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<p><a id="X810D68A7794474F9" name="X810D68A7794474F9"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X810D68A7794474F9">3 <span class="Heading">Graded Rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7A16AC227F0B7E6B">3.1 <span class="Heading">Graded Rings: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8025623B822D44AF">3.1-1 IsHomalgGradedRingRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84BC7F817BF737F7">3.1-2 IsHomalgGradedRingElementRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8538A7C985FE0E46">3.2 <span class="Heading">Graded Rings: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X823140CE8286BA90">3.2-1 HomalgGradedRingElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7CA805077B75EAAD">3.3 <span class="Heading">Graded Rings: Attributes and Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79649FC47A744BD2">3.3-1 DegreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X826C131D83796CC5">3.3-2 CommonNonTrivialWeightOfIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E43637F8431FB69">3.3-3 WeightsOfIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X807A4268816DD69D">3.3-4 IsHomogeneousRingElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7B86A3DD7D2CE35F">3.4 <span class="Heading">Graded Rings: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DA1D14F87DE72D8">3.4-1 UnderlyingNonGradedRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8708576B84997122">3.4-2 UnderlyingNonGradedRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83971F1879017E3D">3.4-3 Name</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X840864B38184FF3E">3.4-4 HomogeneousPartOfRingElement</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Graded Rings</span></h3>

<p>The package <strong class="pkg">GradedRingForHomalg</strong> defines the classes of graded rings, ring elements and 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_mj.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 <strong class="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>


<ul>
<li><p><code class="code">SyzygiesGeneratorsOfRows</code></p>

</li>
<li><p><code class="code">SyzygiesGeneratorsOfColumns</code></p>

</li>
<li><p><code class="code">ReducedSyzygiesGeneratorsOfRows</code></p>

</li>
<li><p><code class="code">ReducedSyzygiesGeneratorsOfColumns</code></p>

</li>
<li><p><code class="code">BasisOfRowModule</code></p>

</li>
<li><p><code class="code">BasisOfColumnModule</code></p>

</li>
<li><p><code class="code">ReducedBasisOfRowModule</code></p>

</li>
<li><p><code class="code">ReducedBasisOfColumnModule</code></p>

</li>
<li><p><code class="code">DecideZeroRows</code></p>

</li>
<li><p><code class="code">DecideZeroColumns</code></p>

</li>
<li><p><code class="code">LeftDivide</code></p>

</li>
<li><p><code class="code">RightDivide</code></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>

<p><a id="X7A16AC227F0B7E6B" name="X7A16AC227F0B7E6B"></a></p>

<h4>3.1 <span class="Heading">Graded Rings: Category and Representations</span></h4>

<p><a id="X8025623B822D44AF" name="X8025623B822D44AF"></a></p>

<h5>3.1-1 IsHomalgGradedRingRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomalgGradedRingRep</code>( <var class="Arg">R</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: true or false</p>

<p>The representation of <strong class="pkg">homalg</strong> graded rings.</p>

<p>(It is a subrepresentation of the <strong class="pkg">GAP</strong> representation <br /> <code class="code">IsHomalgRingOrFinitelyPresentedModuleRep</code>.)</p>


<div class="example"><pre>
DeclareRepresentation( "IsHomalgGradedRingRep",
        IsHomalgGradedRing and
        IsHomalgGradedRingOrGradedModuleRep,
        [ "ring" ] );
</pre></div>

<p><a id="X84BC7F817BF737F7" name="X84BC7F817BF737F7"></a></p>

<h5>3.1-2 IsHomalgGradedRingElementRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomalgGradedRingElementRep</code>( <var class="Arg">r</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: true or false</p>

<p>The representation of elements of <strong class="pkg">homalg</strong> graded rings.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="code">IsHomalgRingElement</code>.)</p>


<div class="example"><pre>
DeclareRepresentation( "IsHomalgGradedRingElementRep",
        IsHomalgGradedRingElement,
        [ ] );
</pre></div>

<p><a id="X8538A7C985FE0E46" name="X8538A7C985FE0E46"></a></p>

<h4>3.2 <span class="Heading">Graded Rings: Constructors</span></h4>

<p><a id="X823140CE8286BA90" name="X823140CE8286BA90"></a></p>

<h5>3.2-1 HomalgGradedRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomalgGradedRingElement</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">‣ HomalgGradedRingElement</code>( <var class="Arg">numer</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a graded ring element</p>

<p>Creates the graded ring element <span class="SimpleMath">\(\textit{numer}/\textit{denom}\)</span> or in the second case <span class="SimpleMath">\(\textit{numer}/1\)</span> for the graded 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>

<p><a id="X7CA805077B75EAAD" name="X7CA805077B75EAAD"></a></p>

<h4>3.3 <span class="Heading">Graded Rings: Attributes and Properties</span></h4>

<p><a id="X79649FC47A744BD2" name="X79649FC47A744BD2"></a></p>

<h5>3.3-1 DegreeGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeGroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a left ℤ-module</p>

<p>The degree Abelian group of the commutative graded ring <var class="Arg">S</var>.</p>

<p><a id="X826C131D83796CC5" name="X826C131D83796CC5"></a></p>

<h5>3.3-2 CommonNonTrivialWeightOfIndeterminates</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommonNonTrivialWeightOfIndeterminates</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a degree</p>

<p>The common nontrivial weight of the indeterminates of the graded ring <var class="Arg">S</var> if it exists. Otherwise an error is issued. WARNING: Since the DegreeGroup and WeightsOfIndeterminates are in some cases bound together, you MUST not set the DegreeGroup by hand and let the algorithm create the weights. Set both by hand, set only weights or use the method WeightsOfIndeterminates to set both. Never set the DegreeGroup without the WeightsOfIndeterminates, because it simply wont work!</p>

<p><a id="X7E43637F8431FB69" name="X7E43637F8431FB69"></a></p>

<h5>3.3-3 WeightsOfIndeterminates</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WeightsOfIndeterminates</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list or listlist of integers</p>

<p>The list of degrees of the indeterminates of the graded ring <var class="Arg">S</var>.</p>

<p><a id="X807A4268816DD69D" name="X807A4268816DD69D"></a></p>

<h5>3.3-4 IsHomogeneousRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomogeneousRingElement</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>returns whether the graded ring element <var class="Arg">r</var> is homogeneous or not.</p>

<p><a id="X7B86A3DD7D2CE35F" name="X7B86A3DD7D2CE35F"></a></p>

<h4>3.4 <span class="Heading">Graded Rings: Operations and Functions</span></h4>

<p><a id="X7DA1D14F87DE72D8" name="X7DA1D14F87DE72D8"></a></p>

<h5>3.4-1 UnderlyingNonGradedRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingNonGradedRing</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> ring</p>

<p>Internally there is a ring, in which computations take place.</p>

<p><a id="X8708576B84997122" name="X8708576B84997122"></a></p>

<h5>3.4-2 UnderlyingNonGradedRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingNonGradedRing</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> ring</p>

<p>Internally there is a ring, in which computations take place.</p>

<p><a id="X83971F1879017E3D" name="X83971F1879017E3D"></a></p>

<h5>3.4-3 Name</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Name</code>( <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a string</p>

<p>The name of the graded ring element <var class="Arg">r</var>.</p>

<p><a id="X840864B38184FF3E" name="X840864B38184FF3E"></a></p>

<h5>3.4-4 HomogeneousPartOfRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomogeneousPartOfRingElement</code>( <var class="Arg">r</var>, <var class="Arg">degree</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a graded ring element</p>

<p>returns the summand of <var class="Arg">r</var> whose monomials have the given degree <var class="Arg">degree</var> and if <var class="Arg">r</var> has no such monomials then it returns the zero element of the ring.</p>


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