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I'm interested in knowing about the first published texts in which sheaf-theoretic methods were used in algebra and/or in algebraic geometry.

The oldest instance I am aware of is J.-P. Serre, Faisceaux algébriques cohérents, which was published five years before the first volume of EGA was published. Nonetheless, I am unaware if other instances showed up before in the literature.

Serre himself in FAC's introduction writes

On sait que les méthodes cohomologiques, et particulièrement la théorie des faisceaux, jouent un rôle croissant, non seulement en théorie des fonctions de plusieurs variables complexes (cf. [5]), mais aussi en géométrie algébrique classique (qu'il me suffise de citer les travaux récents de Kodaira-Spencer sur le théorème de Riemann-Roch)

We know that the cohomological methods, in particular sheaf theory, play an increasing role not only in the theory of several complex variables ([5]), but also in classical algebraic geometry (let me recall the recent works of Kodaira-Spencer on the Riemann-Roch theorem).

(Translation from Piotr Achinger and Łukasz Krupa.)

From the text it seems that Serre points out to Kodaira and Spencer for previous sheaf-theoretic algebraic geometry. However, when I look for "Kodaira-Spencer Riemann-Roch theorem" in google, the only relevant result that shows up is the paper Cohomology and the Riemann-Roch Theorem by Spencer (available here), which has sheaves but seems to be only complex geometry to me, not algebraic geometry. So my questions are:

  1. Is there something in this paper from Kodaria and Spencer I am missing that one can regard as "algebraic geometry" or that has any repercussions to AG?

  2. Am I looking at the wrong published text from Kodaira-Spencer that Serre had on mind when he quoted them on FAC?

  3. In general, do you know any previous instances of sheaf-theoretic methods in algebraic geometry before FAC? (And maybe "the first one"? If there is such).


EDIT: In the introduction of his algebraic geometry book, Milne writes

enter image description here

It seems that to write his paper Serre was fueled by the problem of defining “abstract algebraic varieties.” When Milne says that for his definition Serre was «borrowing ideas from complex analysis,» I think he means the theory of complex analytic spaces (ringed spaces which are locally isomorphic to the zero set $A$ of a finite collection of holomorphic functions in $\mathbb{C}^n$, with $A$ equipped with the sheaf of holomorphic functions). As far as I know, the theory of complex analytic spaces is older than FAC. But I don't know if Serre was the first one to take such an approach.

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    $\begingroup$ Kiyoshi Oka probably deserves to be mentioned. I remember seeing his (in French) "ideals of indefinite extent", which were about sections of sheaves of ideals, etc. $\endgroup$ Commented Mar 18, 2023 at 20:43

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Here is what Dieudonné has to say on contributions other than Serre's in History of Algebraic Geometry (VIII.1.6 and VIII.1.11):

"As early as 1909, Severi, while defining the arithmetic genus of a nonsingular irreducible projective algebraic variety of arbitrary dimension $n$, had conjectured that its arithmetic genus was given by... Inspired by the proofs of Severi and Zariski, Kodaira was able to prove this conjecture in 1952 using Hodge theory and a study of a particular case of the Riemann-Roch problem... In the course of the year 1953, the work of Dolbeault, Serre, Kodaira, Spencer, and Hirzebruch led to a whole series of remarkable results linking Hodge theory, the invariants of the Italian geometry, and sheaf cohomology, and culminating in a formulation of the Riemann-Roch theorem (in the form of an equality) valid in all dimensions."

Kodaira's paper he references is K. Kodaira, The theorem of Riemann-Roch for adjoint systems on three-dimensional varieties, Ann. of Math., 56 (1952), pp. 298-342. Severi's conjecture is proved in §12 (p. 333) titled Arithmetic genera of 3-dimensional algebraic varieties. I am guessing this is, in part, what Serre refers to, since it resolves a conjecture in classical algebraic geometry. He uses algebro-geometric concepts (linear systems, divisors) and results of Zariski in the proof.

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Jean Pierre Serre was awarded the Field medal in 1954. I recommend you his talk for the occasion.

Cohomologie et géométrie algébrique. Congrès int. d’Amsterdam, 1954, vol. III, pp. 515-520

He starts saying "De nombreux problèmes de géométrie algébrique classique peuvent être formulés et étudiés de la façon la plus commode au moyen de la théorie des faisceaux".

It is arguably Jean Pierre Serre who must be considered the person that introduced sheaves in algebraic geometry, but this must be understood in the context of Cartan and using sheaves for several branches of mathematics, and Bourbaki interest in Algebraic geometry in those times.

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  • $\begingroup$ I find a copy here agrothendieck.github.io/divers/serreicm54scan.pdf $\endgroup$ Commented Mar 10, 2023 at 16:57
  • $\begingroup$ Interestingly, Serre cites FAC in Cohomologie et géométrie algébrique. Also, the latter article cites two papers from Kodaira-Spencer, [3] and [4]. But again, in these papers sheaf theory is only used for the analytic topologies and holomorphic sections. $\endgroup$ Commented Aug 14, 2023 at 15:22
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I will reproduce here Dieudonné's History of Algebraic Geometry, VIII.2, first two paragraphs. I think they speak for themselves:

  1. Around 1949, A. Weil had observed that the “Zariski topology” (VII, 45) could be defined on his “abstract varieties” (VII, 38) as well, and not only did it simplify the exposition but it made possible the definition in “abstract” algebraic geometry of the notion of fiber space, modeled on the classical notion (VIII, 2), as well as the extension of the relations between line bundles and divisors (VIII, 4). In 1954, Serre had the idea of similarly generalizing the notion of sheaf to “abstract” varieties by replacing, in the definition of (VIII, 8), the usual topology by the Zariski topology. But, owing to this idea, he realized that the definition and the study of Weil’s “abstract varieties” could even be presented in a much more convenient form, using H. Cartan’s notion of “ringed space,” that is to say, a topological space $X$ on which is given a sheaf of rings, called the structure sheaf (that is, a sheaf where, for each open set $U\subset X$, $H^0(U, \mathcal{O}_X)$ has the structure of a ring such that the restriction homomorphisms $H^0(U, \mathcal{O}_X)\to H^0(V,\mathcal{O}_X)$ for $V\subset U$ are ring homomorphisms). The advantage of this point of view is that this type of structure lends itself very well to “gluing” along open sets, the verification of the “conditions for gluing” ordinarily being trivial.

  2. For the applications he has in mind, Serre is not concerned with questions of the “field of definition” (VII, 36) and he does not use the notion of “generic point”; staying closer to the classical point of view, he fixes, once and for all, an algebraically closed base field $k$ (of arbitrary characteristic). The “pieces” that he glues together to obtain the definition of his varieties are what are called affine varieties over $k$; such a variety $X$ is a subset of a space $k^n$ defined by polynomial equations, and the sheaf $\mathcal{O}_X$ is defined by the condition that, for every open set $U\subset X$, $H^0(U, \mathcal{O}_X)$ is composed of the restrictions to $U$ of the rational functions $x\mapsto P(x)/Q(x)$ on $k^n$ that are defined (that is, such that $Q(x) \neq 0$) at every point $x\in U$. In addition, Serre imposed on his varieties, on the one hand, the condition of having a covering by a finite number of affine open sets (that is, open sets $U$ such that the restriction to $U$ of the sheaf $\mathcal{O}_X$ defines $U$ as an affine variety), and on the other, a “separation condition” allowing “passage to the limit” for the Zariski topology in a reasonable manner: for example, if two rational functions $f,g$ are defined in $X$ and agree on an everywhere dense open set of $X$, then they are necessarily identical.

From my interpretation, Dieudonné's exposition implicitly implies that Serre is the first mathematician doing these kind of things.

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