<?xml version=
"1.0" encoding=
"UTF-8" ?>
<!
DOCTYPE html PUBLIC
"-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >
<
html xmlns=
"http://www.w3.org/1999/xhtml " xml:lang=
"en" >
<
head >
<
title >GAP (Modules) - Chapter
12 : Exterior Algebra and Koszul Complex</
title >
<
meta http-equiv=
"content-type" content=
"text/html; charset=UTF-8" />
<
meta name=
"generator" content=
"GAPDoc2HTML" />
<
link rel=
"stylesheet" type=
"text/css" href=
"manual.css" />
<
script src=
"manual.js" type=
"text/javascript" ></
script >
<
script type=
"text/javascript" >overwriteStyle();</
script >
</
head >
<
body class=
"chap12" onload=
"jscontent()" >
<
div class=
"chlinktop" ><
span class=
"chlink1" >Goto Chapter: </
span ><
a href=
"chap0.html" >Top</
a > <
a href="chap1.html" >1 </a > <a href="chap2.html" >2 </a > <a href="chap3.html" >3 </a > <a href="chap4.html" >4 </a > <a href="chap5.html" >5 </a > <a href="chap6.html" >6 </a > <a href="chap7.html" >7 </a > <a href="chap8.html" >8 </a > <a href="chap9.html" >9 </a > <a href="chap10.html" >10 </a > <a href="chap11.html" >11 </a > <a href="chap12.html" >12 </a > <a href="chap13.html" >13 </a > <a href="chapA.html" >A </a > <a href="chapB.html" >B </a > <a href="chapC.html" >C</a > <a href="chapBib.html" >Bib</a > <a href="chapInd.html" >Ind</a > </div >div class="chlinkprevnexttop" > <a href="chap0.html" >[Top of Book]</a > <a href="chap0.html#contents" >[Contents]</a > <a href="chap11.html" >[Previous Chapter]</a > <a href="chap13.html" >[Next Chapter]</a > </div >p id="mathjaxlink" class="pcenter" ><a href="chap12_mj.html" >[MathJax on]</a ></p >p ><a id="X7BD010F3847B274E" name="X7BD010F3847B274E" ></a ></p >div class="ChapSects" ><a href="chap12.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.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.html#X787BB7FF85F0AD68" >12 1 -1 ExteriorPower</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.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.html#X79C5FE077B58DF82" >12 2 -1 IsExteriorPower</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X87CF59278702A550" >12 2 -2 ExteriorPowerExponent</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X8282D0D7800F63CC" >12 2 -3 ExteriorPowerBaseModule</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.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.html#X7FC4A5DC7B592D04" >12 3 -1 IsExteriorPowerElement</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.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.html#X7C71C3C77F2E225D" >12 4 -1 Wedge</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X8236B4167E79F186" >12 4 -2 ExteriorPowerElementDual</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X85EDBA2783A1E984" >12 4 -3 SingleValueOfExteriorPowerElement</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.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.html#X7D84C7AC809B453F" >12 5 -1 KoszulCocomplex</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X794C601787143D2D" >12 5 -2 CayleyDeterminant</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X7C72190C8331FADD" >12 5 -3 Gcd_UsingCayleyDeterminant</a ></span >div ></div >div >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.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 >12 .1 <span class="Heading" >Exterior Algebra: Constructor</span ></h4>p ><a id="X787BB7FF85F0AD68" name="X787BB7FF85F0AD68" ></a ></p >12 .1 -1 ExteriorPower</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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 >12 .2 <span class="Heading" >Exterior Algebra: Properties and Attributes</span ></h4>p ><a id="X79C5FE077B58DF82" name="X79C5FE077B58DF82" ></a ></p >12 .2 -1 IsExteriorPower</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsExteriorPower</code >( <var class="Arg" >M</var > )</td ><td class="tdright" 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 >12 .2 -2 ExteriorPowerExponent</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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 >12 .2 -3 ExteriorPowerBaseModule</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; ExteriorPowerBaseModule</code >( <var class="Arg" >M</var > )</td ><td "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 >12 .3 <span class="Heading" >Exterior Algebra: Element Properties</span ></h4>p ><a id="X7FC4A5DC7B592D04" name="X7FC4A5DC7B592D04" ></a ></p >12 .3 -1 IsExteriorPowerElement</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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 >12 .4 <span class="Heading" >Exterior Algebra: Element Operations</span ></h4>p ><a id="X7C71C3C77F2E225D" name="X7C71C3C77F2E225D" ></a ></p >12 .4 -1 Wedge</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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" ><var class="Arg" >x</var > ∧ <var class="Arg" >y</var ></span >.</p >p ><a id="X8236B4167E79F186" name="X8236B4167E79F186" ></a ></p >12 .4 -2 ExteriorPowerElementDual</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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" var class="Arg" >x</var >*</span > in the (n-q )-th exterior power, as defined in <a href="chapBib.html#biBCQ11" a >.</p >p ><a id="X85EDBA2783A1E984" name="X85EDBA2783A1E984" ></a ></p >12 .4 -3 SingleValueOfExteriorPowerElement</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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" >[<var class="Arg" >x</var >]</span > as defined in <a href="chapBib.html#biBCQ11" >[CQ11]</a >.</p >p ><a id="X8050EFB77A600595" name="X8050EFB77A600595" ></a ></p >12 .5 <span class="Heading" >Koszul complex and Cayley determinant</span ></h4>p ><a id="X7D84C7AC809B453F" name="X7D84C7AC809B453F" ></a ></p >12 .5 -1 KoszulCocomplex</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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 >12 .5 -2 CayleyDeterminant</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; 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.html#biBCQ11" >[CQ11]</a >.</p >p ><a id="X7C72190C8331FADD" name="X7C72190C8331FADD" ></a ></p >12 .5 -3 Gcd_UsingCayleyDeterminant</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; Gcd_UsingCayleyDeterminant</code >( <var class="Arg" >x</var >, <var "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 >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a > <a href="chap0.html#contents" >[Contents]</a > <a href="chap11.html" >[Previous Chapter]</a > <a href="chap13.html" >[Next Chapter]</a > </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a > <a href="chap1.html" >1 </a > <a href="chap2.html" >2 </a > <a href="chap3.html" >3 </a > <a href="chap4.html" >4 </a > <a href="chap5.html" >5 </a > <a href="chap6.html" >6 </a > <a href="chap7.html" >7 </a > <a href="chap8.html" >8 </a > <a href="chap9.html" >9 </a > <a href="chap10.html" >10 </a > <a href="chap11.html" >11 </a > <a href="chap12.html" >12 </a > <a href="chap13.html" >13 </a > <a href="chapA.html" >A </a > <a href="chapB.html" >B </a > <a href="chapC.html" >C</a > <a href="chapBib.html" >Bib</a > <a href="chapInd.html" >Ind</a > </div >hr />p class="foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a ></p >body >html > Messung V0.5 in Prozent C=100 H=100 G=100
¤ Dauer der Verarbeitung: 0.16 Sekunden
(vorverarbeitet am 2026-09-04)
¤ *© Formatika GbR, Deutschland
