Verfügbarkeit: Lectures on algebraic geometry

Lectures on algebraic geometry: 2 Basic concepts, coherent cohomology, curves and their Jacobians

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1. Verfasser:	Harder, Günter (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Wiesbaden Vieweg 2011
Ausgabe:	1. ed.
Schriftenreihe:	Aspects of mathematics E ; 39
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XIII, 365 S. graph. Darst.
ISBN:	9783834826862 9783834804327

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adam_text	Contents Preface v Contents vii Introduction xii б Basic Concepts of the Theory of Schemes 1 6.1 Affine Schemes ............................... 1 6.1.1 Localization ............................... 1 6.1.2 The Spectrum of a Ring ........................ 2 6.1.3 The Zariski Topology on Spec(A) .................. 6 6.1.4 The Structure Sheaf on Ѕрес(Л) ................... 8 6.1.5 Quasicoherent Sheaves ......................... 11 6.1.6 Schemes as Locally Ringed Spaces .................. 12 Closed Subschemes ........................... 14 Sections ................................. 15 A remark ................................ 15 6.2 Schemes ................................... 16 6.2.1 The Definition of a Scheme ...................... 16 The gluing ............................... 16 Closed subschemes again ........................ 17 Annihilators, supports and intersections ............... 18 6.2.2 Functorial properties .......................... 18 Affine morphisms ............................ 19 Sections again ............................. 19 6.2.3 Construction of Quasi-coherent Sheaves ............... 19 Vector bundles ............................. 20 Vector Bundles Attached to Locally Free Modules ......... 20 6.2.4 Vector bundles and GLn-torsors .................... 21 6.2.5 Schemes over a base scheme S ..................... 22 Some notions of finiteness ....................... 22 Fibered products ............................ 23 Base change ............................... 28 6.2.6 Points, Т -valued Points and Geometric Points ............ 28 Closed Points and Geometric Points on varieties .......... 32 6.2.7 Flat Morphisms ............................. 34 The Concept of Flatness ........................ 35 Representability of functors ...................... 38 6.2.8 Theory of descend ........................... 40 Effectiveness for affine descend data ................. 43 6.2.9 Galois descend ............................. 44 A geometric interpretation ...................... 47 Descend for general schemes of finite type .............. 48 6.2.10 Forms of schemes ............................ 48 6.2.11 An outlook to more general concepts ................. 51 Some Commutative Algebra 55 7.1 Finite A-Algebras ............................. 55 7.1.1 Rings With Finiteness Conditions .................. 58 7.1.2 Dimension theory for finitely generated fc-algebras ......... 59 7.2 Minimal prime ideals and decomposition into irreducibles ....... 61 Associated prime ideals ........................ 63 The restriction to the components .................. 63 Decomposition into irreducibles for noetherian schemes ...... 64 Local dimension ............................ 65 7.2.1 Affine schemes over к and change of scalars ............. 65 What is dim^ Π Ζ2)? ......................... 70 7.2.2 Local Irreducibility ........................... 71 The connected component of the identity of an affine group scheme G/k .............................. 72 7.3 Low Dimensional Rings .......................... 73 Finite /c-Algebras ............................ 73 One Dimensional Rings and Basic Results from Algebraic Number Theory ............................ 74 7.4 Flat morphisms .............................. 80 7.4.1 Finiteness Properties of Tor ...................... 80 7.4.2 Construction of flat families ...................... 82 7.4.3 Dominant morphisms ......................... 84 Birational morphisms ......................... 88 The Artin-Rees Theorem ....................... 89 7.4.4 Formal Schemes and Infinitesimal Schemes ............. 90 7.5 Smooth Points ............................... 91 The Jacobi Criterion .......................... 95 7.5.1 Generic Smoothness .......................... 97 The singular locus ........................... 97 7.5.2 Relative Differentials .......................... 99 7.5.3 Examples ................................ 102 7.5.4 Normal schemes and smoothness in codimension one ........ 109 Regular local rings ........................... 110 7.5.5 Vector fields, derivations and infinitesimal automorphisms ..... Ill Automorphisms ............................. 114 7.5.6 Group schemes ............................. 114 7.5.7 The groups schemes (Ga,(Gřm and μη ................. 116 7.5.8 Actions of group schemes ....................... 117 Projective Schemes 121 8.1 Geometric Constructions ......................... 121 8.1.1 The Projective Space Ψ ....................... 121 Homogenous coordinates ........................ 123 8.1.2 Closed subschemes ........................... 125 8.1.3 Projective Morphisms and Projective Schemes ........... 126 Locally Free Sheaves on P ...................... 129 Opn(ď) as Sheaf of Meromorphic Functions ............. 131 The Relative Differentials and the Tangent Bundle of Pg ..... 132 8.1.4 Seperated and Proper Morphisms ................... 134 8.1.5 The Valuative Criteria ......................... 136 The Valuative Criterion for the Protective Space .......... 136 8.1.6 The Construction Proj(ñ) ....................... 137 A special case of a finiteness result ................... 139 8.1.7 Ample and Very Ample Sheaves ................... 140 8.2 Cohomology of Quasicoherent Sheaves .................. 146 8.2.1 Cech cohomology ............................ 148 8.2.2 The Kimneth-formulae ......................... 150 8.2.3 The cohomology of the sheaves Op» (r) ............... 151 8.3 Cohomology of Coherent Sheaves ..................... 153 The Hubert polynomial ........................ 157 8.3.1 The coherence theorem for proper morphisms ............ 158 Digression: Blowing up and contracting ............... 159 8.4 Base Change ................................ 164 8.4.1 Flat families and intersection numbers ................ 171 The Theorem of Bertini ........................ 179 8.4.2 The hyperplane section and intersection numbers of line bundles . 180 9 Curves and the Theorem of Riemann-Roch 183 9.1 Some basic notions ............................ 183 9.2 The local rings at closed points ...................... 185 9.2.1 The structure of ÔC,P ......................... 186 9.2.2 Base change ..... .......................... 186 9.3 Curves and their function fields ..................... 188 9.3.1 Ramification and the different ideal ................. 190 9.4 Line bundles and Divisors ......................... 193 9.4.1 Divisors on curves .......................... 195 9.4.2 Properties of the degree ....................... 197 Line bundles on non smooth curves have a degree .......... 197 Base change for divisors and line bundles .............. 198 9.4.3 Vector bundles over a curve ...................... 198 Vector bundles on P1 ......................... 199 9.5 The Theorem of Riemann-Roch ...................... 201 9.5.1 Differentials and Residues ...................... 203 9.5.2 The special case С = P l/k ...................... 207 9.5.3 Back to the general case ....................... 211 9.5.4 Riemann-Roch for vector bundles and for coherent sheaves ..... 218 The structure of K (C) ........................ 220 9.6 Applications of the Riemann-Roch Theorem ............... 221 9.6.1 Curves of low genus .......................... 221 9.6.2 The moduli space ........................... 223 9.6.3 Curves of higher genus ......................... 234 The moduli space of curves of genus g ............... 238 9.7 The Grothendieck-Riemann-Roch Theorem ............... 239 9.7.1 A special case of the Grothendieck -Riemann-Roch theorem .... 240 9.7.2 Some geometric considerations .................... 241 9.7.3 The Chow ring ............................. 244 Base extension of the Chow ring ................... 247 9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem . . . 249 9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem . 252 9.7.6 Back to the case p2 : X = C x C —> С ............... 253 9.7.7 Curves over finite fields ......................... 257 Elementary properties of the ζ -function ................ 258 The Riemann hypothesis ........................ 261 10 The Picard functor for curves and their Jacobians 265 Introduction: .............................. 265 10.1 The construction of the Jacobian .................... 265 10.1.1 Generalities and heuristics : ..................... 265 Rigidification of VIC .......................... 267 10.1.2 General properties of the functor VTC ................ 269 The locus of triviality ......................... 269 10.1.3 Infinitesimal properties ........................ 272 Differentiating a line bundle along a vector field ........... 274 The theorem of the cube ........................ 274 10.1.4 The basic principles of the construction of the Picard scheme of a curve ................................... 278 10.1.5 Symmetric powers ........................... 279 10.1.6 The actual construction of the Picard scheme of a curve ...... 284 The gluing ............................... 291 10.1.7 The local representability of VlC9c/k ................. 294 10.2 The Picard functor on X and on J .................... 297 Some heuristic remarks ........................ 297 10.2.1 Construction of line bundles on X and on J ............. 297 The homomorphisms фм ....................... 298 10.2.2 The projectivity of X and J ..................... 301 The morphisms фм are homomorphisms of functors ........ 302 10.2.3 Maps from the curve С to X, local representability oiVlCx/k , VlCj/k and the self duality of the Jacobian .................. 303 10.2.4 The self duality of the Jacobian .................... 310 10.2.5 General abelian varieties ........................ 311 10.3 The ring of endomorphisms End( J) and the f-adic modules Ti( J) . . 314 Some heuristics and outlooks ..................... 314 The study of End( J) ......................... 315 The degree and the trace ....................... 318 The Weil Pairing ........................... 326 The Neron-Severi groups NS{J),NS{J x J) and End(J) ...... 328 The ring of correspondences ...................... 331 10.4 Étale Cohomology ............................. 334 The cyclotomic character ........................ 334 10.4.1 Étale cohomology groups ....................... 335 Galois cohomology ........................... 336 The geometric étale cohomology groups ................ 338 10.4.2 Schemes over finite fields .......................344 The global case ............................346 The degenerating family of elliptic curves ..............350 Bibliography 357 Index 362
adam_txt	Contents Preface v Contents vii Introduction xii б Basic Concepts of the Theory of Schemes 1 6.1 Affine Schemes . 1 6.1.1 Localization . 1 6.1.2 The Spectrum of a Ring . 2 6.1.3 The Zariski Topology on Spec(A) . 6 6.1.4 The Structure Sheaf on Ѕрес(Л) . 8 6.1.5 Quasicoherent Sheaves . 11 6.1.6 Schemes as Locally Ringed Spaces . 12 Closed Subschemes . 14 Sections . 15 A remark . 15 6.2 Schemes . 16 6.2.1 The Definition of a Scheme . 16 The gluing . 16 Closed subschemes again . 17 Annihilators, supports and intersections . 18 6.2.2 Functorial properties . 18 Affine morphisms . 19 Sections again . 19 6.2.3 Construction of Quasi-coherent Sheaves . 19 Vector bundles . 20 Vector Bundles Attached to Locally Free Modules . 20 6.2.4 Vector bundles and GLn-torsors . 21 6.2.5 Schemes over a base scheme S . 22 Some notions of finiteness . 22 Fibered products . 23 Base change . 28 6.2.6 Points, Т -valued Points and Geometric Points . 28 Closed Points and Geometric Points on varieties . 32 6.2.7 Flat Morphisms . 34 The Concept of Flatness . 35 Representability of functors . 38 6.2.8 Theory of descend . 40 Effectiveness for affine descend data . 43 6.2.9 Galois descend . 44 A geometric interpretation . 47 Descend for general schemes of finite type . 48 6.2.10 Forms of schemes . 48 6.2.11 An outlook to more general concepts . 51 Some Commutative Algebra 55 7.1 Finite A-Algebras . 55 7.1.1 Rings With Finiteness Conditions . 58 7.1.2 Dimension theory for finitely generated fc-algebras . 59 7.2 Minimal prime ideals and decomposition into irreducibles . 61 Associated prime ideals . 63 The restriction to the components . 63 Decomposition into irreducibles for noetherian schemes . 64 Local dimension . 65 7.2.1 Affine schemes over к and change of scalars . 65 What is dim^ Π Ζ2)? . 70 7.2.2 Local Irreducibility . 71 The connected component of the identity of an affine group scheme G/k . 72 7.3 Low Dimensional Rings . 73 Finite /c-Algebras . 73 One Dimensional Rings and Basic Results from Algebraic Number Theory . 74 7.4 Flat morphisms . 80 7.4.1 Finiteness Properties of Tor . 80 7.4.2 Construction of flat families . 82 7.4.3 Dominant morphisms . 84 Birational morphisms . 88 The Artin-Rees Theorem . 89 7.4.4 Formal Schemes and Infinitesimal Schemes . 90 7.5 Smooth Points . 91 The Jacobi Criterion . 95 7.5.1 Generic Smoothness . 97 The singular locus . 97 7.5.2 Relative Differentials . 99 7.5.3 Examples . 102 7.5.4 Normal schemes and smoothness in codimension one . 109 Regular local rings . 110 7.5.5 Vector fields, derivations and infinitesimal automorphisms . Ill Automorphisms . 114 7.5.6 Group schemes . 114 7.5.7 The groups schemes (Ga,(Gřm and μη . 116 7.5.8 Actions of group schemes . 117 Projective Schemes 121 8.1 Geometric Constructions . 121 8.1.1 The Projective Space Ψ\ . 121 Homogenous coordinates . 123 8.1.2 Closed subschemes . 125 8.1.3 Projective Morphisms and Projective Schemes . 126 Locally Free Sheaves on P" . 129 Opn(ď) as Sheaf of Meromorphic Functions . 131 The Relative Differentials and the Tangent Bundle of Pg . 132 8.1.4 Seperated and Proper Morphisms . 134 8.1.5 The Valuative Criteria . 136 The Valuative Criterion for the Protective Space . 136 8.1.6 The Construction Proj(ñ) . 137 A special case of a finiteness result . 139 8.1.7 Ample and Very Ample Sheaves . 140 8.2 Cohomology of Quasicoherent Sheaves . 146 8.2.1 Cech cohomology . 148 8.2.2 The Kimneth-formulae . 150 8.2.3 The cohomology of the sheaves Op» (r) . 151 8.3 Cohomology of Coherent Sheaves . 153 The Hubert polynomial . 157 8.3.1 The coherence theorem for proper morphisms . 158 Digression: Blowing up and contracting . 159 8.4 Base Change . 164 8.4.1 Flat families and intersection numbers . 171 The Theorem of Bertini . 179 8.4.2 The hyperplane section and intersection numbers of line bundles . 180 9 Curves and the Theorem of Riemann-Roch 183 9.1 Some basic notions . 183 9.2 The local rings at closed points . 185 9.2.1 The structure of ÔC,P . 186 9.2.2 Base change .'. 186 9.3 Curves and their function fields . 188 9.3.1 Ramification and the different ideal . 190 9.4 Line bundles and Divisors . 193 9.4.1 Divisors on curves . 195 9.4.2 Properties of the degree . 197 Line bundles on non smooth curves have a degree . 197 Base change for divisors and line bundles . 198 9.4.3 Vector bundles over a curve . 198 Vector bundles on P1 . 199 9.5 The Theorem of Riemann-Roch . 201 9.5.1 Differentials and Residues . 203 9.5.2 The special case С = P l/k . 207 9.5.3 Back to the general case . 211 9.5.4 Riemann-Roch for vector bundles and for coherent sheaves . 218 The structure of K'(C) . 220 9.6 Applications of the Riemann-Roch Theorem . 221 9.6.1 Curves of low genus . 221 9.6.2 The moduli space . 223 9.6.3 Curves of higher genus . 234 The ''moduli space" of curves of genus g . 238 9.7 The Grothendieck-Riemann-Roch Theorem . 239 9.7.1 A special case of the Grothendieck -Riemann-Roch theorem . 240 9.7.2 Some geometric considerations . 241 9.7.3 The Chow ring . 244 Base extension of the Chow ring . 247 9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem . . . 249 9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem . 252 9.7.6 Back to the case p2 : X = C x C —> С . 253 9.7.7 Curves over finite fields . 257 Elementary properties of the ζ -function . 258 The Riemann hypothesis . 261 10 The Picard functor for curves and their Jacobians 265 Introduction: . 265 10.1 The construction of the Jacobian . 265 10.1.1 Generalities and heuristics : . 265 Rigidification of VIC . 267 10.1.2 General properties of the functor VTC . 269 The locus of triviality . 269 10.1.3 Infinitesimal properties . 272 Differentiating a line bundle along a vector field . 274 The theorem of the cube . 274 10.1.4 The basic principles of the construction of the Picard scheme of a curve . 278 10.1.5 Symmetric powers . 279 10.1.6 The actual construction of the Picard scheme of a curve . 284 The gluing . 291 10.1.7 The local representability of VlC9c/k . 294 10.2 The Picard functor on X and on J . 297 Some heuristic remarks . 297 10.2.1 Construction of line bundles on X and on J . 297 The homomorphisms фм . 298 10.2.2 The projectivity of X and J . 301 The morphisms фм are homomorphisms of functors . 302 10.2.3 Maps from the curve С to X, local representability oiVlCx/k , VlCj/k and the self duality of the Jacobian . 303 10.2.4 The self duality of the Jacobian . 310 10.2.5 General abelian varieties . 311 10.3 The ring of endomorphisms End( J) and the f-adic modules Ti( J) . . 314 Some heuristics and outlooks . 314 The study of End( J) . 315 The degree and the trace . 318 The Weil Pairing . 326 The Neron-Severi groups NS{J),NS{J x J) and End(J) . 328 The ring of correspondences . 331 10.4 Étale Cohomology . 334 The cyclotomic character . 334 10.4.1 Étale cohomology groups . 335 Galois cohomology . 336 The geometric étale cohomology groups . 338 10.4.2 Schemes over finite fields .344 The global case .346 The degenerating family of elliptic curves .350 Bibliography 357 Index 362
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spelling	Harder, Günter Verfasser aut Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder 1. ed. Wiesbaden Vieweg 2011 XIII, 365 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aspects of mathematics : E 39 Aspects of mathematics : E ... (DE-604)BV023059274 2 Aspects of mathematics E ; 39 (DE-604)BV000018737 39 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016309168&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Harder, Günter Lectures on algebraic geometry Aspects of mathematics
title	Lectures on algebraic geometry
title_auth	Lectures on algebraic geometry
title_exact_search	Lectures on algebraic geometry
title_exact_search_txtP	Lectures on algebraic geometry
title_full	Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder
title_fullStr	Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder
title_full_unstemmed	Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder
title_short	Lectures on algebraic geometry
title_sort	lectures on algebraic geometry basic concepts coherent cohomology curves and their jacobians
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