Preschemes and schemes

Question

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.

Answer 1

The prescheme usage is outdated.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

Answer 2

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

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.

Answer 3

I think that the reason for this old terminology is that the people inventing schemes for the first time were guiding themselves with what Serre had done already in FAC. There Serre defines first an algebraic prevariety as a ringed space which is locally isomorphic to an affine variety (“affine variety” here is a Zariski closed subset of $\mathbb{A}^n_k=k^n$, equipped with the sheaf of regular functions, i.e., functions which are locally given as a quotient of polynomials) along with the additional size restriction of quasi-compactness. After, he defines an (abstract) algebraic variety to be a prevariety that is additionally separated (for the definition of this notion of separatedness, which is not the exactly same as for schemes, see the paper itself from Serre, Chapter II, nº 34; or for a more modern treatment, see Milne's book, Chapter 5, sections c and h).

[What follows is a guess of myself, and not a completely informed historical comment. I would appreciate if someone corrected me if there is something inaccurate in the following ideas.]

The people that were inventing schemes generalized prevarieties to preschemes, as ringed spaces that are locally isomorphic to an affine scheme, and then generalized algebraic varieties to schemes, as preschemes that are separated (in the modern sense). My supposition is that people realized that once one achieves such a broad generalization there is no need to fixate oneself with an additional axiom of separation, and the true interesting object of general study is what previously was called a “prescheme” and today is called a scheme.

I don't know the precise time when mathematicians moved on to the new terminology. As the OP pointed out, Mumford's Red Book, that was published in 1967, still used then the old terminology. On the other hand, as user717 indicated, in the 2nd edition from EGA I, published in 1971, the new terminology was already adopted by Grothendieck.

Note: along this answer I purposely use the expression “people inventing schemes” instead of just saying “Grothendieck” because I'm not completely familiar with the precise historical details of how schemes got born.