# What is the history of the Scheme theory before Grothendieck?

If you read here, you will be able to read paragraphs like this:

Grothendieck created a synthesis out of these ideas, based in essence on the conceptual presentation of Zariski-Chevalley-Nagata. Schemes are thus a way of encoding systems of equations as well as the transformations to which one may subject them.

So, I first looked into the work of the mathematicians who appear here, Zariski, Chevalley, and Nagata, on their schemes. The references I have referenced are here1 and here2.

Zariski

The term itself was coined by Chevalley, although accepted in a more restrictive sense than the term as used by Grothendieck. (...) Chevalley wondered what was invariant, in Weil’s sense of variety. The answer, inspired by Zariski’s work, was simple and elegant: the scheme of an algebraic variety is the collection of local rings of the sub-varieties found inside the rational function field. (here1)

Cartier defines a spectrum $$Ω_A$$ for each finite type algebra $$A$$ over a field $$k$$, with a Zariski topology. This comes close to Grothendieck’s sense of “spectrum”. The elements of the algebra $$A$$ are construed as “functions” from the spectrum to a field extension of k as in current scheme theory. Cartier parallels Serre’s use of sheaves when he defines “algebraic sets” by pasting together spectra. He proves various theorems familiar to anyone who knows current scheme theory, though with the restriction that his spectra are all of finite type over a field.(here2)

I don't think Zariski himself put much effort into developing the scheme.

Chevalley-Nagata

An “affine scheme”, in Chevalley-Nagata’s sense amounts to what Cartier called the spectrum of a semi-simple algebra $$A$$, only reworded in ideal-theoretic terms. That makes a difference. This is Grothendieck’s spectrum. Chevalley schemes are gotten by pasting together affines. So they are schemes in Grothendieck’s sense, with all the apparatus for the general case, but not stated in all generality. This published record well supports what Cartier says about anticipating Grothendieck’s theory of schemes.(here2)

And after researching, I found that in addition to the three mathematicians above, a mathematician named Serre also seems to have influenced the development of scheme theory. So I will write about it additionally.

There is no need for an explicit topology, a point of distinction between Chevalley and Serre, who at roughly the same time introduced his algebraic varieties using Zariski topologies and sheaves. Each of the two approaches had advantages, but also limitations: Serre had an algebraically closed base field; Chevalley had to work only with irreducible varieties. In both cases, the two fundamental problems of products of varieties and base change could only be approached indirectly. All the same, Chevalley’s point of view was better suited to future extensions to arithmetic, as Nagata soon observed.(here1)

I've been researching the work of four mathematicians, Zariski, Chevalley-Nagata, and Serre, who were influential in the development of scheme before Grothendieck developed scheme, but I'm curious about what theorems they specifically studied and discovered, and what it is about those theorems that influenced the emergence of scheme.

Edit:

In the past, Mathematicians first created a geometric space and then thought about functions on it. For example, we defined a vector space and then thought about a linear map, defined a topological space and then thought about a continuous map. However, as time passed and functional analysis and category theory developed, the perspective of first thinking about a function space and then reconstructing a geometric space to match it developed. However, Mathematicians in 19th century wondered if it is really a feasible idea to think about a function first and then create a geometric space. Fortunately, if we look at the example of a vector space, it seems that it is really possible. In this case, the function space becomes a dual space, and if we take dual again from the function space, it becomes a double dual, so we get something that is really similar to the original space.

I read that the achievement of Grothendieck, a mathematician in the 20th century, actually mathematically created this idea that was widely spread among mathematicians. Is this true? And if this is true, did mathematicians like Serre and Zariski not know about this?

I don't think I can give any substantive answer but here are relevant places to study this topic at least for historical traces of development:

The word "scheme" does appear in Zariski's original 1935 edition of his book:

Zariski, O. (1935). Algebraic surfaces. Springer. p. 10.

We add that the scheme of a given singularity, as analyzed in this section, can be graphically represented by a very convenient diagram. See the quoted treatise of ENRIQUES-CHISINI. (my italics for emphasis)

The second edition (1971) of this book may be particularly valuable as it contains commentary by Zariski students Abhyankar, Lipman, and Mumford who of course understood schemes in the post-Grothendieckian sense and they draw attention to many connections and rephrasings.

A brief summary by Maruyama, Miyanishi, Mori, and Oda of Nagata's work on schemes can be found in the summary of his work in the Notices of the AMS 56:1 (2009) p. 58:

A series of papers in the late 1950s on algebraic geometry over Dedekind domains laid the foundation for later developments of algebraic geometry in terms of schemes. The concept of the Henselization of rings, developed in a series of papers in the 1950s, turned out to be fundamental for algebraic spaces and étale topology.