概型的几何

出版时间：2010-1  出版社：世界图书出版公司  作者：（美）艾森邦德 著  页数：294  
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内容概要

概型理论是代数几何的基础，在代数几何的经典领域不变理论和曲线模中有了较好的发展。将代数数论和代数几何有机的结合起来，实现了早期数论学者们的愿望。这种结合使得数论中的一些主要猜测得以证明。    本书旨在建立起经典代数几何基本教程和概型理论之间的桥梁。例子讲解详实，努力挖掘定义背后的深层次东西。练习加深读者对内容的理解。学习本书的起点低，了解交换代数和代数变量的基本知识即可。本书揭示了概型和其他几何观点，如流形理论的联系。了解这些观点对学习本书是相当有益的，虽然不是必要。目次：基本定义；例子；射影概型；经典结构；局部结构；概型和函子。

书籍目录

I Basic Definitions   I.1 Affine Schemes     I.1.1 Schemes as Sets     I.1.2 Schemes as Topological Spaces     I.1.3 An Interlude on Sheaf Theory References for the Theory of Sheaves     I.1.4 Schemes as Schemes (Structure Sheaves)   I.2 Schemes in General     I.2.1 Subschemes     I.2.2 The Local Ring at a Point     I.2.3 Morphisms     I.2.4 The Gluing Construction Projective Space   I.3 Relative Schemes     I.3.1 Fibered Products     I.3.2 The Category of S-Schemes     I.3.3 Global Spec   I.4 The Functor of Points II Examples   II.1 Reduced Schemes over Algebraically Closed Fields     II. 1.1 Affine Spaces     II.1.2 Local Schemes   II.2 Reduced Schemes over Non-Algebraically Closed Fields   II.3 Nonreduced Schemes     II.3.1 Double Points     II.3.2 Multiple Points  Degree and Multiplicity     II.3.3 Embedded Points Primary Decomposition     II.3.4 Flat Families of Schemes       Limits       Examples       Flatness     II.3.5 Multiple Lines   II.4 Arithmetic Schemes     II.4.1 Spec Z     II.4.2 Spec of the Ring of Integers in a Number Field     II.4.3 Affine Spaces over Spec Z     II.4.4 A Conic over Spec Z     II.4.5 Double Points in Al III Projective Schemes   III.1 Attributes of Morphisms     III.1.1 Finiteness Conditions     III.1.2 Properness and Separation   III.2 Proj of a Graded Ring     III.2.1 The Construction of Proj S     III.2.2 Closed Subschemes of Proj R     III.2.3 Global Proj       Proj of a Sheaf of Graded 0x-Algebras       The Projectivization P(ε) of a Coherent Sheaf ε     III.2.4 Tangent Spaces and Tangent Cones       Affine and Projective Tangent Spaces       Tangent Cones     III.2.5 Morphisms to Projective Space     III.2.6 Graded Modules and Sheaves     III.2.7 Grassmannians     III.2.8 Universal Hypersurfaces   III.3 Invariants of Projective Schemes     III.3.1 Hilbert Functions and Hilbert Polynomials     1II.3.2 Flatness Il: Families of Projective Schemes     III.3.3 Free Resolutions     III.3.4 Examples       Points in the Plane       Examples: Double Lines in General and in p3     III.3.5 BEzout's Theorem       Multiplicity of Intersections     III.3.6 Hilbert Series IV Classical Constructions V Local Constructions VI Schemes and Functors References Index

章节摘录

　　1.4 The Functor of PointsOne of the intriguing things about schemes is precisely that they have somuch structure that is not conveyed by their underlying sets, so that thefamiliar operations on sets such as taking direct products require vigilantscrutiny lest they turn out not to make sense. It is therefore remarkable thatmany of the set-theoretic ideas can be restored through a simple device,the functor of points. This point of view, while initially adding a layer ofcomplication to the subject, is often extremely illuminating; as a result itand its attendant terminology have become pervasive. We will give a briefintroduction to the necessary definitions here and use them occasionally inthe following chapters before returning to them in detail in Chapter VI.　　We start with the observation that the points Of a scheme do not ingeneral look anything like one another: we have nonclosed points as well asclosed ones; and if we are working over a non-algebraically closed field, theneven closed points may be distinguished by having different residue fields.Similarly, if we are working over Z, different points may have residue fieldsof different characteristic; and if we extend the notion of point to "closedsubscheme whose underlying topological space is a point," we have an evengreater variety. And, of course, a morphism between schemes will not at allbe determined by the associated map on underlying point sets.　　There is, however, a way of looking at a scheme——via its functor ofpoints- that reduces it in effect to a set. More precisely, we may think ofa scheme as an organized collection of sets, a functor on the category ofschemes, on which the familiar operations on sets behave as usual. In thissection we will examine this functorial descripti　　n. A big payoff is that wewill see the category of schemes embedded in a larger category of functors,in which many constructions are much easier. The advantage of this issomething like the advantage in analysis of working with distributions, notjust ordinary functions; it shifts the problem of making constructions inthe category of schemes to the problem of understanding which functorscome from schemes. Further, many geometric constructions that arise inthe category of schemes can be extended to larger categories of functors ina useful way.

用户评论 (总计4条)

•   代数几何的经典是Hartshorne的《代数几何》，每个人应该买几本来学习。但是《代数几何》只有一般的抽象概念和逻辑推理。作为逻辑推理起点的形象却很少出现。我在学习概型的概念的时候觉得应该写几个简单的例子来支持自己的直觉，然而老师没有给我写出这样的例子，我自己写了一个很小的例子给老师看，老师却说我这不是代数几何。一个偶人的机会在网上找到这本书的电子版，发现我想做的事情作者都做了。而且做得还非常漂亮！
  我的建议是，学习代数几何要从Hartshorne的《代数几何》开始，但是手边一定要备一本《概型的几何》作为补充参考资料。如果把这里的例子都看懂了，那我们的动手能力就足够强了！
  相信这本书的出版一定会制造出一大批懂得代数几何的小怪物的！
•   慢慢啃吧。
•   此书很适合初学者，很易读，包装也很好，印刷也很好，排版也很好。
  这本书内容非常好，我向大家推介这本书。
•   又是Eisenbud的作品