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Foundations of algebraic geometry. (English) Zbl 0063.08198
Advances in the more arithmetic branches of modern algebra and their application to number theory naturally lead, as we may venture to say today, to problems which to the well-informed mathematician either appeared familiar as part of the heritage of classical algebraic geometry or seemed to be intrinsically adapted to a solution by more conceptual geometric methods. Furthermore, since major parts of the theory of algebraic functions of one variable had been fitted into the system of algebra it was sensible that similar interpretations and attempts at solutions were (and had to be) tried for higher dimensional problems. In order to understand and appreciate the ultimate significance of this book the reader may well keep in mind the preceding twofold motivation for the interest in algebraic geometry. Classical algebraic geometry made free use of a type and mode of reasoning with which the modern mathematician often feels uncomfortable, though the experience based on a rich and intricate source of examples made the founders of this discipline avoid serious mistakes in final results which lesser men might have been prone to make.
The main purpose of this treatise is to formulate the broad principles of the intersection theory for algebraic varieties. We find those fundamental facts without which, for example, a good treatment of the theory of linear series would be difficult. The doctrine of this book is that an unassailable foundation (and thereby justification) of the basic concepts and results of algebraic geometry can be furnished by certain elementary methods of algebra. Thus, the reader will agree after some time that he is finding a delicate tool which can serve him to remove the traces of insecurity which occasionally accompany geometric reasoning. Incidentally, the term “elementary” used here and by the author is to be understood in a restricted technical sense, in the sense that general ideal theory and the theory of power series rings are not brought into play too often. The proofs require the general plan of using the “principles of specialization”, as formulated algebraically by van der Waerden; and they are by no means elementary in the customary connotation. To some readers the adherence to a definite type of approach, where another author may have deemed it more instructive or appropriate to use slightly different methods, may tend to cloud occasionally immediate understanding by the less adept. However, once the reader has grasped the real geometric meaning of a definition or theorem (he then has to forget occasionally the fine points resulting from the facts that the author imposes no restriction on the characteristic of the underlying field of quantities) he will recognize how skilfully the language and methods of algebra are used to overcome certain limitations of spatial intuition.
The author begins his work with judiciously selected results from the theory of algebraic and transcendental extensions of fields [chapter I, Algebraic preliminaries]. Special emphasis has to be placed on inseparable extensions, which incidentally means a more complete account than is found in books on algebra. The further plan of the book is perhaps best appreciated if one starts to ponder over a more or less heuristic definition of “algebraic variety”, and then asks one’s self informally how one should define “intersections with multiplicities” of “subvarieties”. Then, in view of the principle of local linearization in classical analysis, the author’s arrangements of topics is more or less dictated by the ultimate subject under discussion, provided one does not place the interpretation of geometrical concepts by ideal theory at the head of the discussion.
Therefore the technical definitions of point, variety, generic point and point set attached to a variety [chapter IV, The geometric language] must be preceded by suitable algebraic preparations [essentially in chapter II, Algebraic theory of specializations] and more arithmetic studies [chapter III, Analytic theory of specializations]. Crucial results in this connection, based on arithmetical considerations, are found in proposition 7 on page 60 and theorem 4 on page 62, where the existence of a well-defined multiplicity is proved for specializations. For further work, the author next introduces the concept of simple point of a variety in affine space by means of the linear variety attached to the point. [See the significant propositions 19 to 21 on pages 97–99.]
Next, the intersection theory of varieties in affine space is presented through the following stages of increasing complexity: (i) intersection with a linear subspace of complementary dimension, the 0-dimensional case, with the important criterion for multiplicity 1 in proposition 7 on page 122, and ultimately the criterion for simple points in theorem 6 on page 136; (ii) intersection with a linear subspace of arbitrary dimension, with theorem 4 on page 129 which justifies the invariant meaning of the term “intersection multiplicity of a variety with a linear variety along a variety” [chapter V, Intersection multiplicities, special case].
In chapter VI, entitled ‘General intersection theory’, the results for the linear case are extended so as to culminate in the important theorem 2 on page 146 concerning the proper components of the intersection of two subvarieties in a given variety. Furthermore, all important properties of intersection multiplicities are established. Later, in appendix III, it is shown that the properties established for a certain symbol are characteristic for intersection multiplicities and uniquely define that concept. It may be mentioned that the topological definition of the chain intersections on manifolds coincides with the algebraically defined concept of this book. Of course, the underlying coefficient field has to be the field of all complex numbers and further simplifying assumptions on the variety have to be made. However, this comparison cannot be made at the level of chapters V and VI, since there one deals with affine varieties to which the ordinary topological considerations are not directly applicable.
The subsequent chapter VII, Abstract varieties, provides the necessary background for the aforementioned connections and also contains complete proofs of those results which one might have formulated first had one deliberately adopted ideal-theoretic intentions at an early stage. The abstract varieties of this chapter are obtained by piecing together varieties in affine spaces by means of suitably restricted birational transformations. This definition of the author has turned out to be very fruitful for the work on the Riemann hypothesis for function fields and the study of Abelian varieties in general. In the course of the work, the results of the preceding chapters are extended so as to lead up to the important theorem 8 on page 193 related to Hopf’s “inverse homomorphism”. The chapter ends with a theory of cycles of dimension \(s\), that is, formal integral combinations of simple abstract subvarieties of dimension \(s\). The notion of the intersection product of cycles is also introduced here [page 202], by means of which the investigation of equivalence theories can be initiated.
This is done more explicitly in chapter IX, Comments and discussion; apparently the Riemann-Roch theorem for surfaces should now be accessible to a careful re-examination. As a further result, the theory of quasi-divisibility of Artin and van der Waerden is developed in theorems 3 and 4 on pages 224–225 and theorem 6 on page 230. These theorems exhibit the relations between the theory of cycles of highest dimension and the theory of quasi-divisibility, where naturally some of the results in appendix II, Normalization of varieties, are to be added for the necessary integral closure of the required rings of functions. In this appendix the author relates his results on the normalization of algebraic varieties to those of Zariski. At this point the individual reader may well compare the elementary and the ideal-theoretic approach to a group of theorems. In appendix I, Projective spaces, often used properties and facts concerning projective spaces are quickly developed on the basis of the preceding work. This brief discussion not only deals with results which are generally useful in algebraic geometry, but also contains one of the theorems on linear series of divisors which was frequently used in the classical work [see page 266].
Because of the wealth of material and the excellent “advice to the reader” prefacing this rich and important book the reviewer feels that he should mention some of the highlights and not delve into a discussion of technical details. In short, the only way to appreciate this treatise is actually to read it.
See also the review of the second ed. (1962) in Zbl 0168.18701.

MSC:
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry