概型的幾何

內(nèi)容概要

概型理論是代數(shù)幾何的基礎(chǔ)，在代數(shù)幾何的經(jīng)典領(lǐng)域不變理論和曲線模中有了較好的發(fā)展。將代數(shù)數(shù)論和代數(shù)幾何有機(jī)的結(jié)合起來(lái)，實(shí)現(xiàn)了早期數(shù)論學(xué)者們的愿望。這種結(jié)合使得數(shù)論中的一些主要猜測(cè)得以證明。    本書旨在建立起經(jīng)典代數(shù)幾何基本教程和概型理論之間的橋梁。例子講解詳實(shí)，努力挖掘定義背后的深層次東西。練習(xí)加深讀者對(duì)內(nèi)容的理解。學(xué)習(xí)本書的起點(diǎn)低，了解交換代數(shù)和代數(shù)變量的基本知識(shí)即可。本書揭示了概型和其他幾何觀點(diǎn)，如流形理論的聯(lián)系。了解這些觀點(diǎn)對(duì)學(xué)習(xí)本書是相當(dāng)有益的，雖然不是必要。目次：基本定義；例子；射影概型；經(jīng)典結(jié)構(gòu)；局部結(jié)構(gòu)；概型和函子。

書籍目錄

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

章節(jié)摘錄

　　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.

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