<h4>1.1 <span class="Heading">What is the Role of the <strong class="pkg">LocalizeRingForHomalg</strong> Package in the <strong class="pkg">homalg</strong> Project?</span></h4>
<p>The <strong class="pkg">homalg</strong> project <a href="chapBib.html#biBhomalg-project">[tpa22]</a> aims at providing a general and abstract framework for homological computations. The package <strong class="pkg">LocalizeRingForHomalg</strong> enables the <strong class="pkg">homalg</strong> project to construct localizations from commutative rings in <strong class="pkg">homalg</strong> at their maximal ideals.</p>
<p>The package <strong class="pkg">LocalizeRingForHomalg</strong> on the one hand builds on the package <strong class="pkg">MatricesForHomalg</strong> and on the other hands adds functionality to <strong class="pkg">MatricesForHomalg</strong>. It uses the computability (i.e. capability to solve linear systems) of a commutative ring <span class="SimpleMath">R</span> declared in <strong class="pkg">MatricesForHomalg</strong> to construct the localization <span class="SimpleMath">R_m</span> of <span class="SimpleMath">R</span> at a maximal ideal <span class="SimpleMath">m</span> (given by a finite set of generators). This localized ring <span class="SimpleMath">R_m</span> is again computable and can thus be used by <strong class="pkg">MatricesForHomalg</strong>.</p>
<p>Furthermore, via the package <strong class="pkg">RingsForHomalg</strong>, an interface to <strong class="pkg">Singular</strong> is used to compute in localized polynomial rings with the help of Mora's algorithm.
<h4>1.3 <span class="Heading">The Math Behind This Package</span></h4>
<p>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 <strong class="pkg">homalg</strong> <a href="chapBib.html#biBhomalg-package">[BLH20]</a>) without need of implementing new low level algorithms. Details can be found in the paper <a href="chapBib.html#biBBL">[BLH11]</a>. This ring can be constructed by <code class="func">LocalizeAt</code> (<a href="chap4.html#X7E4E70FE82978F9C"><span class="RefLink">4.3-14</span></a>) and <code class="func">LocalizeAtZero</code> (<a href="chap4.html#X7D910AA785CEED34"><span class="RefLink">4.3-15</span></a>).</p>
<p>Furthermore we use the package <strong class="pkg">RingsForHomalg</strong> to communicate with <strong class="pkg">Singular</strong> 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 LocalizePolynomialRingAtZeroWithMora (4.3-16).
<h4>1.4 <span class="Heading">Which Ring to Use?</span></h4>
<p>Since there are two kinds of rings included in this package, we want to offer a short comparison of these.</p>
<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>
<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.</p>
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