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. Contents......Page 5 Introduction......Page 7 Conventions and notation......Page 9 1. 1 Definition of presheaves......Page 11 1. 2 Examples of presheaves......Page 12 1. 3 Interlude: direct limits......Page 13 1. 4 Stalks of presheaves......Page 18 1. 5 Morphisms of presheaves......Page 19 Exercises on Chapter 1......Page 21 2.1 The sheaf axiom......Page 24 2. 3 Sheaf spaces......Page 27 2. 4 The sheafification of a presheaf......Page 32 2. 5 Sheaf spaces of abelian groups......Page 35 Exercises on Chapter 2......Page 37 3. 1 Categories and functors......Page 41 3. 2 The categories of sheaves and presheaves......Page 45 3. 3 Kernels and monomorphisms......Page 47 3. 4 Cokernels and epimorphisms......Page 51 3. 5 Biproducts and the abelianness of Presh and Shv......Page 57 3. 6 Exact sequences......Page 59 3. 7 Change of base space......Page 63 3. 8 Restriction and extension......Page 72 Exercises on Chapter 3......Page 78 4.1 The category of ringed spaces over a ring R......Page 83 4. 2 The prime spectrum of a ring......Page 91 4. 3 Geometric spaces and manifolds......Page 97 4. 4 Modules over ringed spaces......Page 104 4. 5 Locally free Modules......Page 112 Exercises on Chapter 4......Page 117 5.1 Injective objects......Page 125 5. 2 Derived functors......Page 129 5. 3 Sheaf cohomology......Page 141 5.4 Cech cohomology......Page 150 Exercises on Chapter 5......Page 162 The way ahead: further reading......Page 164 References......Page 166 Hints and answers to some exercises......Page 167 Index of terminology......Page 169 Index of notation......Page 173 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.