Derived Categories and Derived Functors

Abstract

Homological algebra was founded by D. Hilbert. He considered, in particular, the following problem. Let $\sum\nolimits_{j = 1}^m {{a_{ij}}{x_j} = 0;\;i = 1, \ldots ,n;{a_{ij}}} \in k\left[ {{t_1}, \ldots {t_r}} \right]$, be a system of linear homogeneous equations with coefficients lying in the polynomial ring over a field. All polynomial solutions are linear combinations (with polynomial coefficients) of a finite subset of solutions. However, in general there exists no basis of solutions that are linearly independent over k[t ₁..., t _r]. Linear relations among elements of a generating system of solutions are, in turn, linear combinations of some finite set of relations, and again it might happen that there exists no free system of generators for relations.