Algebraic Geometry 1. Schemes : with examples and exercises

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. Cover......Page 1 Algebraic Geometry I: Schemes With Examples and Exercises......Page 4 9783834806765......Page 5 Contents......Page 6 Introduction......Page 10 Leitfaden......Page 12 Acknowledgements......Page 15 1 Prevarieties......Page 16 Affine algebraic sets......Page 17 Affine algebraic sets as spaces with functions......Page 26 Prevarieties......Page 32 Projective varieties......Page 35 Exercises......Page 45 2 Spectrum of a Ring......Page 49 Spectrum of a ring as a topological space......Page 50 Excursion: Sheaves......Page 56 Spectrum of a ring as a locally ringed space......Page 66 Exercises......Page 71 Schemes......Page 75 Examples of schemes......Page 81 Basic properties of schemes and morphisms of schemes......Page 83 Prevarieties as Schemes......Page 87 Subschemes and Immersions......Page 92 Exercises......Page 97 Schemes as functors......Page 102 Fiber products of schemes......Page 106 Base change, Fibers of a morphism......Page 114 Exercises......Page 123 Schemes over a field which is not algebraically closed......Page 127 Dimension of schemes over a field......Page 129 Schemes over fields and extensions of the base field......Page 142 Intersections of plane curves......Page 147 Exercises......Page 150 6 Local Properties of Schemes......Page 154 The tangent space......Page 155 Smooth morphisms......Page 162 Regular schemes......Page 167 Normal schemes......Page 171 Exercises......Page 173 Excursion: OX-modules......Page 178 Quasi-coherent modules on a scheme......Page 190 Properties of quasi-coherent modules......Page 198 Exercises......Page 208 8 Representable Functors......Page 214 Representable Functors......Page 215 The example of the Grassmannian......Page 218 Brauer-Severi schemes......Page 228 Exercises......Page 231 9 Separated morphisms......Page 235 Diagonal of scheme morphisms and separated morphisms......Page 236 Rational maps and function fields......Page 241 Exercises......Page 247 10 Finiteness Conditions......Page 250 Finiteness conditions (noetherian case)......Page 251 Finiteness conditions in the non-noetherian case......Page 258 Schemes over inductive limits of rings......Page 267 Constructible properties......Page 279 Exercises......Page 287 11 Vector bundles......Page 295 Vector bundles and locally free modules......Page 296 Flattening stratification for modules......Page 306 Divisors......Page 307 Vector bundles on P1......Page 322 Exercises......Page 325 Affine morphisms......Page 329 Finite and quasi-finite morphisms......Page 333 Serre’s and Chevalley’s criteria to be affine......Page 343 Normalization......Page 348 Proper morphisms......Page 352 Zariski’s main theorem......Page 358 Exercises......Page 370 13 Projective morphisms......Page 375 Projective spectrum of a graded algebra......Page 376 Embeddings into projective space......Page 393 Blowing-up......Page 415 Exercises......Page 427 Flat morphisms......Page 432 Properties of flat morphisms......Page 438 Faithfully flat descent......Page 448 Dimension and fibers of morphisms......Page 472 Dimension and regularity conditions......Page 482 Hilbert schemes......Page 487 Exercises......Page 489 Morphisms into and from one-dimensional schemes......Page 494 Valuative criteria......Page 496 Curves over fields......Page 500 Divisors on curves......Page 505 Exercises......Page 510 Determinantal varieties......Page 512 Cubic surfaces and a Hilbert modular surface......Page 529 Cyclic quotient singularities......Page 538 Abelian varieties......Page 542 Exercises......Page 549 A The language of categories......Page 550 B Commutative Algebra......Page 556 C Permanence for properties of morphisms of schemes......Page 582 D Relations between properties of morphisms of schemes......Page 585 E Constructible and open properties......Page 587 Bibliography......Page 592 Detailed List of Contents......Page 597 Index of Symbols......Page 607 Index......Page 611 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 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. Ulrich Görtz, Torsten Wedhorn. Includes Bibliographical References (p. 583-587) And Index. Annotation 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 This comprehensive introduction to schemes is complemented by many exercises that serve to check the comprehension of the text, treat further examples and give an outlook on further results. Includes details from commutative algebra in an appendix.