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Algebraic geometry: 1 Schemes : with examples and exercises

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Bibliographic Details
Main Authors:	Görtz, Ulrich 1973- (Author), Wedhorn, Torsten 1970- (Author)
Format:	Book
Language:	English
Published:	Wiesbaden Vieweg + Teubner 2010
Edition:	1st edition
Series:	Advanced lectures in mathematics Springer Studium Mathematik - Master
Subjects:	Lehrbuch
Online Access:	Inhaltsverzeichnis
Physical Description:	vii, 615 Seiten Diagramme
ISBN:	9783834806765

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MARC


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adam_text	v Contents Introduction 1 1 Prevarieties 7 Affine algebraic sets ................................ 8 Affine algebraic sets as spaces with functions .................. 17 Prevarieties ..................................... 22 Projective varieties ................................. 26 Exercises ...................................... 35 2 Spectrum of a Ring 40 Spectrum of a ring as a topological space .................... 41 Excursion: Sheaves ................................. 46 Spectrum of a ring as a locally ringed space ................... 57 Exercises ...................................... 62 3 Schemes 66 Schemes ....................................... 66 Examples of schemes ................................ 72 Basic properties of schemes and morphisms of schemes ............. 74 Prevarieties as Schemes .............................. 78 Subschemes and Immersions ............................ 83 Exercises ...................................... 88 4 Fiber products 93 Schemes as functors ................................ 93 Fiber products of schemes ............................. 97 Base change, Fibers of a morphism ........................ 105 Exercises ...................................... 114 5 Schemes over fields 118 Schemes over a field which is not algebraically closed .............. 118 Dimension of schemes over a field ......................... 120 Schemes over fields and extensions of the base field ............... 133 Intersections of plane curves ............................ 138 Exercises ...................................... 141 6 Local Properties of Schemes 145 The tangent space ................................. 146 Smooth morphisms ................................. 153 Regular schemes .................................. 158 Normal schemes .................................. 161 Exercises ...................................... 164 VI Contents 7 Quasi-coherent modules 169 Excursion: ^χ -modules .............................. 169 Quasi-coherent modules on a scheme ....................... 181 Properties of quasi-coherent modules ....................... 189 Exercises ...................................... 199 8 Representadle Functors 205 Representable Functors .............................. 206 The example of the Grassmannian ........................ 209 Brauer-Severi schemes ............................... 219 Exercises ...................................... 222 9 Separated morphisms 226 Diagonal of scheme morphisms and separated morphisms ........... 226 Rational maps and function fields ......................... 232 Exercises ...................................... 238 10 Finiteness Conditions 241 Finiteness conditions (noetherian case) ...................... 242 Finiteness conditions in the non-noetherian case ................ 249 Schemes over inductive limits of rings ...................... 258 Constructible properties .............................. 270 Exercises ...................................... 278 11 Vector bundles 286 Vector bundles and locally free modules ..................... 286 Flattening stratification for modules ....................... 296 Divisors ....................................... 298 Vector bundles on P1 ................................ 313 Exercises ...................................... 316 12 Affine and proper morphisms 320 Affine morphisms .................................. 320 Finite and quasi-finite morphisms ......................... 324 Serre s and Chevalley s criteria to be affine ................... 334 Normalization .................................... 339 Proper morphisms ................................. 343 Zariski s main theorem ............................... 349 Exercises ...................................... 361 13 Projectíve morphisms 366 Projective spectrum of a graded algebra ..................... 367 Embeddings into projective space ......................... 384 Blowing-up ..................................... 406 Exercises ...................................... 418 __________________________________________________________________ VII 14 Flat morphisms and dimension 423 Flat morphisms ................................... 423 Properties of flat morphisms ............................ 429 Faithfully flat descent ............................... 439 Dimension and fibers of morphisms ........................ 463 Dimension and regularity conditions ....................... 473 Hubert schemes ................................... 478 Exercises ...................................... 480 15 One-dimensional schemes 485 Morphisms into and from one-dimensional schemes ............... 485 Valuative criteria .................................. 487 Curves over fields .................................. 491 Divisors on curves ................................. 496 Exercises ...................................... 500 16 Examples 503 Determinantal varieties .............................. 503 Cubic surfaces and a Hubert modular surface .................. 520 Cyclic quotient singularities ............................ 528 Abelian varieties .................................. 532 Exercises ...................................... 539 A The language of categories 541 В Commutative Algebra 547 С Permanence for properties of morphisms of schemes 573 D Relations between properties of morphisms of schemes 576 E Constructible and open properties 578 Bibliography 583 Detailed List of Contents 588 Index of Symbols 598 Index 602
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series2	Advanced lectures in mathematics Springer Studium Mathematik - Master
spelling	Görtz, Ulrich 1973- Verfasser (DE-588)122267540 aut Algebraic geometry 1 Schemes : with examples and exercises Ulrich Görtz, Torsten Wedhorn 1st edition Wiesbaden Vieweg + Teubner 2010 vii, 615 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Advanced lectures in mathematics Springer Studium Mathematik - Master (DE-588)4123623-3 Lehrbuch gnd-content Wedhorn, Torsten 1970- Verfasser (DE-588)120349183 aut (DE-604)BV035958399 1 Erscheint auch als Online-Ausgabe 978-3-8348-9722-0 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018852622&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Görtz, Ulrich 1973- Wedhorn, Torsten 1970- Algebraic geometry
subject_GND	(DE-588)4123623-3
title	Algebraic geometry
title_auth	Algebraic geometry
title_exact_search	Algebraic geometry
title_full	Algebraic geometry 1 Schemes : with examples and exercises Ulrich Görtz, Torsten Wedhorn
title_fullStr	Algebraic geometry 1 Schemes : with examples and exercises Ulrich Görtz, Torsten Wedhorn
title_full_unstemmed	Algebraic geometry 1 Schemes : with examples and exercises Ulrich Görtz, Torsten Wedhorn
title_short	Algebraic geometry
title_sort	algebraic geometry schemes with examples and exercises
topic_facet	Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018852622&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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