<p>The package <strong class="pkg">LocalizeRingForHomalg</strong> defines the classes of local(ized) rings, local ring elements and local matrices. These three objects can be used as data structures defined in <strong class="pkg">MatricesForHomalg</strong> on which the <strong class="pkg">homalg</strong> project can rely to do homological computations over localized rings.</p>
<p>A <strong class="pkg">homalg</strong> local ring element contains two <strong class="pkg">homalg</strong> ring elements, a numerator (--> <code class="func">Numerator</code> (<a href="chap4.html#X86B86F85831145D0"><span class="RefLink">4.3-4</span></a>)) and a denominator (--> <code class="func">Denominator</code> (<a href="chap4.html#X83E87130831148A4"><span class="RefLink">4.3-6</span></a>)). A <strong class="pkg">homalg</strong> local matrix contains a global <strong class="pkg">homalg</strong> matrix as a numerator (--> <code class="func">Numerator</code> (<a href="chap4.html#X8613D7A27D3D770D"><span class="RefLink">4.3-5</span></a>)) and a ring element as a denominator (--> <code class="func">Denominator</code> (<a href="chap4.html#X863DB91A7DEDEA22"><span class="RefLink">4.3-7</span></a>)). New constructors for ring elements and matrices are <code class="func">HomalgLocalRingElement</code> (<a href="chap4.html#X8741B0AE787624CB"><span class="RefLink">4.3-17</span></a>) and <code class="func">HomalgLocalMatrix</code> (<a href="chap4.html#X808317A88773E967"><span class="RefLink">4.3-18</span></a>) in addition to the standard contructors introduced in other packages of the <strong class="pkg">homalg</strong> project.</p>
<p>The local rings most prominently can be used with methods known from general <strong class="pkg">homalg</strong> rings. The methods for doing the computations are presented in the appendix (<a href="chapA.html#X7C7697867CBF79C9"><span class="RefLink">A</span></a>), since they are not for external use. The new attributes and operations are documented here.</p>
<p>Since the objects inplemented here are representations from objects elsewhere in the <strongclass="pkg">homalg</strong> project (i.e. <strong class="pkg">MatricesForHomalg</strong>), we want to stress that there are many other operations in <strong class="pkg">homalg</strong>, which can be used in connection with the ones presented here. A few of them can be found in the examples and the appendix of this documentation.</p>
<p>The representation of <strong class="pkg">homalg</strong> local rings.</p>
<p>(It is a subrepresentation of the <strong class="pkg">GAP</strong> representation <br /> <code class="code">IsHomalgRingOrFinitelyPresentedModuleRep</code>.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Numerator</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a (global) <strong class="pkg">homalg</strong> ring element</p>
<p>The numerator from a local ring element <var class="Arg">r</var>, which is a <strong class="pkg">homalg</strong> ring element from the computation ring.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Denominator</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a (global) <strong class="pkg">homalg</strong> ring element</p>
<p>The denominator from a local ring element <var class="Arg">r</var>, which is a <strong class="pkg">homalg</strong> ring element from the computation ring.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Denominator</code>( <var class="Arg">mat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a (global) <strong class="pkg">homalg</strong> ring element</p>
<p>The denominator from a local matrix <var class="Arg">mat</var>, which is a <strong class="pkg">homalg</strong> matrix from the computation ring.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetMatElm</code>( <var class="Arg">mat</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">r</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Changes the entry (<var class="Arg">i,j</var>) of the local matrix <var class="Arg">mat</var> to the value <var class="Arg">r</var>. Here <var class="Arg">R</var> is the (local) <strong class="pkg">homalg</strong> ring involved in these computations.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddToMatElm</code>( <var class="Arg">mat</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">r</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Changes the entry (<var class="Arg">i,j</var>) of the local matrix <var class="Arg">mat</var> by adding the value <var class="Arg">r</var> to it. Here <var class="Arg">R</var> is the (local) <strong class="pkg">homalg</strong> ring involved in these computations.</p>
<p>Returns the entry (<var class="Arg">i,j</var>) of the local matrix <var class="Arg">mat</var> as a string. Here <var class="Arg">R</var> is the (local) <strong class="pkg">homalg</strong> ring involved in these computations.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatElm</code>( <var class="Arg">mat</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a local ring element</p>
<p>Returns the entry (<var class="Arg">i,j</var>) of the local matrix <var class="Arg">mat</var>. Here <var class="Arg">R</var> is the (local) <strong class="pkg">homalg</strong> ring involved in these computations.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cancel</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: two ring elements</p>
<p>For <span class="SimpleMath"><var class="Arg">a</var>=a'*c and b=b'*c</span> return <span class="SimpleMath">a' and b'</span>. The exact form of <span class="SimpleMath">c</span> depends on whether a procedure for gcd computation is included in the ring package.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalizeAt</code>( <var class="Arg">R</var>, <var class="Arg">l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a local ring</p>
<p>If <var class="Arg">l</var> is a list of elements of the global ring <var class="Arg">R</var> generating a maximal ideal, the method creates the corresponding localization of <var class="Arg">R</var> at the complement of the maximal ideal.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalizeAtZero</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a local ring</p>
<p>This method creates the corresponding localization of <var class="Arg">R</var> at the complement of the maximal ideal generated by the indeterminates ("at zero").</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalizePolynomialRingAtZeroWithMora</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a local ring</p>
<p>This method localizes the ring <var class="Arg">R</var> at zero and this localized ring is returned. The ring table uses Mora's algorithm as implemented Singular for low level computations.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomalgLocalRingElement</code>( <var class="Arg">numer</var>, <var class="Arg">denom</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomalgLocalRingElement</code>( <var class="Arg">numer</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a local ring element</p>
<p>Creates the local ring element <span class="SimpleMath"><var class="Arg">numer</var>/<var class="Arg">denom</var></span> or in the second case <span class="SimpleMath"><var class="Arg">numer</var>/1</span> for the local ring <var class="Arg">R</var>. Both <var class="Arg">numer</var> and <var class="Arg">denom</var> may either be a string describing a valid global ring element or from the global ring or computation ring.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomalgLocalMatrix</code>( <var class="Arg">numer</var>, <var class="Arg">denom</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomalgLocalMatrix</code>( <var class="Arg">numer</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a local matrix</p>
<p>Creates the local matrix <span class="SimpleMath"><var class="Arg">numer</var>/<var class="Arg">denom</var></span> or in the second case <span class="SimpleMath"><var class="Arg">numer</var>/1</span> for the local ring <var class="Arg">R</var>. Both <var class="Arg">numer</var> and <var class="Arg">denom</var> may either be from the global ring or the computation ring.</p>
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