anywhere in Mathematics, than global.

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In algebraic geometry being an "open immersion" is a global property, hence my question: Why is this called an immersion rather than an embedding? – Charles Staats Dec 16 '15 at 2:49

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