What's so "schematic" about schemes? - MathOverflow

What's so "schematic" about schemes?

2011-08-20T22:11:18Z

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...

Answer by Andreas Blass for What's so "schematic" about schemes?

2011-08-20T22:35:06Z

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.

Answer by quid for What's so "schematic" about schemes?

2011-08-21T00:31:42Z

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.

Answer by ACL for What's so "schematic" about schemes?

2011-08-21T09:43:39Z

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.