<p>The package <strong class="pkg">GradedRingForHomalg</strong> defines the classes of graded rings, ring elements and homogeneous matrices over such rings. 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 graded rings.</p>
<p>The graded 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="chapB.html#X78C55DF7875560DD"><span class="RefLink">B</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">MatricesForHomalg</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>Operations within <strong class="pkg">MatricesForHomalg</strong> that take matrices as input and produce a matrix as an output produce homogeneous output for homogeneous input in the following cases: the graded ring in question is either a polynomial ring or the exterior algebra residing in <strong class="pkg">Singular</strong>, and the called operation is one of the following listed below:</p>
</li>
</ul>
<p>These operation trigger Gröbner bases computations in <strong class="pkg">Singular</strong>, which are always forced to be performed with a tail reduction by <strong class="pkg">homalg</strong>. In particular, the resulting elements of the Gröbner bases have to be homogeneous.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixOverGradedRing</code>( <var class="Arg">mat</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a matrix over a graded ring</p>
<p>Creates a matrix for the graded ring <var class="Arg">S</var>, where <var class="Arg">mat</var> is a matrix over <code class="code">UnderlyingNonGradedRing</code>(<var class="Arg">S</var>).</p>
<p>The output is the homogeneous part of the matrix <var class="Arg">A</var> with respect to the given degrees <var class="Arg">degrees</var>. See <var class="Arg">HomogeneousPartOfRingElement</var>.</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 matrix <var class="Arg">mat</var> to the value <var class="Arg">r</var>. Here <var class="Arg">R</var> is the graded <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 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 (graded) <strong class="pkg">homalg</strong> ring involved in these computations.</p>
<p>Returns the entry (<var class="Arg">i,j</var>) of the matrix <var class="Arg">mat</var> as a string. Here <var class="Arg">R</var> is the (graded) <strong class="pkg">homalg</strong> ring involved in these computations.</p>
<p>Returns the entry (<var class="Arg">i,j</var>) of the matrix <var class="Arg">mat</var>. Here <var class="Arg">R</var> is the (graded) <strong class="pkg">homalg</strong> ring involved in these computations.</p>
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