Characterizing closed immersions

A universally closed, universally injective, and unramified morphism is a closed immersion.

Here are some references. The result itself is here

SCHEMES: Lemma Tag 04XV
SPACES: Lemma Tag 05W8

The definition of an unramified morphisms is here

RINGS: Definition Tag 00UT
SCHEMES: Definition Tag 02G4
SPACES: Definition Tag 03ZH

Formally unramified morphisms are defined here

RINGS: Definition Tag 00UN
SCHEMES: Definition Tag 02H8
SPACES: Definition Tag 04G7

and a morphism which is formally unramified and locally of finite type is unramified, see here

SCHEMES: Lemma Tag 02HE
SPACES: Lemma Tag 04GA


2 thoughts on “Characterizing closed immersions

  1. This was for my benefit. My question was whether there is any scheme-theoretic version of the criterion that, roughly, something is an immersion if it separates points and tangent vectors.

    Johan pointed out to me that the normalization of a nodal curve, minus one of the two points over the node, provides a counterexample. So one needs some kind of properness. (See also

    So, “universally closed” corresponds to properness, “universally injective” corresponds to separating points, and “unramified” corresponds to separating tangent vectors. Thanks!

  2. By the way, what is the status of the term “embedding” in algebraic geometry? In differential topology, we have the notions of “immersion,” “injective immersion,” and “embedding,” corresponding to “unramified,” “universally injective and unramified,” and “immersion” respectively. An embedding is an injective immersion which is homeomorphic onto its image. The classic counterexample to show that an injective immersion need not be an embedding is the so-called 6-figure injectively immersing an open interval in the plane. It is a lot like the counterexample mentioned above.

    I’ve noticed Hartshorne use the word “embedding” here and there and wondered what it means. Is there a comparable French word in EGA? I don’t believe the word “embedding” appears in the stacks project to describe any kind of morphism of schemes (though it is ubiquitous in other contexts).

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