Abstract algebraic geometry

The branch of algebraic geometry dealing with the general properties of algebraic varieties (cf. Algebraic variety) over arbitrary fields and with schemes (cf. Scheme), which are their generalizations. The first studies in abstract algebraic geometry appeared as early as the 19th century, but the main development of the subject dates back to the 1950s, with the creation of the general theory of schemes by A. Grothendieck. Interest in algebraic geometry over arbitrary fields arose in the context of number-theoretical problems and, in particular, in the theory of equations with two unknowns. Very important in the development of abstract algebraic geometry was the introduction of the concept of the zeta-function of an algebraic curve by E. Artin in 1924 (cf. Zeta-function in algebraic geometry), and the proof of the analogue of the Riemann hypothesis for elliptic curves by H. Hasse in 1933. The theory of algebraic curves over an arbitrary field of constants, which was developed at the same time, was of fundamental importance in this proof.

The general development of the theory of rings and fields in the first two decades of the 20th century prepared the ground for a systematic development of higher-dimensional algebraic geometry over arbitrary fields. In his series of articles (1933–1938), B.L. van der Waerden based abstract algebraic geometry on the theory of polynomial ideals. In particular, he developed the intersection theory on a non-singular projective algebraic variety. The results of the studies conducted in this field are summarized in [4].

It was noted in 1940 by A. Weil that a proof of the Riemann hypothesis for algebraic curves of arbitrary genus involves the theory of higher-dimensional varieties over arbitrary fields. He accordingly developed the theory of abstract algebraic varieties (not necessarily projective) over an arbitrary ground field, the theory of divisors, the intersection theory for such varieties, and the general theory of Abelian varieties, which had been previously studied only from the analytic point of view. The appearance in 1946 of the book by Weil [9] made the theory of valuations and field theory (Weil's "general points" ( "generic points" ) language) to be generally accepted as a base of abstract algebraic geometry for a long time thereafter.

In the early 1950s powerful methods of commutative algebra [6], [8] were brought to bear on abstract algebraic geometry. The subject was further modified by J.-P. Serre in his study [7] on coherent algebraic sheaves (cf. Coherent algebraic sheaf). The ideas and methods of homological algebra were here introduced into algebraic geometry for the first time. The development of abstract algebraic geometry paralleled that of the concept of an algebraic variety. Following the definition of an abstract algebraic variety given by Weil, various generalizations of this concept were proposed, and the concept of a scheme proved to be the most useful. A systematic exposition of these ideas and the development of the theory of schemes was initiated in 1960 by A. Grothendieck in a series of memoirs [5], in which the language of functors and of the theory of categories was introduced into abstract algebraic geometry, and many classical constructions in the field were radically transformed.

The rapid development of abstract algebraic geometry was due to the recognition of the fact that it is possible, in the framework of the theory of schemes, to apply to the "abstract case" practically all of the known concepts of the classical complex case and, in particular, the homology theory of complex-analytic manifolds. An important role in the development of abstract algebraic geometry was played by the Weil conjecture (1947), which postulated the existence of a cohomology theory in which the Lefschetz formula for the number of fixed points of a mapping would be valid, and which established an intimate connection of this hypothesis with purely arithmetical problems of algebraic varieties (cf. Zeta-function in algebraic geometry).

The concept of a topologized category (a Grothendieck topology) found numerous applications, the development of which laid the foundations of new branches of abstract algebraic geometry: the theory of representable functors (cf. Representable functor), formal geometry, Weil cohomology; $ K $- theory, and the theory of group schemes (cf. Group scheme). The ideas and methods developed in this way had their influence in many branches of mathematics (commutative algebra, category theory, the theory of analytic spaces and topology).

The recent generalization of the concept of an algebraic variety to that of an algebraic space, which was proposed in the late 1960s, made it possible to extend the framework of abstract algebraic geometry, and to connect it even more closely with other branches of algebraic geometry.

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The systematic development of the concept of a topologized category and its applications was carried out by Grothendieck, P. Deligne and many others (cf. [a1], [a2]).

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