# What's so “schematic” about schemes?

Well, the title clearly follows the title of this question.

Why the objects so successfully defined by Grothendieck have been called "schemes"? In my opinion the original French word (schéma) doesn't help, by itself, to understand the motivations behind such a choice of nomenclature...

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This is what Grothendieck says -- no difference with Andreas Blass, I just thought it might interest some as additional information -- in Récoltes et semailles (p. 31/32) [my emaphasize]:

La notion de schéma est la plus naturelle, la plus "évidente" imaginable, pour englober en une notion unique la série infinie de notions de "variété" (algébrique) qu’on maniait précédemment (une telle notion pour chaque nombre premier (39)...). De plus, un seul et même "schéma" (ou "variété" nouveau style) donne naissance, pour chaque nombre premier p, à une "variété (algébrique) de caractéristique p" bien déterminée. La collection de ces différentes variétés des différentes caractéristiques peut alors être visualisée comme une sorte d’ "éventail (infini) de variétés" (une pour chaque caractéristique). Le "schéma" est cet éventail magique, qui relie entre eux, comme autant de "branches" différentes, ses "avatars" ou "incarnations" de toutes les caractéristiques possibles.

My (poor) translation:

The notion of scheme is the most natural, the most "obvious" imaginable, to encompass in one unique notion the infinite series of notions of (algebraic) "variety", which one used before (such a notion for each prime number(39)...) Moreover, one and the same "scheme" (or "variety" of a new form) gives rise, for each prime number p, to a well-determined "(algebraic) variety of characteristic p." The collection of these different varieties of different characteristics can thus be seen as a sort of "(infinite) fan of varieties" (one for each characteristic). The "scheme" is this magic fan, which ties together, as many different "branches", its "avatars" or "incarnations" of all the possible characteristics.

End of translation. [In particular the end might be a bit messed up, as éventail also has a botanic meaning and this might be the better one with the branches, but not sure.]

Footnote 39 merely mentions that this is to include primes at infinity.

P.S. In case somebody has suggestions for improvements of the translation, I'd appreciate them.

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i.e. geometry without specifying a base –  Robert K Aug 21 '11 at 1:35
I've always had issues translating éventail ; 'fan' is accurate but it also means ventilateur, which can be very misleading. I've used 'spread' and 'range' but these are not quite as accurate as they are both closer to étendue. As far as I can tell, there is no good solution here. –  François G. Dorais Aug 21 '11 at 2:07
To clarify: here is an éventail upload.wikimedia.org/wikipedia/commons/9/92/… and here is a ventilateur and a chien zone.wallpaper.free.fr/galleries/Humour/… Does anyone know a good way to distinguish the two in English? –  François G. Dorais Aug 21 '11 at 2:16
@François: according to Wikipedia a more complete term for the former is "hand fan," although I've never heard anyone actually use this term in English (though perhaps I have just forgotten). –  Qiaochu Yuan Aug 21 '11 at 5:24
Thank you everybody for the feedback. Bruce, thank you for the suggestion, but I meanwhile looked in more detail and the botanic meaning of éventail seems to be merely the figurative one of fan (for cutting trees like this; used like 'the tree was cut to (the form of) a fan' so it does not even seem to be a standalone word.) –  quid Aug 21 '11 at 11:36

I have always assumed that the motivation was more or less as follows. Consider, for example, the projective planes over various fields $k$ (in the sense of classical, long-before-Grothendieck geometry). As the field $k$ varies, these are different spaces, but there is clearly something common about them; they are constructed according to a single scheme (in the intuitive sense of the word) from the various fields. The "projective plane over $\mathbb Z$", as a scheme (now in the technical sense) encapsulates what is common to all these planes, and "produces" them as the sets of $k$-valued points. More generally, a scheme (in the technical sense) is a way of producing a family of related varieties, by instantiating the scheme over various fields, and so the scheme captures what is common to those varieties, the "scheme" (in the intuitive sense) according to which they are constructed.

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This sounds reasonable; on the other hand, this is just the relative point of view which is somehow not special for algebraic geometry. I like the terminology, but it is somewhat very general. –  Martin Brandenburg Aug 21 '11 at 7:14

In this context, the introduction of the word schéma is due to Claude Chevalley.

According to Dieudonné (The historical development of algebraic geometry, Amer. Math. Monthly, vol 1979, 1972, p. 827-866), nobody had ever given "an intrinsic definition of an affine variety'' until the 1950s, ie "independent of any imbedding". For general varieties, Weil's definition used local charts, as in differential geometry and Chevalley had asked himself what was invariant in Weil's definition. Cartier (Grothendieck et les motifs. Notes sur l’histoire et la philosophie des mathématiques IV, 2000), himself quoted by Ralf Krömer (Tool and object: a history and philosophy of category theory, §4.1.1.1, footnote 319, page 164) explains that

La réponse, inspirée des travaux antérieurs de Zariski, était simple et élégante: le schéma de la variété algébrique est la collection des anneaux locaux des sous-variétés, à l'intérieur du corps des fonctions rationnelles. (Krömer's translation: The answer, inspired by previous work by Zariski, was simple and elegant: the scheme of the algebraic variety is the collection of local rings of the subvarieties inside the field of the rational functions.)

Soon after, Grothendieck's schémas pushed this idea even further.

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I think this is very helpful in terms of what Martin was asking. Zariski had already come up with the "Zariski Riemann surface"; which in modern terminology is a locale construction (comment of Martin Hyland). To become more intrinsic you need to throw out the function field inside which you are working. You can't just have any old "collection of (commutative) local rings", though. Sheaf theory allows you to say what "hangs together" as a collection of local rings, via Spec of an arbitrary commutative ring. With hindsight this was the right answer (classifying topos for local rings). –  Charles Matthews Aug 21 '11 at 11:12