Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben

2010 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Algebraic Geometry I

Schemes With Examples and Exercises

verfasst von: Ulrich Görtz, Torsten Wedhorn

Verlag: Vieweg+Teubner

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

Algebraic geometry has its origin in the study of systems of polynomial equations f (x ,. . . ,x )=0, 1 1 n . . . f (x ,. . . ,x )=0. r 1 n Here the f ? k[X ,. . . ,X ] are polynomials in n variables with coe?cients in a ?eld k. i 1 n n ThesetofsolutionsisasubsetV(f ,. . . ,f)ofk . Polynomialequationsareomnipresent 1 r inandoutsidemathematics,andhavebeenstudiedsinceantiquity. Thefocusofalgebraic geometry is studying the geometric structure of their solution sets. n If the polynomials f are linear, then V(f ,. . . ,f ) is a subvector space of k. Its i 1 r “size” is measured by its dimension and it can be described as the kernel of the linear n r map k ? k , x=(x ,. . . ,x ) ? (f (x),. . . ,f (x)). 1 n 1 r For arbitrary polynomials, V(f ,. . . ,f ) is in general not a subvector space. To study 1 r it, one uses the close connection of geometry and algebra which is a key property of algebraic geometry, and whose ?rst manifestation is the following: If g = g f +. . . g f 1 1 r r is a linear combination of the f (with coe?cients g ? k[T ,. . . ,T ]), then we have i i 1 n V(f ,. . . ,f)= V(g,f ,. . . ,f ). Thus the set of solutions depends only on the ideal 1 r 1 r a? k[T ,. . . ,T ] generated by the f .

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
Here the \( f_i \in k[X_1 ,...,X_n ] \) are polynomials in n variables with coefficients in a field k. The set of solutions is a subset \( V(f_1 ,...,f_r ) \) of \( k^n \). Polynomial equations are omnipresent in and outside mathematics, and have been studied since antiquity. The focus of algebraic geometry is studying the geometric structure of their solution sets.
Ulrich Görtz, Torsten Wedhorn
1. Prevarieties
Abstract
The fundamental topic of algebraic geometry is the study of systems of polynomial equation in several variables. In the end we would like to study polynomial equations with coefficients in an arbitrary ring but as a motivation and a guide line we will assume in this chapter that our ring of coefficients is an algebraically closed field k. In this case the theory has a particularly nice geometric flavor.
Ulrich Görtz, Torsten Wedhorn
2. Spectrum of a Ring
Ulrich Görtz, Torsten Wedhorn
3. Schemes
Abstract
In the current chapter, we will define the notion of scheme. In a sense, the remainder of this book is devoted to the study of schemes, so this notion is fundamental for all which follows. Schemes arise by “gluing affine schemes”, similarly as prevarieties are obtained by gluing affine varieties. Therefore after the preparations in the previous chapter, the definition is very simple, see (3.1). As for varieties we define projective space (3.6) by gluing copies of affine spaces. This is an example of a scheme which is not affine.
Ulrich Görtz, Torsten Wedhorn
4. Fiber products
Abstract
In this chapter we study one of the central technical tools of algebraic geometry: If S is a scheme and X and Y are S-schemes we define the product X × S Y of X and Y over S which is also called fiber product. We do this by defining X × S Y as an S-scheme which satisfies a certain universal property (and by proving that such a scheme always exists).
Ulrich Görtz, Torsten Wedhorn
5. Schemes over fields
Abstract
A very important special case are schemes that are of finite type over a field. Thus before we progress with the general abstract theory of schemes we focus in this and the next chapter on the case of schemes of finite type over a field (although some of the definitions and results are formulated and proved in greater generality). In fact this is also an important building block for the study of arbitrary morphism of schemes f : X → S because we have seen how we may attach to each s ∈ S its fiber Xs = f−1 (s) (4.8). Thus f yields a family of schemes over various fields and we may study f by first studying its fibers and then how these fibers vary.
Ulrich Görtz, Torsten Wedhorn
6. Local Properties of Schemes
Abstract
Consider a scheme X of finite type over an algebraically closed field k. If X is reduced then “locally” around almost all closed points X looks like affine space. Compare Figure 1.1: zooming in sufficiently, this is true for the pictured curve in all points except for the point where it self-intersects. However, while in differential geometry this can be used as the definition of a manifold, the Zariski topology is too coarse to capture appropriately what should be meant by “local”. Instead, one should look at whether X can be “well approximated by a linear space”.
Ulrich Görtz, Torsten Wedhorn
7. Quasi-coherent modules
Ulrich Görtz, Torsten Wedhorn
8. Representable Functors
Abstract
In Chapter 4 we attached to a scheme X a contravariant functor h X from the category of schemes to the category of sets. The Yoneda Lemma 4.6 tells us that we obtain an embedding of the category of schemes into the category of such functors and thus we can consider schemes also as functors. Functors F that lie in the essential image of this embedding are called representable. We say that a scheme X represents F if \( h_X \cong F \). It is one of the central problems within algebraic geometry to study functors that classify certain interesting objects and to decide whether they are representable, i.e., whether they are “geometric objects”. For general functors F and G it may be difficult to envisage them as geometric objects. But it makes sense to say that a morphism f : F → G is “geometric” (called representable), even if F and G are not necessarily representable. Thus we may speak of immersions or of open coverings of functors. We will show that a functor that is a sheaf for the Zariski topology and has an open covering by representable functors is itself representable.
Ulrich Görtz, Torsten Wedhorn
9. Separated morphisms
Abstract
Recall that a topological space X is Hausdorff if and only if the following equivalent conditions are satisfied.
Ulrich Görtz, Torsten Wedhorn
10. Finiteness Conditions
Ulrich Görtz, Torsten Wedhorn
11. Vector bundles
Ulrich Görtz, Torsten Wedhorn
12. Affine and proper morphisms
Abstract
In this chapter, we will study properties of morphisms of schemes which distinguish important subclasses of morphisms. The emphasis in this chapter is on properties that are not local on the source. We start with a relative version of being affine and then study finite and quasi-finite morphisms.
Ulrich Görtz, Torsten Wedhorn
13. Projective morphisms
Ulrich Görtz, Torsten Wedhorn
14. Flat morphisms and dimension
Abstract
The main topic of this chapter is a detailed study of flat morphisms. Although it is impossible to fully capture the algebraic notion of flatness in geometric terms, there is a useful heuristic: If f : X → S is a flat morphisms, then the fibers f−1(s) form a continuously varying family of varieties, as s varies in S. In simple situations, the converse to this statement is true, see Proposition 14.14, Theorem 14.32 and Theorem 14.126 below. One could say that flatness is the correct generalization of this naive point of view to the general case.
Ulrich Görtz, Torsten Wedhorn
15. One-dimensional schemes
Abstract
In this chapter we will apply the results obtained so far to noetherian schemes of dimension one. Arbitrary one-dimensional noetherian schemes will be called absolute curves. Examples for absolute curves are rings of integers in number fields (i.e., finite extensions of ℚ) or schemes of finite type over a field k of pure dimension one. The latter we will call curves over k.
Ulrich Görtz, Torsten Wedhorn
16. Examples
Abstract
In this chapter we consider several examples. Each example is given in such a way that it progresses along the theory introduced in the book and that it is possible to study the examples in parallel to the main text. We indicate in the section titles up to which chapter definitions and results are used in that particular section.
Ulrich Görtz, Torsten Wedhorn
Backmatter
Metadaten
Titel
Algebraic Geometry I
verfasst von
Ulrich Görtz
Torsten Wedhorn
Copyright-Jahr
2010
Verlag
Vieweg+Teubner
Electronic ISBN
978-3-8348-9722-0
Print ISBN
978-3-8348-0676-5
DOI
https://doi.org/10.1007/978-3-8348-9722-0

Premium Partner

    Bildnachweise