Preface
Contents
Introduction
1 categories,Froducts,Projective and Inductive Limits
1.1 The Notion of a Category and Examples
1.2 Functors
1.3 Products,Projective Limits and Direct Limits in a Category
1.3.1 The Projective Limit
1.3.2 The Yoneda Lemma
1.3.3 Examples
1.3.4 Representable Functors
1.3.5 Direct Limits
1.4 Exercises
2 Basic Concepts of Homological AlgSbra
2.1 The Category Modr of r-modules
2.2 More Functors
2.2.1 Invariants,Coinvariants and Bxactnes
2.2.2 The First Cohomology Group
2.2.3 Some Notation
2.2.4 Exercises
2.3 The Derived Functors
2.3.1 The Simple Principle
2.3.2 Functoriality
2.3.3 Other Resolutions
2.3.4 Injective Resolutions of Short Exact Sequences
A Fundamental Remark
The Cohomology and the Long Bxact Sequence
The Homology of Groups
2.4 The Functors Ext and Tor
2.4.1 The Functor Ext
2.4.2 The Derived Functor for the Tensor Product
2.4.3 Exercise
3 Sheaves
3.1 Presheaves and Sheaves
3.1.1 What is a Presheaf
3.1.2 A Remark about Products and Presheaf
3.1.3 What is a Sheaf
3.1.4 Examples
3.2 Manifolds as Locally Ringed Spaces
3.2.1 What Are Manifolds
3.2.2 Examples and Exercise
3.3 Stalks and Sheafification
3.3.1 Stalks
3.3.2 The Process of Sheafification of a Presheaf
3.4 The Functors f*and f
3.4.1 The Adjunction Formula
3.4.2 Bxtensions and Restrictiona
3.5 Constructions of Sheaves
4 Cohomology of Sheaves
4.1 Examples
