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¡¡¡¡This Volume is the first part of a work designed to provide a convenient account of the foundations and methods of modern algebraic geometry. Since nearly every topic of algebraic geometry has some claim for inclusion it has been necessary£¬ in order to keep the size of this volume within reasonable limits£¬ to confine ourselves strictly to general methods£¬ and to stop short of any detailed development of geometrical properties. ¡¡¡¡We have thought it de8irable to begin with a section devoted to pure algebra£¬ since the necessary algebraic topics are not easily accessible in Engl18h texts. After a preliminary chapter on the basic notions of algebra£¬ we develop the theory of matrices. Some novelty has been given to this work by the fact that the ground field is not assumed to be commutative. The more general results obtained are used in Chapters V and VI to analyse the concepts on which projective geometry is based. Chapters III and IV£¬ which will be required in a later volume£¬ are devoted to a study of algebraic equations. ¡¡¡¡Book II is concerned with the definition and basic properties of projective space of n dimensions. Both the algebraic and the 8ynthetic definitions are discussed£¬ and the theory of matrices over a non-commutative field is used to show that a space based on the propositions of incidence can be represented by coordinates£¬ without the introduction of any assumption equivalent to Pappus\' theorem. The necessity of considering a large number of special case8 has made Chapter VI rather long£¬ but some space has been 8aved in the later parts of the chapter by merely mentioning the special cases and leaving the proofs to the reader£¬ when they are sufficiently simple. It is hoped that this will not cause any difficulty. This Book concludes with a purely algebraic account of collineations and correlations. Certain elementary geometrical consequences are indicated£¬ but a complete study of the geometrical problems involved would have taken us beyond our present objective. ¡¡¡¡这卷是一个工作的第一部分，旨在提供一个方便的帐户，现代代数几何的基础和方法。由于几乎每一个代数几何的主题都有一些主张包含，它是必要的£¬为了保持这个体积的大小在合理的限制£¬严格限制一般方法£¬和停止任何几何性质的详细发展。¡¡¡¡我们认为它不能从专门纯代数£¬的部分开始，因为必要的代数主题在Engl18h文本中不容易获得。在关于代数£¬的基本概念的初步章节之后，我们发展了矩阵的理论。由于地面场不是可交换的，这项工作有了一些新颖之处。在第五章和第六章中，所得到的更一般的结果被用来分析射影几何所基于的概念。第三章和第四章£¬将需要在后面的卷£¬是专门为代数方程的研究。 |
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