Elements of Homological Algebra

Abstract

Homological algebra is one of the most important tools if not the most important of all for the theory of representations of algebras. It gathers information of different points of views about its main subject, which is that of short exact sequences. First an equivalence relation is established on short exact sequences with fixed end terms. Then a left and a right multiplication with certain morphisms are defined and it is shown that this two multiplications are compatible. This enables to define a structure of vector space on the equivalence classes of short exact sequences with fixed end terms. Finally, for each module and each short exact sequence two so-called “long exact” sequences are associated. This makes it possible to compare the extension spaces for different end terms. Homological algebra prepares the basics for the second step for understanding the structure of module categories, which is called Auslander-Reiten theory and developed in the next chapter.