The Mathematical Idea behind &homalg;

As finite dimensional constructions in linear algebra over a field k boil down to solving (in)homogeneous linear systems over k, the Gaussian algorithm makes the whole theory perfectly computable.

Hence, for homological algebra (viewed as linear algebra over general rings) to be constructive one needs to find apropriate substitutes for the Gaussian algorithm, where finite dimensionality has to be replaced by finite generatedness.

Luckily such substitutes exist for many rings of interest. Beside the well-known Hermite normal form algorithm for principal ideal rings it turns out that apropriate generalizations of the classical Gröbner basis algorithm for polynomial rings provide the desired substitute for a wide class of commutative and noncommutative rings. Note that for noncommutative rings the above discussion has to be restricted to homological constructions leading to one-sided linear systems XA=B resp. AX=B (&see; ).

The two appendices and provide the list of matrix operations needed by &homalg; to perform homological computations. Subsection explains how these matrix operations can be delegated to external systems.