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In scheme theory, an algebraic scheme is the data of a scheme + a morphism of finite type to the spectrum of a field. Where does the term "algebraic scheme" come from? It does not seem intuitive to me (to me all schemes are equally algebraic, others may have a different opinion). Are there any historical accounts regarding this matter?

Here is one mention of the category of algebraic schemes without an explicit reference to the base (though it appears that in that terminology all schemes are understood to come with a morphism to the spectrum of a field). In the Stacks Project, they use "algebraic $k$-schemes" (with $k$ being a field).

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  • $\begingroup$ @NajibIdrissi that is kind of funny, I did not think of that. Maybe so, maybe so. $\endgroup$ – user141498 Jun 7 at 7:48
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    $\begingroup$ It's not a great terminology, and I actually never saw it (or didn't pay attention). +1 if you prefer "scheme of finite type over a field". $\endgroup$ – YCor Jun 7 at 7:54
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    $\begingroup$ But actually now that I'm looking, I can see in Grothendieck (1960) numdam.org/article/SB_1958-1960__5__193_0.pdf the use of "schéma algébrique sur $A$" (algebraic scheme over $A$), where $A$ does not have to be a field (yet it seems to mean finite type). $\endgroup$ – YCor Jun 7 at 7:57
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    $\begingroup$ In the context of field extensions, finite type does not imply algebraic. $\endgroup$ – François Brunault Jun 7 at 17:03
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    $\begingroup$ finite type schemes over a field are exactly the schemes which correspond to algebraic varieties, so algebraic scheme = algebraic variety. I think that is the reason for the name. $\endgroup$ – GLe Jun 9 at 12:59

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