intro.xml

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<!-- 

  intro.xml            LocalizeRingForHomalg package documentation             Mohamed Barakat
                                                                        Markus Lange-Hegermann

         Copyright (C) 2009, Mohamed Barakat, Universität des Saarlandes
                      Markus Lange-Hegermann, RWTH-Aachen University

This chapter gives a short introduction and explains the philosophy
behind the package.

-->

<Chapter Label="intro">
  <Heading>Introduction</Heading>

  <Section Label="role">
    <Heading>What is the Role of the &LocalizeRingForHomalg; Package in the &homalg; Project?</Heading>
    The &homalg; project <Cite Key="homalg-project"/> aims at providing a general and abstract framework for homological computations. The package &LocalizeRingForHomalg; enables the &homalg; project to construct localizations from commutative rings in &homalg; at their maximal ideals.
  </Section>

  <Section Label="Functionality">
    <Heading>Functionality</Heading>
    The package &LocalizeRingForHomalg; on the one hand builds on the package &MatricesForHomalg; and on the other hands adds functionality to &MatricesForHomalg;. It uses the computability (i.e. capability to solve linear systems) of a commutative ring <M>R</M> declared in &MatricesForHomalg; to construct the localization <M>R_m</M> of <M>R</M> at a maximal ideal <M>m</M> (given by a finite set of generators). This localized ring <M>R_m</M> is again computable and can thus be used by &MatricesForHomalg;.
    <P/>Furthermore, via the package &RingsForHomalg;, an interface to &Singular; is used to compute in localized polynomial rings with the help of Mora's algorithm.
  </Section>

  <Section Label="math">
    <Heading>The Math Behind This Package</Heading>
    The math behind this package is a simple trick in allowing global computation to be done instead of local computations. This works on any commutative computable ring (in the sense of &homalg; <Cite Key="homalg-package"/>) without need of implementing new low level algorithms. Details can be found in the paper <Cite Key="BL"/>. This ring can be constructed by <Ref Oper="LocalizeAt" Label= "for a commutative ring and a maximal ideal"/> and <Ref Oper="LocalizeAtZero" Label= "for a free polynomial ring"/>.
    <P/>Furthermore we use the package &RingsForHomalg; to communicate with &Singular; and use the implementation of Mora's algorithm there. This is restricted to polynomial rings and needs the package &RingsForHomalg;. This ring can be constructed by <Ref Oper="LocalizePolynomialRingAtZeroWithMora" Label= "constructor for homalg localized rings using Mora's algorithm"/>.
  </Section>

  <Section Label="ring comparison">
    <Heading>Which Ring to Use?</Heading>
    Since there are two kinds of rings included in this package, we want to offer a short comparison of these.
    <P/>As usually one important part of such a comparison is the computation time. In our experience the general localization is much faster than Mora's algorithm for large examples.
    <P/>The main advantage of using local bases with Mora's algorithm is the possibility of computing Hilbert polynomials and other combinatorical invariants. This is not possible with our localization algorithm. But it is possible to do a large computation without Mora's algorithm, which perhaps would not terminate in acceptable time, and afterwards compute a local standard basis of the - in comparison to intermediate computations usually much smaller - result to get the combinatorical information and invariants.
    <P/>Furthermore we remark, that our localization algorithm works on any maximal ideal in any computable commutative ring, whereas Mora's algorithm only works for polynomial rings at the maximal ideal generated by the indeterminates. Of course by affine transformation Mora's algorithm will work on any maximal ideal in a polynomial ring where the residue class field is isomorphic to the ground field.
  </Section>

</Chapter>