Scheme theoretic interpretations of the Weil's foundations of algebraic geometry

Question

For more than 10 years before the apearance of the Grothendieck's theory on schemes, the Weil's foundations of algebraic geometry had been the standard language of algebraic geometry. There were important works written in the language. So a natural question is: How can they be translated in the language of schemes? There is no general solution to this problem and one has to prove each one of them from scratch?

For example, Serre wrote in his book "Algebraic groups and class fields":

This chapter contains the construction and elementary study of the generalized Jacobians of an algebraic curve. We will follow closely the paper of Rosenlicht [64] on this subject, itself inspired by Weil's Varietes abeliennes [89], where the case of the usual Jacobian is treated. We will make use, as they did, of the method of "generic points". This obliges us to renounce the point of view of the preceding chapters (where all points had their coordinates in a fixed base field), and to adopt that of Foundations [87]. It is certain that the generic points could be replaced by divisors on product varieties, after first developing in detail the properties of these divisors (that is to say essentially the cohomology of coherent algebraic sheaves on a product variety); that would take us too far afield.

How the generic points could be replaced by divisors on product varieties?

Answer 1

The existence of the Néron model is one famous example of a basic result which was first proved in the language of Weil's Foundations and was reproved in Grothendieck's language of schemes after it became the standard language of algebraic geometry. As it happens, when André Néron wrote his Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publications Mathématiques de l'IHÉS, 21 (1964), p. 5-128, schemes had been around for some time, but he chose to use the more familiar language of Weil and Shimura. The theory was recast in the language of schemes by Michel Raynaud, at the suggestion of Grothendieck.

Addendum. Here is what James Milne writes in MR1045822 (91i:14034) about the book by Bosch, Lütkebohmert and Raynaud:

Consider an abelian variety $A_K$ over the field of fractions $K$ of a discrete valuation ring $R$. If $A_K$ extends to a smooth projective scheme over $R$, then that scheme is unique. In the early 1960s, Néron proved that $A_K$ always has a canonical extension to a smooth separated group scheme $A$ over $R$ that is characterized by the following universal property: for any smooth scheme $X$ over $R$, every $K$-morphism $X_K\to A_K$ extends uniquely to an $R$-morphism $X\to A$. The group scheme $A$ is now called the Néron model of $A_K$.

Although this is a very basic and widely used result, very few mathematicians have ever read Néron's proof. There are two reasons for this: first, the proof is difficult and for most purposes it is enough to know that $A$ exists and has the stated universal property; second, although Grothendieck had already developed the theory of schemes at the time he was working, Néron expressed the proof in his own relative form of the language of Weil's "Foundations'', a language which has not been adopted by other mathematicians. The authors' main purpose in writing their book has been to give a modern scheme-theoretic treatment of Néron's theorem. At the same time they have simplified his proof, removed an unnecessary condition from its statement (that the residue field be perfect), and included some more recent work on Néron models.

Answer 2

I would only add to Chandan's answer that Mike Artin, when he refereed Neron's article, translated it into scheme language in order to understand it. His lectures

Artin, M. Néron models. Arithmetic geometry (Storrs, Conn., 1984), 213--230, Springer, New York, 1986. MR0861977

are based on his personal notes from the time when he refereed Neron. Unfortunately, they are still very concise.