LMS: 20 Sheaf Theory (London Mathematical Society Lecture Note Series, Series Number 20)

Sheaf Theory Provides A Means Of Discussing Many Different Kinds Of Geometric Objects In Respect Of The Connection Between Their Local And Global Properties. It Finds Its Main Applications In Topology And Modern Algebraic Geometry Where It Has Been Used As A Tool For Solving, With Great Success, Several Long-standing Problems. This Text Is Based On A Lecture Course For Graduate Pure Mathematicians Which Builds Up Enough Of The Foundations Of Sheaf Theory To Give A Broad Definition Of Manifold, Covering As Special Cases The Algebraic Geometer's Schemes As Well As The Topological, Differentiable And Analytic Kinds, And To Define Sheaf Cohomology For Application To Such Objects. Exercises Are Provided At The End Of Each Chapter And At Various Places In The Text. Hints And Solutions To Some Of Them Are Given At The End Of The Book. Presheaves And Their Stalks -- Sheaves And Sheaf Spaces -- Morphisms Of Sheaves And Presheaves -- Ringed Spaces -- Cohomology. B. R. Tennison. Includes Indexes. Bibliography: P. 154-155. Cambridge University Press Contents 5 Introduction 7 Conventions and notation 9 Chapter 1: Presheaves and their stalks 11 1. 1 Definition of presheaves 11 1. 2 Examples of presheaves 12 1. 3 Interlude: direct limits 13 1. 4 Stalks of presheaves 18 1. 5 Morphisms of presheaves 19 Exercises on Chapter 1 21 Chapter 2: Sheaves and sheaf spaces 24 2.1 The sheaf axiom 24 2. 2 Examples of sheaves 27 2. 3 Sheaf spaces 27 2. 4 The sheafification of a presheaf 32 2. 5 Sheaf spaces of abelian groups 35 Exercises on Chapter 2 37 Chapter 3: Morphisms of sheaves and presheaves 41 3. 1 Categories and functors 41 3. 2 The categories of sheaves and presheaves 45 3. 3 Kernels and monomorphisms 47 3. 4 Cokernels and epimorphisms 51 3. 5 Biproducts and the abelianness of Presh and Shv 57 3. 6 Exact sequences 59 3. 7 Change of base space 63 3. 8 Restriction and extension 72 Exercises on Chapter 3 78 Chapter 4: Ringed spaces 83 4.1 The category of ringed spaces over a ring R 83 4. 2 The prime spectrum of a ring 91 4. 3 Geometric spaces and manifolds 97 4. 4 Modules over ringed spaces 104 4. 5 Locally free Modules 112 Exercises on Chapter 4 117 Chapter 5: Cohomology 125 5.1 Injective objects 125 5. 2 Derived functors 129 5. 3 Sheaf cohomology 141 5.4 Cech cohomology 150 Exercises on Chapter 5 162 The way ahead: further reading 164 References 166 Hints and answers to some exercises 167 Index of terminology 169 Index of notation 173 0521207843,9780521207843 Some of the results on automatic continuity of intertwining operators and homomorphisms that were obtained between 1960 and 1973 are here collected together to provide a detailed discussion of the subject. The book will be appreciated by graduate students of functional analysis who already have a good foundation in this and in the theory of Banach algebras.