Categories and Homological Algebra

The aim of these Notes is to introduce the reader to the language of categories with emphazis on homological algebra. We treat with some details basic homological algebra, that is, categories of complexes in additive and abelian categories and construct with some care the derived functors. We also introduce the reader to the more sophisticated concepts of triangulated and derived categories. Our exposition on these topics is rather sketchy, and the reader is encouraged to consult the literature. These Notes are extracted from [13]. Other references are [15], [2] for the general theory of categories, [7], [18] and [12], Ch I for homological algebra, including derived categories. The book [14] provides a nice elementary introduction to the classical homological algebra. For further developements, see [10], [13]. Let us briefly describe the contents of these Notes. Chapter 1 is a survey of linear algebra over a ring. It serves as a guide for the theory of additive and abelian categories. First, we study the functors Hom and ⊗ on the category Mod(A) of modules over a (non necessarily commutative) ring A. Then we introduce the inductive and projective limits of modules and study the exactness of the functors lim − → and lim ← −. Finally we introduce Koszul complexes. In Chapter 2 we expose the basic language of categories and functors, including the Yoneda Lemma, and the notions of representable and adjoint functors. In Chapter 3 we construct the projective and inductive limits and, as a particular case, the kernels and cokernels, products and coproducts. We introduce the notions filtrant category and cofinal functors, and study with some care filtrant inductive limits in the category Set of sets. Finally, we define right or left exact functors and give some examples. Chapter 4 is devoted to the study of additive categories and complexes in such categories. In the category C(C) of complexes of an additive category C, we introduce some basic constructions and notions such as the shift func-tor, the mapping cone of a morphism, the morphisms homotopic to zero, the 5 6 CONTENTS