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On Prop. 1.7 (a) on page 5 of Milne's Etale Cohomology book states:

Any immersion is quasi-finite.

A google search turned up definitions for "open immersion" and "closed immersion", never just "immersion".

Question: what does it mean for a morphism to be an "immersion"?

Could it be

  1. A morphism of schemes which as a map on the topological spaces is a homeomorphism onto the image?

  2. Short hand for either an open or a closed immersion?

  3. Short hand for one but not the other?

  4. Something else?

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    $\begingroup$ Often a synonym for "locally closed immersion", so a closed embedding into an open subscheme. (I feel like I've seen the opposite order but in Milne's setting it shouldn't matter.) $\endgroup$
    – Hoot
    Aug 28, 2015 at 22:33
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    $\begingroup$ EGA I, §4, def. 4.2.1: a morphism $Y\rightarrow X$ is an immersion if it factors as $Y\xrightarrow{g} Z\xrightarrow{ j } X$, where $g$ is an isomorphism, $Z$ a subscheme of $X$ and $j$ the canonical injection. $\endgroup$
    – abx
    Aug 29, 2015 at 5:01
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    $\begingroup$ Or see stacks.math.columbia.edu/tag/01IO. $\endgroup$
    – pbelmans
    Aug 29, 2015 at 5:24
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    $\begingroup$ Indeed, the opposite order is the same as the usual order, except in the case of non-quasicompact morphisms from a non-reduced scheme. stacks.math.columbia.edu/tag/01QV stacks.math.columbia.edu/tag/03DQ The opposite order appears in Hartshorne and, if I remember correctly, causes some strife by making some of the exercises unnecessarily difficult. $\endgroup$
    – Will Sawin
    Aug 29, 2015 at 13:12

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