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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chapA_mj.html">A</a>  <a href="chapB_mj.html">B</a>  <a href="chapC_mj.html">C</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap11_mj.html">[Previous Chapter]</a>    <a href="chap13_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap12.html">[MathJax off]</a></p>
<p><a id="X7BD010F3847B274E" name="X7BD010F3847B274E"></a></p>
<div class="ChapSects"><a href="chap12_mj.html#X7BD010F3847B274E">12 <span class="Heading">Exterior Algebra and Koszul Complex</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7A005D4E870C281D">12.1 <span class="Heading">Exterior Algebra: Constructor</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X787BB7FF85F0AD68">12.1-1 ExteriorPower</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7E09B9C5844FC31E">12.2 <span class="Heading">Exterior Algebra: Properties and Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X79C5FE077B58DF82">12.2-1 IsExteriorPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X87CF59278702A550">12.2-2 ExteriorPowerExponent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X8282D0D7800F63CC">12.2-3 ExteriorPowerBaseModule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7A2AC54B87C85695">12.3 <span class="Heading">Exterior Algebra: Element Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7FC4A5DC7B592D04">12.3-1 IsExteriorPowerElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X80D7B36379182854">12.4 <span class="Heading">Exterior Algebra: Element Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7C71C3C77F2E225D">12.4-1 Wedge</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X8236B4167E79F186">12.4-2 ExteriorPowerElementDual</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X85EDBA2783A1E984">12.4-3 SingleValueOfExteriorPowerElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X8050EFB77A600595">12.5 <span class="Heading">Koszul complex and Cayley determinant</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7D84C7AC809B453F">12.5-1 KoszulCocomplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X794C601787143D2D">12.5-2 CayleyDeterminant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7C72190C8331FADD">12.5-3 Gcd_UsingCayleyDeterminant</a></span>
</div></div>
</div>

<h3>12 <span class="Heading">Exterior Algebra and Koszul Complex</span></h3>

<p>What follows are several operations related to the exterior algebra of a free module:</p>


<ul>
<li><p>A constructor for the graded parts of the exterior algebra (<q>exterior powers</q>)</p>

</li>
<li><p>Several Operations on elements of these exterior powers</p>

</li>
<li><p>A constructor for the <q>Koszul complex</q></p>

</li>
<li><p>An implementation of the <q>Cayley determinant</q> as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>, which allows calculating greatest common divisors from finite free resolutions.</p>

</li>
</ul>
<p><a id="X7A005D4E870C281D" name="X7A005D4E870C281D"></a></p>

<h4>12.1 <span class="Heading">Exterior Algebra: Constructor</span></h4>

<p><a id="X787BB7FF85F0AD68" name="X787BB7FF85F0AD68"></a></p>

<h5>12.1-1 ExteriorPower</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorPower</code>( <var class="Arg">k</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> module</p>

<p>Construct the <var class="Arg">k</var>-th exterior power of module <var class="Arg">M</var>.</p>

<p><a id="X7E09B9C5844FC31E" name="X7E09B9C5844FC31E"></a></p>

<h4>12.2 <span class="Heading">Exterior Algebra: Properties and Attributes</span></h4>

<p><a id="X79C5FE077B58DF82" name="X79C5FE077B58DF82"></a></p>

<h5>12.2-1 IsExteriorPower</h5>

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

<p>Marks a module as an exterior power of another module.</p>

<p><a id="X87CF59278702A550" name="X87CF59278702A550"></a></p>

<h5>12.2-2 ExteriorPowerExponent</h5>

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

<p>The exponent of the exterior power.</p>

<p><a id="X8282D0D7800F63CC" name="X8282D0D7800F63CC"></a></p>

<h5>12.2-3 ExteriorPowerBaseModule</h5>

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

<p>The module that <var class="Arg">M</var> is an exterior power of.</p>

<p><a id="X7A2AC54B87C85695" name="X7A2AC54B87C85695"></a></p>

<h4>12.3 <span class="Heading">Exterior Algebra: Element Properties</span></h4>

<p><a id="X7FC4A5DC7B592D04" name="X7FC4A5DC7B592D04"></a></p>

<h5>12.3-1 IsExteriorPowerElement</h5>

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

<p>Checks if the element <var class="Arg">x</var> is from an exterior power.</p>

<p><a id="X80D7B36379182854" name="X80D7B36379182854"></a></p>

<h4>12.4 <span class="Heading">Exterior Algebra: Element Operations</span></h4>

<p><a id="X7C71C3C77F2E225D" name="X7C71C3C77F2E225D"></a></p>

<h5>12.4-1 Wedge</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Wedge</code>( <var class="Arg">x</var>, <var class="Arg">y</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an element of an exterior power</p>

<p>Calculate <span class="SimpleMath">\(\textit{x} \wedge \textit{y}\)</span>.</p>

<p><a id="X8236B4167E79F186" name="X8236B4167E79F186"></a></p>

<h5>12.4-2 ExteriorPowerElementDual</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorPowerElementDual</code>( <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an element of an exterior power</p>

<p>For <var class="Arg">x</var> in a q-th exterior power of a free module of rank n, return <span class="SimpleMath">\(\textit{x}*\)</span> in the (n-q)-th exterior power, as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>.</p>

<p><a id="X85EDBA2783A1E984" name="X85EDBA2783A1E984"></a></p>

<h5>12.4-3 SingleValueOfExteriorPowerElement</h5>

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

<p>For <var class="Arg">x</var> in a highest exterior power, returns its single coordinate in the canonical basis; i.e. <span class="SimpleMath">\([\textit{x}]\)</span> as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>.</p>

<p><a id="X8050EFB77A600595" name="X8050EFB77A600595"></a></p>

<h4>12.5 <span class="Heading">Koszul complex and Cayley determinant</span></h4>

<p><a id="X7D84C7AC809B453F" name="X7D84C7AC809B453F"></a></p>

<h5>12.5-1 KoszulCocomplex</h5>

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

<p>Calculate the <var class="Arg">E</var>-valued Koszul complex of <var class="Arg">a</var>.</p>

<p><a id="X794C601787143D2D" name="X794C601787143D2D"></a></p>

<h5>12.5-2 CayleyDeterminant</h5>

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

<p>Calculate the Cayley determinant of the complex <var class="Arg">C</var>, as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>.</p>

<p><a id="X7C72190C8331FADD" name="X7C72190C8331FADD"></a></p>

<h5>12.5-3 Gcd_UsingCayleyDeterminant</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gcd_UsingCayleyDeterminant</code>( <var class="Arg">x</var>, <var class="Arg">y</var>[, <var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a ring element</p>

<p>Returns the greatest common divisor of the given ring elements, calculated using the Cayley determinant.</p>


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