## Preschemes and schemes

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.

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

The prescheme usage is outdated.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

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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

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.

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