Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry

The main topics of this book are cohomology, sheaves, and sheaf cohomology. Why? Mostly because for more than thirty years the senior author has been trying to learn algebraic geometry. To his dismay, he realized that since 1960, under the in uence and vision of A. Grothendieck and his collaborators, in particular Serre, the foundations of algebraic geometry were built on sheaves and cohomology. But the invasion of these theories was not restricted to algebraic geometry. Cohomology was already a pillar of algebraic topology but sheaves and sheaf cohomology sneaked in too. For a novice the situation seems hopeless. Even before one begins to discuss curves or surfaces, one has to spend years learning (1) Some algebraic topology (especially homology and cohomology). (2) Some basic homological algebra (chain complexes, cochain complexes, exact sequences, chain maps, etc.). Some commutative algebra (injective and projective modules, injective and projective resolutions). (3) Some sheaf theory. This book represents the result of an un nished journey in attempting to accomplish the above. What we discovered on the way is that algebraic topology is a captivating and beautiful subject. We also believe that it is hard to appreciate sophisticated concepts such as sheaf cohomology without prior exposure to fundamentals of algebraic topology, simplicial homology, singular homology, and CW complexes, in particular. UPLOAD BY ZAMPIVA 1 Introduction 1.1 Exact Sequences, Chain Complexes, Homology, Cohomology . . . . . . . . . 15 1.2 Relative Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . 24 1.3 Duality; Poincare, Alexander, Lefschetz . . . . . . . . . . . . . . . . . . . . . 26 1.4 Presheaves, Sheaves, and Cech Cohomology . . . . . . . . . . . . . . . . . . 32 1.5 Shea cation and Stalk Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.6 Cokernels and Images of Sheaf Maps . . . . . . . . . . . . . . . . . . . . . . 41 1.7 Injective and Projective Resolutions; Derived Functors . . . . . . . . . . . . 42 1.8 Universal -Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.9 Universal Coecient Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.10 Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.11 Alexander and Alexander{Lefschetz Duality . . . . . . . . . . . . . . . . . . 55 1.12 Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.13 Suggestions On How to Use This Book . . . . . . . . . . . . . . . . . . . . . 56 2 Homology and Cohomology 59 2.1 Exact Sequences and Short Exact Sequences . . . . . . . . . . . . . . . . . . 59 2.2 The Five Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.3 Duality and Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.4 The Functors Hom(􀀀;A), Hom(A;􀀀), and 􀀀 2.5 Abstract Cochain Complexes and Their Cohomology . . . . . . . . . . . . . 78 2.6 Chain Maps and Chain Homotopies . . . . . . . . . . . . . . . . . . . . . . . 82 2.7 The Long Exact Sequence of Cohomology or Zig-Zag Lemma . . . . . . . . . 84 2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3 de Rham Cohomology 95 3.1 Review of de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2 The Mayer{Vietoris Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3 Poincare Duality on an Orientable Manifold . . . . . . . . . . . . . . . . . . 104 3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 9 10 CONTENTS 4 Singular Homology and Cohomology 109 4.1 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 Homotopy Equivalence and Homology . . . . . . . . . . . . . . . . . . . . . 117 4.3 Relative Singular Homology Groups . . . . . . . . . . . . . . . . . . . . . . . 120 4.4 Good Pairs and Reduced Homology . . . . . . . . . . . . . . . . . . . . . . . 125 4.5 Excision and the Mayer{Vietoris Sequence . . . . . . . . . . . . . . . . . . . 128 4.6 Some Applications of Singular Homology . . . . . . . . . . . . . . . . . . . . 134 4.7 Singular Homology with G-Coecients . . . . . . . . . . . . . . . . . . . . . 143 4.8 Singular Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.9 Relative Singular Cohomology Groups . . . . . . . . . . . . . . . . . . . . . 155 4.10 The Cup Product and the Cohomology Ring . . . . . . . . . . . . . . . . . . 159 4.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5 Simplicial Homology and Cohomology 165 5.1 Simplices and Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . 167 5.2 Simplicial Homology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.3 Simplicial and Relative Homology with G-Coecients . . . . . . . . . . . . . 187 5.4 Equivalence of Simplicial and Singular Homology . . . . . . . . . . . . . . . 189 5.5 The Euler{Poincare Characteristic of a Simplicial Complex . . . . . . . . . . 195 5.6 Simplicial Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6 Homology and Cohomology of CW Complexes 205 6.1 CW Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.2 Homology of CW Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.3 The Euler{Poincare Characteristic of a CW Complex . . . . . . . . . . . . . 224 6.4 Cohomology of CW Complexes . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7 Poincare Duality 237 7.1 Orientations of a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.2 The Cap Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.3 Cohomology with Compact Support . . . . . . . . . . . . . . . . . . . . . . . 256 7.4 The Poincare Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 258 7.5 The Poincare Duality Theorem with Coecients in G . . . . . . . . . . . . . 269 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 8 Presheaves and Sheaves; Basics 275 8.1 Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 8.3 Direct Mapping Families and Direct Limits . . . . . . . . . . . . . . . . . . . 288 8.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 CONTENTS 11 9 Cech Cohomology with Values in a Presheaf 297 9.1 Cech Cohomology of a Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 9.2 Cech Cohomology with Values in a Presheaf . . . . . . . . . . . . . . . . . . 305 9.3 Equivalence of Cech Cohomology to Other Cohomologies . . . . . . . . . . . 309 9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 10 Presheaves and Sheaves; A Deeper Look 319 10.1 Stalks and Maps of Stalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 10.2 Shea cation of a Presheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 10.3 Stalk Spaces (or Sheaf Spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10.4 The Equivalence of Sheaves and Stalk Spaces . . . . . . . . . . . . . . . . . . 339 10.5 Stalk Spaces of Modules or Rings . . . . . . . . . . . . . . . . . . . . . . . . 342 10.6 Kernels of Presheaves and Sheaves . . . . . . . . . . . . . . . . . . . . . . . . 344 10.7 Cokernels of Presheaves and Sheaves . . . . . . . . . . . . . . . . . . . . . . 347 10.8 Presheaf and Sheaf Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 351 10.9 Exact Sequences of Presheaves and Sheaves . . . . . . . . . . . . . . . . . . 354 10.10Categories, Functors, Additive Categories . . . . . . . . . . . . . . . . . . . . 357 10.11Abelian Categories and Exactness . . . . . . . . . . . . . . . . . . . . . . . . 367 10.12Ringed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 10.13Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 11 Derived Functors, -Functors, and @-Functors 379 11.1 Projective, Injective, and Flat Modules . . . . . . . . . . . . . . . . . . . . . 385 11.2 Projective and Injective Resolutions . . . . . . . . . . . . . . . . . . . . . . . 396 11.3 Comparison Theorems for Resolutions . . . . . . . . . . . . . . . . . . . . . 403 11.4 Left and Right Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . 413 11.5 Left-Exact and Right-Exact Derived Functors . . . . . . . . . . . . . . . . . 423 11.6 Long Exact Sequences Induced by Derived Functors . . . . . . . . . . . . . . 426 11.7 T-Acyclic Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 11.8 Universal -Functors and @-Functors . . . . . . . . . . . . . . . . . . . . . . 439 11.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 12 Universal Coecient Theorems 457 12.1 Universal Coecient Theorems for Homology . . . . . . . . . . . . . . . . . 458 12.2 Computing Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 12.3 Universal Coecient Theorems for Cohomology . . . . . . . . . . . . . . . . 471 12.4 Computing Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 12.5 Kunneth Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 12.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 12 CONTENTS 13 Cohomology of Sheaves 497 13.1 Cohomology Groups of a Sheaf of Modules . . . . . . . . . . . . . . . . . . . 498 13.2 Flasque Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 13.3 Comparison of Cech Cohomology and Sheaf Cohomology . . . . . . . . . . . 507 13.4 Singular Cohomology and Sheaf Cohomology . . . . . . . . . . . . . . . . . . 515 13.5 Soft Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 13.6 Fine Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 13.7 de Rham Cohomology and Sheaf Cohomology . . . . . . . . . . . . . . . . . 525 13.8 Alexander{Spanier Cohomology and Sheaf Cohomology . . . . . . . . . . . . 526 13.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 14 Alexander and Alexander{Lefschetz Duality 531 14.1 Relative Alexander{Spanier Cohomology . . . . . . . . . . . . . . . . . . . . 531 14.2 Alexander{Spanier Cohomology as a Direct Limit . . . . . . . . . . . . . . . 535 14.3 Alexander{Spanier Cohomology with Compact Support . . . . . . . . . . . . 540 14.4 Relative Classical Cech Cohomology . . . . . . . . . . . . . . . . . . . . . . 542 14.5 Alexander{Lefschetz Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 14.6 Alexander Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 14.7 Alexander{Lefschetz Duality for Cohomology with Compact Support . . . . 556 14.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 15 Spectral Sequences 559 15.1 Case 1: Filtered Di erential Modules . . . . . . . . . . . . . . . . . . . . . . 561 15.2 Graded Modules and Their Cohomology . . . . . . . . . . . . . . . . . . . . 566 15.3 Construction of the Spectral Sequence . . . . . . . . . . . . . . . . . . . . . 569 15.4 Case 2: Filtered Di erential Complexes . . . . . . . . . . . . . . . . . . . . . 575 15.5 Some Graded Modules of a Filtered and Graded Complex . . . . . . . . . . . 581 15.6 Construction of a Spectral Sequence; Serre{Godement . . . . . . . . . . . . . 589 15.7 Convergence of Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . 597 15.8 Degenerate Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 607 15.9 Spectral Sequences De ned by Double Complexes . . . . . . . . . . . . . . . 612 15.10Spectral Sequences of a Di erential Sheaf . . . . . . . . . . . . . . . . . . . . 627 15.11Spectral Sequences of Cech Cohomology, I . . . . . . . . . . . . . . . . . . . 632 15.12Spectral Sequences of Cech Cohomology, II . . . . . . . . . . . . . . . . . . . 642 15.13Grothendieck's Spectral Sequences; Composed Functors ~ . . . . . . . . . . 647 15.14Exact Couples ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 15.15Construction of a Spectral Sequence; Cartan{Eilenberg ~ . . . . . . . . . . . 659 15.16More on the Degeneration of Spectral Sequences ~ . . . . . . . . . . . . . . 669 15.17Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 "For more than thirty years the senior author has been trying to learn algebraic geometry. In the process he discovered that many of the classic textbooks in algebraic geometry require substantial knowledge of cohomology, homological algebra, and sheaf theory. In an attempt to demystify these abstract concepts and facilitate understanding for a new generation of mathematicians, he along with co-author wrote this book for an audience who is familiar with basic concepts of linear and abstract algebra, but who never has had any exposure to the algebraic geometry or homological algebra. As such this book consists of two parts. The first part gives a crash-course on the homological and cohomological aspects of algebraic topology, with a bias in favor of cohomology. The second part is devoted to presheaves, sheaves, Cech cohomology, derived functors, sheaf cohomology, and spectral sequences. All important concepts are intuitively motivated and the associated proofs of the quintessential theorems are presented in detail rarely found in the standard texts"-- Provided by publisher