Skip to main content
MathOverflow

You are not logged in. Your edit will be placed in a queue until it is peer reviewed.

We welcome edits that make the post easier to understand and more valuable for readers. Because community members review edits, please try to make the post substantially better than how you found it, for example, by fixing grammar or adding additional resources and hyperlinks.

Required fields*

Why "open immersion" rather than "open embedding"?

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.]

Answer*

Cancel
2
  • 4
    $\begingroup$ 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). $\endgroup$ Commented Nov 27, 2017 at 17:24
  • 2
    $\begingroup$ 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. $\endgroup$ Commented Nov 27, 2017 at 17:29