Format:
Online-Ressource (IV, 615p, digital)
ISBN:
9783834897220
Series Statement:
SpringerLink
Content:
Prevarieties -- Spectrum of a Ring -- Schemes -- Fiber products -- Schemes over fields -- Local Properties of Schemes -- Quasi-coherent modules -- Representable Functors -- Separated morphisms -- Finiteness Conditions -- Vector bundles -- Affine and proper morphisms -- Projective morphisms -- Flat morphisms and dimension -- One-dimensional schemes -- Examples.
Content:
This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get startet, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes. Prevarieties - Spectrum of a Ring - Schemes - Fiber products - Schemes over fields - Local properties of schemes - Quasi-coherent modules - Representable functors - Separated morphisms - Finiteness Conditions - Vector bundles - Affine and proper morphisms - Projective morphisms - Flat morphisms and dimension - One-dimensional schemes - Examples Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen Prof. Dr. Torsten Wedhorn, Department of Mathematics, University of Paderborn.
Note:
Includes bibliographical references (p. 583-587) and indexes
,
""Contents""; ""Introduction""; ""Leitfaden""; ""Notation""; ""Corrigenda and addenda""; ""Acknowledgements""; ""1 Prevarieties""; ""Affine algebraic sets""; ""Affine algebraic sets as spaces with functions""; ""Prevarieties""; ""Projective varieties""; ""Exercises""; ""2 Spectrum of a Ring""; ""Spectrum of a ring as a topological space""; ""Excursion: Sheaves""; ""Spectrum of a ring as a locally ringed space""; ""Exercises""; ""3 Schemes""; ""Schemes""; ""Examples of schemes""; ""Basic properties of schemes and morphisms of schemes""; ""Prevarieties as Schemes""
,
""Subschemes and Immersions""""Exercises""; ""4 Fiber products""; ""Schemes as functors""; ""Fiber products of schemes""; ""Base change, Fibers of a morphism""; ""Exercises""; ""5 Schemes over fields""; ""Schemes over a field which is not algebraically closed""; ""Dimension of schemes over a field""; ""Schemes over fields and extensions of the base field""; ""Intersections of plane curves""; ""Exercises""; ""6 Local Properties of Schemes""; ""The tangent space""; ""Smooth morphisms""; ""Regular schemes""; ""Normal schemes""; ""Exercises""; ""7 Quasi-coherent modules""
,
""Excursion: OX-modules""""Quasi-coherent modules on a scheme""; ""Properties of quasi-coherent modules""; ""Exercises""; ""8 Representable Functors""; ""Representable Functors""; ""The example of the Grassmannian""; ""Brauer-Severi schemes""; ""Exercises""; ""9 Separated morphisms""; ""Diagonal of scheme morphisms and separated morphisms""; ""Rational maps and function fields""; ""Exercises""; ""10 Finiteness Conditions""; ""Finiteness conditions (noetherian case)""; ""Finiteness conditions in the non-noetherian case""; ""Schemes over inductive limits of rings""; ""Constructible properties""
,
""Exercises""""11 Vector bundles""; ""Vector bundles and locally free modules""; ""Flattening stratification for modules""; ""Divisors""; ""Vector bundles on P1""; ""Exercises""; ""12 Affine and proper morphisms""; ""Affine morphisms""; ""Finite and quasi-finite morphisms""; ""Serre�s and Chevalley�s criteria to be affine""; ""Normalization""; ""Proper morphisms""; ""Zariski�s main theorem""; ""Exercises""; ""13 Projective morphisms""; ""Projective spectrum of a graded algebra""; ""Embeddings into projective space""; ""Blowing-up""; ""Exercises""; ""14 Flat morphisms and dimension""
,
""Flat morphisms""""Properties of flat morphisms""; ""Faithfully flat descent""; ""Dimension and fibers of morphisms""; ""Dimension and regularity conditions""; ""Hilbert schemes""; ""Exercises""; ""15 One-dimensional schemes""; ""Morphisms into and from one-dimensional schemes""; ""Valuative criteria""; ""Curves over fields""; ""Divisors on curves""; ""Exercises""; ""16 Examples""; ""Determinantal varieties""; ""Cubic surfaces and a Hilbert modular surface""; ""Cyclic quotient singularities""; ""Abelian varieties""; ""Exercises""; ""A The language of categories""; ""B Commutative Algebra""
,
""C Permanence for properties of morphisms of schemes""
Additional Edition:
ISBN 9783834806765
Additional Edition:
Buchausg. u.d.T. Görtz, Ulrich, 1973 - Algebraic geometry ; 1: Schemes Wiesbaden : Vieweg + Teubner, 2010 ISBN 9783834806765
Additional Edition:
Erscheint auch als Druck-Ausgabe Görtz, Ulrich, 1973 - Algebraic geometry ; 1: Schemes with examples and exercises Wiesbaden : Vieweg + Teubner, 2010 ISBN 9783834806765
Language:
English
Subjects:
Mathematics
Keywords:
Varietät
;
Ringspektrum
;
Vektorraumbündel
;
Lehrbuch
DOI:
10.1007/978-3-8348-9722-0
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Görtz, Ulrich 1973-
Author information:
Wedhorn, Torsten 1970-
