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  Idea.xml            Modules package documentation

         Copyright (C) 2007-2009, Mohamed Barakat, RWTH-Aachen

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<Appendix Label="homalg-Idea">
<Heading>The Mathematical Idea behind &Modules;</Heading>

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

Hence, for homological algebra (viewed as linear algebra over
general rings) to be computable one needs to find appropriate
substitutes for the Gaussian algorithm, where finite dimensionality
has to be replaced by finite generatedness. <P/>

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 appropriate generalizations of the classical Gröbner
basis algorithm for polynomial rings provide the desired substitute
for a wide class of commutative <E>and</E> noncommutative rings. Note
that for noncommutative rings the above discussion has to be
restricted to homological constructions leading to one-sided linear
systems <M>XA=B</M> resp. <M>AX=B</M> (&see; <Ref
Label="Modules-limitation" Text="Principal limitation"/>). <P/>

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The two appendices <Ref Chap="Basic_Operations"/> and <Ref
Chap="Tool_Operations"/> provide the list of matrix operations needed
by &homalg; to perform homological computations. Subsection <Ref
Sect="homalg-delegates"/> explains how these matrix operations can be
delegated to external systems.
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</Appendix>