## Sheaf TheorySheaf 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. |

### Contents

1 | 3 |

Exercises on Chapter | 11 |

2 | 17 |

5 | 25 |

Morphisms of sheaves and presheaves | 31 |

Cohomology | 115 |

further reading | 154 |

163 | |

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### Common terms and phrases

abelian category abelian groups Abgp Algebraic analytic bijection Cą(U cohomology cokernel commutes condition constant sheaf construct d-functor Definition denotes derived functor diagram direct limit direct system epimorphism example finite flasque functions geometric space germ given gives Hence Hom(F homeomorphism injective inverse kernel Lemma Let F local homeomorphism locally free Modules monomorphism monopresheaf morphism f morphism of presheaves morphism of sheaves natural isomorphism open cover open neighbourhood open set phism Presh Presh/X presheaf presheaf F presheaf of abelian Proof Proposition R-algebras R-module restriction map ring morphism ringed space SCok(f sections of F sequence of sheaves sets or abelian sheaf F sheaf of abelian sheaf space Show Shv/X spaces of abelian Spec stalks subspace surjective system of abelian Theorem topological space unique universal property zero α α ΕΛ λελ