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PREFACE
Algebraic geometry is the study of systems of algebraic equations in several variables, and of the structure which one can give to the solutions of such equations. There are four ways in which this study can be carried out: analytic, topological, algebraico-geometric, and arithmetic. It almost goes without saying that these four ways are by no means independent of each other, although each one can be pushed forward by methods appropriate to its point of view.
To use analytic and topological methods, one starts with equations whose coefficients are complex numbers. One may then consider the set of zeros of the equations as a manifold, topological, or analytic, provided one makes suitable assumptions of non-singularity.
The algebraico-geometric methods are applied in dealing with equations having coefficients in an arbitrary field, the solutions of the equations being taken to lie in its algebraic closure, or in a "universal domain." The arguments used are geometric, and are supplemented by as much algebra as the taste of the geometer will allow.
One frequently meets a problem which consists in relating invariants arising from topological, analytic, or algebraic methods. For instance, the genus of a curve may alternatively be defined as the number of holes in its Riemann surface, the number of differentials of first kind, the dimension of the Jacobian variety, or that integer which makes the Riemann-Roch formula valid. One must then prove that all these numbers are equal.
Finally, in arithmetic, one has equations whose coefficients lie in the field of rational numbers, or fields canonically derived from it. One then studies properties which depend essentially on the special nature of the coefficients field selected: algebraic number fields (finite extensions of the rationals), finite fields obtained from those by reducing modulo p, and more generally fields of finite type, generated from those by a finite number of elements, which may be transcendental. This fourth approach to algebraic equations includes of course all of diophantine analysis. The purpose of the present book is to give a rapid, concise, and self-contained introduction to the algebraic aspects of the third approach, the algebraico-geometric, without presupposing any extensive knowledge of algebra (local algebra in particular). It is not meant as a complete treatise, but we hope that after becoming acquainted with it, the reader will find the door open to a more thorough study of the literature. With this in mind, we have appended to each chapter a short list of papers which the reader may find stimulating. We have also made some historical comments when it seemed appropriate to do so.
We have not touched any topic related to the intersection theory. This would have taken us beyond the intended scope of our book, which we hope will be used not only as an introduction to algebraic geometry, but also as an introduction to Weil's Foundations. There are many basic results contained in Foundations which still cannot be found anywhere else, especially in Chapters VII and VIII; however, we have tried to include all the results which don't pertain directly to intersection theory (i.e. all the qualitative results). A discussion of the Chow coordinates is omitted, as belonging properly to intersection theory. This theory is about to undergo an extensive recasting, because, as Weil hoped, one is now able to deal with linear equivalence and the algebraic homology ring.
We have included all the theorems of a general nature which have been used recently in laying the foundations of the theory of algebraic groups. For a bibliography, the reader is referred to the end of Chapter IX. The Riemann-Roch theorem has been given in order to provide the background for a sketch of the construction of the Jacobian variety, a typical example of complete group varieties.
The reader should keep in mind that today one can develop a more abstract algebraic geometry than that given here. It is known as the arithmetic case, and is of great importance for number theory. For instance, an affine variety V in the arithmetic case is identified with a finitely generated ring over the ordinary integers, and the absolutely algebraic point s of V are the homomorphisms of this ring into finite fields. This idea, which, as Weil has pointed out, goes as far back as Kronecker, has been rediscovered in recent times by Kahler and Weil, and given new impetus by Weil in his address to the International Congress of 1950. For a systematic attempt at laying the foundations of the arithmetic case, we refer the reader to M. Nagata, "A general theory of algebraic geometry over Dedekind domains," American Journal of Mathematics, January 1956. We have never brought out this theory explicitly here, but it is a profitable activity to translate theorems into their analogues in the arithmetic case.
S. L.
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Excerpted from "Introduction to Algebraic Geometry"
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Copyright © 2019 Serge Lang.
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