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<p><a id="X7B8C95CA7DA733B4" name="X7B8C95CA7DA733B4"></a></p>
<div class="ChapSects"><a href="chap1_mj.html#X7B8C95CA7DA733B4">1 <span class="Heading">Module Presentations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X78D1062D78BE08C1">1.1 <span class="Heading">Functors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F7AC44478418555">1.1-1 FunctorStandardModuleLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87AEC1177DB7F50D">1.1-2 FunctorStandardModuleRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8427C0B17A445822">1.1-3 FunctorGetRidOfZeroGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F1E779D8003146B">1.1-4 FunctorGetRidOfZeroGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X819A04517B3601C0">1.1-5 FunctorLessGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F80AE9B7EC07198">1.1-6 FunctorLessGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X877B7ACE87E1BEC2">1.1-7 FunctorDualLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D56611D7BF91B54">1.1-8 FunctorDualRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C9901F8851FD24A">1.1-9 FunctorDoubleDualLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8016306881444DCA">1.1-10 FunctorDoubleDualRight</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7D03633A7D98026B">1.2 <span class="Heading">GAP Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X79DBCB747E91FB70">1.2-1 IsLeftOrRightPresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85E26CFF86855B6B">1.2-2 IsLeftPresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X873EFE29849F6998">1.2-3 IsRightPresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BB95B7A7EB96854">1.2-4 IsLeftOrRightPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C71B8D17C60C6B5">1.2-5 IsLeftPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7DBF478D7EE3FE63">1.2-6 IsRightPresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X86EC0F0A78ECBC10">1.3 <span class="Heading">Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87010AB6819736C8">1.3-1 PresentationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C9B36AD7B9CCC8D">1.3-2 AsMorphismBetweenFreeLeftPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F1AD55C852BE617">1.3-3 AsMorphismBetweenFreeRightPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BE01A1381744627">1.3-4 AsLeftPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X780443B07F43AA1C">1.3-5 AsRightPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F345A2A87ABE417">1.3-6 FreeLeftPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85536E4E85D15252">1.3-7 FreeRightPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X86F926B27C579E66">1.3-8 UnderlyingMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D8E30E486A08439">1.3-9 UnderlyingHomalgRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B5539AB8541F618">1.3-10 Annihilator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87946F997AD1005A">1.3-11 LeftPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BDF988F7FFEAB8C">1.3-12 RightPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8145816A85BA2680">1.3-13 LeftPresentations_as_FreydCategory_CategoryOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7ED9DFEB82DE9653">1.3-14 RightPresentations_as_FreydCategory_CategoryOfColumns</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7C701DBF7BAE649A">1.4 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8157F3E8847B15E1">1.4-1 UnderlyingHomalgRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X83CA6F06832162B7">1.4-2 UnderlyingMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X81CDBC6E7DBB4EA0">1.5 <span class="Heading">Non-Categorical Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8508310C7E908093">1.5-1 StandardGeneratorMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7EF1493A7D341F5E">1.5-2 CoverByFreeModule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X836749D8814FEEE6">1.6 <span class="Heading">Natural Transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85D3DB1F856D05EF">1.6-1 NaturalIsomorphismFromIdentityToStandardModuleLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B64AF718133C945">1.6-2 NaturalIsomorphismFromIdentityToStandardModuleRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7CA2B84E7F933125">1.6-3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7A2DCBD6844093E3">1.6-4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B331B0A86010185">1.6-5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X834AC0FD825FCD2F">1.6-6 NaturalIsomorphismFromIdentityToLessGeneratorsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E37CB058378CBEE">1.6-7 NaturalTransformationFromIdentityToDoubleDualLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F92B4448041A68C">1.6-8 NaturalTransformationFromIdentityToDoubleDualRight</a></span>
</div></div>
</div>

<h3>1 <span class="Heading">Module Presentations</span></h3>

<p><a id="X78D1062D78BE08C1" name="X78D1062D78BE08C1"></a></p>

<h4>1.1 <span class="Heading">Functors</span></h4>

<p><a id="X7F7AC44478418555" name="X7F7AC44478418555"></a></p>

<h5>1.1-1 FunctorStandardModuleLeft</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is a functor which takes a left presentation as input and computes its standard presentation.</p>

<p><a id="X87AEC1177DB7F50D" name="X87AEC1177DB7F50D"></a></p>

<h5>1.1-2 FunctorStandardModuleRight</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is a functor which takes a right presentation as input and computes its standard presentation.</p>

<p><a id="X8427C0B17A445822" name="X8427C0B17A445822"></a></p>

<h5>1.1-3 FunctorGetRidOfZeroGeneratorsLeft</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is a functor which takes a left presentation as input and gets rid of the zero generators.</p>

<p><a id="X7F1E779D8003146B" name="X7F1E779D8003146B"></a></p>

<h5>1.1-4 FunctorGetRidOfZeroGeneratorsRight</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is a functor which takes a right presentation as input and gets rid of the zero generators.</p>

<p><a id="X819A04517B3601C0" name="X819A04517B3601C0"></a></p>

<h5>1.1-5 FunctorLessGeneratorsLeft</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is functor which takes a left presentation as input and computes a presentation having less generators.</p>

<p><a id="X7F80AE9B7EC07198" name="X7F80AE9B7EC07198"></a></p>

<h5>1.1-6 FunctorLessGeneratorsRight</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is functor which takes a right presentation as input and computes a presentation having less generators.</p>

<p><a id="X877B7ACE87E1BEC2" name="X877B7ACE87E1BEC2"></a></p>

<h5>1.1-7 FunctorDualLeft</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span> that has an involution function. The output is functor which takes a left presentation <var class="Arg">M</var> as input and computes its Hom(M, R) as a left presentation.</p>

<p><a id="X7D56611D7BF91B54" name="X7D56611D7BF91B54"></a></p>

<h5>1.1-8 FunctorDualRight</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span> that has an involution function. The output is functor which takes a right presentation <var class="Arg">M</var> as input and computes its Hom(M, R) as a right presentation.</p>

<p><a id="X7C9901F8851FD24A" name="X7C9901F8851FD24A"></a></p>

<h5>1.1-9 FunctorDoubleDualLeft</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span> that has an involution function. The output is functor which takes a left presentation <var class="Arg">M</var> as input and computes its <var class="Arg">Hom( Hom(M, R), R )</var> as a left presentation.</p>

<p><a id="X8016306881444DCA" name="X8016306881444DCA"></a></p>

<h5>1.1-10 FunctorDoubleDualRight</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span> that has an involution function. The output is functor which takes a right presentation <var class="Arg">M</var> as input and computes its <var class="Arg">Hom( Hom(M, R), R )</var> as a right presentation.</p>

<p><a id="X7D03633A7D98026B" name="X7D03633A7D98026B"></a></p>

<h4>1.2 <span class="Heading">GAP Categories</span></h4>

<p><a id="X79DBCB747E91FB70" name="X79DBCB747E91FB70"></a></p>

<h5>1.2-1 IsLeftOrRightPresentationMorphism</h5>

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

<p>The GAP category of morphisms in the category of left or right presentations.</p>

<p><a id="X85E26CFF86855B6B" name="X85E26CFF86855B6B"></a></p>

<h5>1.2-2 IsLeftPresentationMorphism</h5>

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

<p>The GAP category of morphisms in the category of left presentations.</p>

<p><a id="X873EFE29849F6998" name="X873EFE29849F6998"></a></p>

<h5>1.2-3 IsRightPresentationMorphism</h5>

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

<p>The GAP category of morphisms in the category of right presentations.</p>

<p><a id="X7BB95B7A7EB96854" name="X7BB95B7A7EB96854"></a></p>

<h5>1.2-4 IsLeftOrRightPresentation</h5>

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

<p>The GAP category of objects in the category of left presentations or right presentations.</p>

<p><a id="X7C71B8D17C60C6B5" name="X7C71B8D17C60C6B5"></a></p>

<h5>1.2-5 IsLeftPresentation</h5>

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

<p>The GAP category of objects in the category of left presentations.</p>

<p><a id="X7DBF478D7EE3FE63" name="X7DBF478D7EE3FE63"></a></p>

<h5>1.2-6 IsRightPresentation</h5>

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

<p>The GAP category of objects in the category of right presentations.</p>

<p><a id="X86EC0F0A78ECBC10" name="X86EC0F0A78ECBC10"></a></p>

<h4>1.3 <span class="Heading">Constructors</span></h4>

<p><a id="X87010AB6819736C8" name="X87010AB6819736C8"></a></p>

<h5>1.3-1 PresentationMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PresentationMorphism</code>( <var class="Arg">A</var>, <var class="Arg">M</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(A,B)\)</span></p>

<p>The arguments are an object <span class="SimpleMath">\(A\)</span>, a homalg matrix <span class="SimpleMath">\(M\)</span>, and another object <span class="SimpleMath">\(B\)</span>. <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span> shall either both be objects in the category of left presentations or both be objects in the category of right presentations. The output is a morphism <span class="SimpleMath">\(A \rightarrow B\)</span> in the the category of left or right presentations whose underlying matrix is given by <span class="SimpleMath">\(M\)</span>.</p>

<p><a id="X7C9B36AD7B9CCC8D" name="X7C9B36AD7B9CCC8D"></a></p>

<h5>1.3-2 AsMorphismBetweenFreeLeftPresentations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsMorphismBetweenFreeLeftPresentations</code>( <var class="Arg">m</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(F^r,F^c)\)</span></p>

<p>The argument is a homalg matrix <span class="SimpleMath">\(m\)</span>. The output is a morphism <span class="SimpleMath">\(F^r \rightarrow F^c\)</span> in the the category of left presentations whose underlying matrix is given by <span class="SimpleMath">\(m\)</span>, where <span class="SimpleMath">\(F^r\)</span> and <span class="SimpleMath">\(F^c\)</span> are free left presentations of ranks given by the number of rows and columns of <span class="SimpleMath">\(m\)</span>.</p>

<p><a id="X7F1AD55C852BE617" name="X7F1AD55C852BE617"></a></p>

<h5>1.3-3 AsMorphismBetweenFreeRightPresentations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsMorphismBetweenFreeRightPresentations</code>( <var class="Arg">m</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(F^c,F^r)\)</span></p>

<p>The argument is a homalg matrix <span class="SimpleMath">\(m\)</span>. The output is a morphism <span class="SimpleMath">\(F^c \rightarrow F^r\)</span> in the the category of right presentations whose underlying matrix is given by <span class="SimpleMath">\(m\)</span>, where <span class="SimpleMath">\(F^r\)</span> and <span class="SimpleMath">\(F^c\)</span> are free right presentations of ranks given by the number of rows and columns of <span class="SimpleMath">\(m\)</span>.</p>

<p><a id="X7BE01A1381744627" name="X7BE01A1381744627"></a></p>

<h5>1.3-4 AsLeftPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsLeftPresentation</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>

<p>The argument is a homalg matrix <span class="SimpleMath">\(M\)</span> over a ring <span class="SimpleMath">\(R\)</span>. The output is an object in the category of left presentations over <span class="SimpleMath">\(R\)</span>. This object has <span class="SimpleMath">\(M\)</span> as its underlying matrix.</p>

<p><a id="X780443B07F43AA1C" name="X780443B07F43AA1C"></a></p>

<h5>1.3-5 AsRightPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsRightPresentation</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>

<p>The argument is a homalg matrix <span class="SimpleMath">\(M\)</span> over a ring <span class="SimpleMath">\(R\)</span>. The output is an object in the category of right presentations over <span class="SimpleMath">\(R\)</span>. This object has <span class="SimpleMath">\(M\)</span> as its underlying matrix.</p>

<p><a id="X7F345A2A87ABE417" name="X7F345A2A87ABE417"></a></p>

<h5>1.3-6 FreeLeftPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeLeftPresentation</code>( <var class="Arg">r</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>

<p>The arguments are a non-negative integer <span class="SimpleMath">\(r\)</span> and a homalg ring <span class="SimpleMath">\(R\)</span>. The output is an object in the category of left presentations over <span class="SimpleMath">\(R\)</span>. It is represented by the <span class="SimpleMath">\(0 \times r\)</span> matrix and thus it is free of rank <span class="SimpleMath">\(r\)</span>.</p>

<p><a id="X85536E4E85D15252" name="X85536E4E85D15252"></a></p>

<h5>1.3-7 FreeRightPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeRightPresentation</code>( <var class="Arg">r</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>

<p>The arguments are a non-negative integer <span class="SimpleMath">\(r\)</span> and a homalg ring <span class="SimpleMath">\(R\)</span>. The output is an object in the category of right presentations over <span class="SimpleMath">\(R\)</span>. It is represented by the <span class="SimpleMath">\(r \times 0\)</span> matrix and thus it is free of rank <span class="SimpleMath">\(r\)</span>.</p>

<p><a id="X86F926B27C579E66" name="X86F926B27C579E66"></a></p>

<h5>1.3-8 UnderlyingMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingMatrix</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg matrix</p>

<p>The argument is an object <span class="SimpleMath">\(A\)</span> in the category of left or right presentations over a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the underlying matrix which presents <span class="SimpleMath">\(A\)</span>.</p>

<p><a id="X7D8E30E486A08439" name="X7D8E30E486A08439"></a></p>

<h5>1.3-9 UnderlyingHomalgRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingHomalgRing</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg ring</p>

<p>The argument is an object <span class="SimpleMath">\(A\)</span> in the category of left or right presentations over a homalg ring <span class="SimpleMath">\(R\)</span>. The output is <span class="SimpleMath">\(R\)</span>.</p>

<p><a id="X7B5539AB8541F618" name="X7B5539AB8541F618"></a></p>

<h5>1.3-10 Annihilator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Annihilator</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(I, F)\)</span></p>

<p>The argument is an object <span class="SimpleMath">\(A\)</span> in the category of left or right presentations. The output is the embedding of the annihilator <span class="SimpleMath">\(I\)</span> of <span class="SimpleMath">\(A\)</span> into the free module <span class="SimpleMath">\(F\)</span> of rank <span class="SimpleMath">\(1\)</span>. In particular, the annihilator itself is seen as a left or right presentation.</p>

<p><a id="X87946F997AD1005A" name="X87946F997AD1005A"></a></p>

<h5>1.3-11 LeftPresentations</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the category of left presentations over <span class="SimpleMath">\(R\)</span>.</p>

<p><a id="X7BDF988F7FFEAB8C" name="X7BDF988F7FFEAB8C"></a></p>

<h5>1.3-12 RightPresentations</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the category of right presentations over <span class="SimpleMath">\(R\)</span>.</p>

<p><a id="X8145816A85BA2680" name="X8145816A85BA2680"></a></p>

<h5>1.3-13 LeftPresentations_as_FreydCategory_CategoryOfRows</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the category of left presentations over <span class="SimpleMath">\(R\)</span>, constructed internally as the <code class="code">FreydCategory</code> of the <code class="code">CategoryOfRows</code> of <var class="Arg">R</var>. Only available if the package <code class="code">FreydCategoriesForCAP</code> is available.</p>

<p><a id="X7ED9DFEB82DE9653" name="X7ED9DFEB82DE9653"></a></p>

<h5>1.3-14 RightPresentations_as_FreydCategory_CategoryOfColumns</h5>

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

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the category of right presentations over <span class="SimpleMath">\(R\)</span>, constructed internally as the <code class="code">FreydCategory</code> of the <code class="code">CategoryOfColumns</code> of <var class="Arg">R</var>. Only available if the package <code class="code">FreydCategoriesForCAP</code> is available.</p>

<p><a id="X7C701DBF7BAE649A" name="X7C701DBF7BAE649A"></a></p>

<h4>1.4 <span class="Heading">Attributes</span></h4>

<p><a id="X8157F3E8847B15E1" name="X8157F3E8847B15E1"></a></p>

<h5>1.4-1 UnderlyingHomalgRing</h5>

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

<p>The argument is a morphism <span class="SimpleMath">\(\alpha\)</span> in the category of left or right presentations over a homalg ring <span class="SimpleMath">\(R\)</span>. The output is <span class="SimpleMath">\(R\)</span>.</p>

<p><a id="X83CA6F06832162B7" name="X83CA6F06832162B7"></a></p>

<h5>1.4-2 UnderlyingMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingMatrix</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg matrix</p>

<p>The argument is a morphism <span class="SimpleMath">\(\alpha\)</span> in the category of left or right presentations. The output is its underlying homalg matrix.</p>

<p><a id="X81CDBC6E7DBB4EA0" name="X81CDBC6E7DBB4EA0"></a></p>

<h4>1.5 <span class="Heading">Non-Categorical Operations</span></h4>

<p><a id="X8508310C7E908093" name="X8508310C7E908093"></a></p>

<h5>1.5-1 StandardGeneratorMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardGeneratorMorphism</code>( <var class="Arg">A</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(F, A)\)</span></p>

<p>The argument is an object <span class="SimpleMath">\(A\)</span> in the category of left or right presentations over a homalg ring <span class="SimpleMath">\(R\)</span> with underlying matrix <span class="SimpleMath">\(M\)</span> and an integer <span class="SimpleMath">\(i\)</span>. The output is a morphism <span class="SimpleMath">\(F \rightarrow A\)</span> given by the <span class="SimpleMath">\(i\)</span>-th row or column of <span class="SimpleMath">\(M\)</span>, where <span class="SimpleMath">\(F\)</span> is a free left or right presentation of rank <span class="SimpleMath">\(1\)</span>.</p>

<p><a id="X7EF1493A7D341F5E" name="X7EF1493A7D341F5E"></a></p>

<h5>1.5-2 CoverByFreeModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoverByFreeModule</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(F,A)\)</span></p>

<p>The argument is an object <span class="SimpleMath">\(A\)</span> in the category of left or right presentations. The output is a morphism from a free module <span class="SimpleMath">\(F\)</span> to <span class="SimpleMath">\(A\)</span>, which maps the standard generators of the free module to the generators of <span class="SimpleMath">\(A\)</span>.</p>

<p><a id="X836749D8814FEEE6" name="X836749D8814FEEE6"></a></p>

<h4>1.6 <span class="Heading">Natural Transformations</span></h4>

<p><a id="X85D3DB1F856D05EF" name="X85D3DB1F856D05EF"></a></p>

<h5>1.6-1 NaturalIsomorphismFromIdentityToStandardModuleLeft</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalIsomorphismFromIdentityToStandardModuleLeft</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{StandardModuleLeft}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural isomorphism from the identity functor to the left standard module functor.</p>

<p><a id="X7B64AF718133C945" name="X7B64AF718133C945"></a></p>

<h5>1.6-2 NaturalIsomorphismFromIdentityToStandardModuleRight</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalIsomorphismFromIdentityToStandardModuleRight</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{StandardModuleRight}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural isomorphism from the identity functor to the right standard module functor.</p>

<p><a id="X7CA2B84E7F933125" name="X7CA2B84E7F933125"></a></p>

<h5>1.6-3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsLeft}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of left modules.</p>

<p><a id="X7A2DCBD6844093E3" name="X7A2DCBD6844093E3"></a></p>

<h5>1.6-4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsRight}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of right modules.</p>

<p><a id="X7B331B0A86010185" name="X7B331B0A86010185"></a></p>

<h5>1.6-5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalIsomorphismFromIdentityToLessGeneratorsLeft</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{LessGeneratorsLeft}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural morphism from the identity functor to the left less generators functor.</p>

<p><a id="X834AC0FD825FCD2F" name="X834AC0FD825FCD2F"></a></p>

<h5>1.6-6 NaturalIsomorphismFromIdentityToLessGeneratorsRight</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalIsomorphismFromIdentityToLessGeneratorsRight</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{LessGeneratorsRight}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural morphism from the identity functor to the right less generator functor.</p>

<p><a id="X7E37CB058378CBEE" name="X7E37CB058378CBEE"></a></p>

<h5>1.6-7 NaturalTransformationFromIdentityToDoubleDualLeft</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalTransformationFromIdentityToDoubleDualLeft</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualLeft}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural morphism from the identity functor to the double dual functor in left Presentations category.</p>

<p><a id="X7F92B4448041A68C" name="X7F92B4448041A68C"></a></p>

<h5>1.6-8 NaturalTransformationFromIdentityToDoubleDualRight</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalTransformationFromIdentityToDoubleDualRight</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a natural transformation <span class="SimpleMath">\(\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualRight}\)</span></p>

<p>The argument is a homalg ring <span class="SimpleMath">\(R\)</span>. The output is the natural morphism from the identity functor to the double dual functor in right Presentations category.</p>


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