Complexes in Abelian Categories

Abstract

In this chapter, we study complexes (and double complexes) in abelian categories and give tools to compute their cohomology. In particular, we prove the classical “Snake lemma” and we construct the long exact sequence associated with a short exact sequence of complexes.

As an application, we discuss Koszul complexes associated to functors defined on a category of finite subsets of a set S, with values in an abelian category C. The main result asserts that such a complex may be obtained as the mapping cone of a morphism acting on a simpler Koszul complex. We apply these results to the study of distributive families of subobjects of an object X in C.

We postpone the introduction of derived categories to the next chapter.

Note that we avoid the use of spectral sequences, using instead systematically the “truncation functors”.