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This is a very minor point, but one which had been grating me for a while. I apologize for asking a relatively trivial question, but nevertheless hope that it is suitable for MO since it should have a definite answer.

In Mumford's books, for instance Curves on Surfaces or Red Book, there is thing called "prescheme" which looks like a scheme, and scheme is something else.

But this terminology does not seem to be used elsewhere, and if at all is the case, prescheme seems to be something cruder than scheme.

I will be grateful for clarifications regarding this terminology. "Curves on surfaces" is a nice book, but whenever I pick it up I find myself wondering about this without any avail.

edited Jan 10 2010 at 15:59

asked Jan 10 2010 at 14:40

Anweshi
4,226●2●25●56

Perhaps, this is a soft question? – Anweshi Jan 10 2010 at 14:42

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No, this is called you not checking google or wikipedia. =( – Harry Gindi Jan 10 2010 at 14:56

True enough.... – Anweshi Jan 10 2010 at 15:06

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@Harry et al: I had the same question just now, googled, and this is now the first relevant hit! – Frank Thorne Feb 5 at 15:48

Ho Chung Siu · Accepted Answer · 2010-01-10 14:43:17Z UTC

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The prescheme usage is outdated. As indicated in nLab,

our schemes are in EGA called preschemes; EGA’s schemes are what we call separated schemes

answered Jan 10 2010 at 14:43

Ho Chung Siu
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	Is there any connection between the notion of a separated presheaf and a separated scheme? – Harry Gindi Jan 10 2010 at 14:56
	Well, both have a diagonal that is "closed" relative to some ambient closure operator. I'm not sure that the closure operators are related in any way---one is closure in the point-set-topological sense, while the other is closure of sieves relative to a Grothendieck topology. – Mike Shulman Jan 11 2010 at 4:56

S1 · Answer 2 · 2010-01-10 15:44:05Z UTC

In the 1971 edition of EGA (this is a revised version of the original 1960 EGA) you can find the following remark in the introduction:

Signalons enfin, par rapport à la première édition, un changement important de terminologie: le mot «schéma» désigne maintenant ce qui était appelé «préschéma» dans la première édition, et les mots «schéma séparé» ce qui était appelé «schéma».

The 1971 terminology should be standard today.

Preschemes and schemes

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