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< div class= "ChapSects">< a href= "chap3_mj.html#p>There is only one candidate, so we have found the permutation character.</p>
< div class= "ContSect">< span class= "tocline">< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7E3651DF87064E72"> 3. 1 < span class= "Heading">Objects: Category and Representations</ span></ a>
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< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7E610FA77A49B9EC"> 3. 1- 1 IsHomalgObject</ a></ span>
< span class= "ContSS">
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7F1BC3F77949E779"> 3. 1- 3 IsFinitelyPresentedObjectRep</ a></ span>
<class= "ContSS"> br/< span class= "nocss"> </ span>< a href= "chap3_mj.html#X79ED26577A1C2E09"> 3. 1- 4 IsStaticFinitelyPresentedObjectOrSubobjectRep</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7B645ADA876153F2"> 3. 1- 5 IsStaticFinitelyPresentedObjectRep</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> &java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 0
</ div></ div>
< div class= "ContSect">< span class= "tocline">< span class= "nocss"> </ span>< a href= "java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 26
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< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X810D3BFB7D9FE47E"> 3. 2- 1 Subobject</ a></ span>
</ div></ div>
< div class= "ContSect">< span class= "tocline">< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7B3E8C797D15F0B7"> 3. 3 < span class= "Heading">Objects: Properties</ span></ a>
</ span>
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< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7CD2A77778432E7B"> 3. 3- 1 IsFree</ a></ span>
<panclass=ContSS "<br/><pan ""> </span><a href="chap3_mj. html#X7D49FC85781256AB ">3.3-2 IsStablyFree</a></span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7EC041A77E7E46D2"> 3. 3- 3 IsProjective</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X84A8AB217E8F4611"> 3. 3- 4 IsProjectiveOfConstantRank</ a></ span>
<href "hap3_mj.html#". 3- 5 IsInjective< a< span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7FCE608683CCDC6B"> 3. 3- 6 IsInjectiveCogenerator</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X8784F151844F01FA"> 3. 3- 7 FiniteFreeResolutionExists</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7A6A34C283332F60"> 3. 3- 8 IsReflexive</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X86D92DA17DCE22DD"> 3. 3- 9 IsTorsionFree</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7D8F8A0B81EFD22A"> 3. 3- 10 IsArtinian</ a></ span>
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 12
< span class= "ContSS">< br />< span class= "nocss"> </ span>< a href= "chap3_mj.html#X7B894ED27D38E4B5"> 3. 3- 12 IsPure</ a></ span>
< span class= "ContSS">< br /><java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X83CBA38E81DC4A72">3.3-14 IsGorenstein</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><ap< = name="7>/>/p>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7A20E4597A707218">3.3-16 HasConstantRank</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7CD026F185A5E41E">3.3-17 ConstructedAsAnIdeal</a></span>
</div></div>
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 175
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7E6C8ED2865B6F35">3.4-1 TorsionSubobject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X82BCEE867CBE84E5">3.4-2 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mjFinite <a href"hapBib_mjhtmlbiBCCN85>CCN+,p 234]</a> <">(\</pan hasa class of maximal subgroups <span class="SimpleMath">\(H\)</span> of the type <span class="SimpleMath">\(2^{{10+16}}.O_{10}^+(2)\)</span>. Currently the character table of <span class="SimpleMath">\(H\)</span> and the class fusion into <span class="SimpleMath">\(M\)</span> are not available in <strong class="pkg">GAP</strong>. We are interested in the permutation character <span class="SimpleMath">\(1_H^M\)</span>, and we will compute it without this information.</p>
<span"ontSS">br>classnocss&; << "."34- />/span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X81679BB58541E235">3.4-5 ZeroSubobject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7C16CBCC78C56CDC">3.4-6 EmbeddingInSuperObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7ADC5B647C8E6D8C">3.4-7 SuperObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7A23EAD67E6B85C1">3.4-9 UnderlyingSubobject</a></span>
<span class="ContSS"><br<iv="java.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 26
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#span class="GAPprompt">gap></span> <span class="GAPinput">Hbar:= CharacterTable( "O10+(2)" );;</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X809A7C3882912EFD">3.4-12 EndomorphismRing</a></span>
<span class">br /< class="nocss"> </span><a href="chap3_mj.html#X85F3D7CF81E85423">3.4-13 UnitObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7E192147807E66DA">3.4-14 RankOfObject</a></span>
<span class="ContSS"><br /><span class="nocss"> S82)" java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X807BA3C583D3F1EB">3.span java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 91
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7E32A9FC81E0E101">3.4-17 Grade</a></span>
<span class="ContSS"><br /><span496
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8021C33D85444081">3.4-19 CodegreeOfPurity</a></span>
<span class="ContSS"><br /><span span class="GAPprompt">gap></spanspan =GAPinput>=( "O10() ;/span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7BC36CC67CB09858">3.4-21 AffineDimension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X82A1B55879AB1742">3.4-22 ProjectiveDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X791B772A7E368A88">3.4-23 ConstantTermOfHilbertPolynomialn</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [66, 5) out of bounds for length 152
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X81024DAF8695083E">3.4-25 ChernPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X79942F6187DF4434">3.4-26 ChernCharacter</a></span>
</div></div>
<div class="ContSect"><span527
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X87AEDF2985D65DCC">3.5-1 CurrentResolution</a></span>
<class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X81FACFAC828CA2F9">3.5-2 UnderlyingObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X82AE15AF82136AE0">3.5-3 Saturate</a></span>
</div></div>
</div>
<h3>3 <span class="Heading">Objects</span></h3>
<p><a id="X7E3651DF87064E72" name="X7E3651DF87064E72"></a></p>
<h4>3.1 <span class="Heading">Objects: Category and Representations</span></h4>
<p><a id="X7E610FA77A49B9EC" name="X7E610FA77A49B9EC"></a></p>
<h5>3.1-1 IsHomalgObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>This is the super <strong class="pkg">GAP</strong>-category which will include the <strong class="pkg">GAP</strong>-categories <code class="func">IsHomalgStaticObject</code> (<a href="chap3_mj.html#X79FC4A848517AF55"><span class="RefLink">3.1-2</span></a>), <code class="func">span>
<div class="example"><pre>
DeclareCategory( "IsHomalgObject",
IsHomalgObjectOrMorphism and
IsStructureObjectOrObject and
CharacterTable( "B" java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
</pre></div>
<p><a id="X79FC4A848517AF55" name="X79FC4A848517AF55"></a></p>
<h5>3.1-2 IsHomalgStaticObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgStaticObject</code>( <var class="Arg">F</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>This is the super <strong class="pkg">GAP</strong>-category which will include the <strong class="pkg">GAP</strong>-categories <code class="code">IsHomalgModule</code>, etc.</p>
<div class="example"><pre>
DeclareCategory( "IsHomalgStaticObject",
IsHomalgStaticObjectOrMorphism and
IsHomalgObject );
</pre></div>
<p><a id="X7F1BC3F77949E779" name="X7F1BC3F77949E779"></a></p>
<h5>3.1-3 IsFinitelyPresentedObjectRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsFinitelyPresentedObjectRep</code>( <var class="Arg">M</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>The <strong class="pkg">GAP</strong> representation of finitley presented <strong class="pkg">homalg</strong> objects.</p>
<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgObject</code> (<a href="chap3_mj.html#java.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 38
<div class="example"><pre>
span class="GAPprompt">gap></span> <span class="GAPinput">List( filt, t -> Size( t ) / order );</span>
IsHomalgObject and
IsStructureObjectOrFinitelyPresentedObjectRep,
[ ] );
</pre></div>
<p><a id="X79ED26577A1C2E09" name="X79ED26577A1C2E09"></a></p>
<h5>3.1-4 IsStaticFinitelyPresentedObjectOrSubobjectRep</h5>
<div class="func"><table class="func" width="100%"><tr><[ 1]
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>The <strong class="pkg">GAP</strong> representation of finitley presented <strong class="pkg">homalg</strong> static objects.</p>
<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgStaticObject</code> (<a href="chap3_mj.html#X79FC4A848517AF55"><span class="RefLink">3.1-2</span></a>).)</p>
<div class="example"><pre>
DeclareRepresentation( "IsStaticFinitelyPresentedObjectOrSubobjectRep",
IsHomalgStaticObject,
[ ] );
</pre></div>
<p><a id="X7B645ADA876153F2" name="X7B645ADA876153F2"></a></p>
<h5>3.1-5 IsStaticFinitelyPresentedObjectRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsStaticFinitelyPresentedObjectRep</code>( <var class="Arg">M</var> )</td><td class="tdright">( representation )</td></tr></table>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>The <p>Analogouslywe fixa code class=">2B</code> involution <span class="SimpleMath">\(_\</span> <pan java.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 1111
<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgStaticObject</code> (<a href="chap3_mj.html#X79FC4A848517AF55"><span class="RefLink">3.1-2</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representations <code class="code">IsStaticFinitelyPresentedObjectOrSubobjectRep</code> and <code class="code">IsFinitelyPresentedObjectRep</code>.)</p>
<div class="example"><pre>
DeclareRepresentation( "IsStaticFinitelyPresentedObjectRep",
java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 57
IsFinitelyPresentedObjectRep,
[ ] );
</pre></div>
<p><a id="X837C31E38502E580" name="X837C31E38502E580"></a></p>
< IsStaticFinitelyPresentedSubobjectRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsStaticFinitelyPresentedSubobjectRep</code>( <var class="Arg">M</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>The <strong class="pkg">GAP</strong> representation of finitley presented <strong class="pkg">homalg</strong> subobjects of static objects.</p>
<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgStaticObject</code> (<a href="chap3_mj.html#X79FC4A848517AF55"><span class="RefLink">3.1-2</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representations <code class="code">java.lang.StringIndexOutOfBoundsException: Index 313 out of bounds for length 11
<div class="example"><pre>
DeclareRepresentation( "IsStaticFinitelyPresentedSubobjectRep",
IsStaticFinitelyPresentedObjectOrSubobjectRep span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( maxes, t -> Size( t ) mod order = 0 );</span>
IsFinitelyPresentedObjectRep,
[ ] );
</pre></div>
<p><a id="X7BD901538362C36E" name="X7BD901538362C36E"></a java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 39
<h4>3.2 <span class="Heading">Objects: Constructors</span></h4>
<p><a id="X810D3BFB7D9FE47E" name="X810D3BFB7D9FE47E"></a></p>
<h5>3.2-1 Subobject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Subobject</code>( <var class="Arg">phi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>A synonym of <code class="func">ImageSubobject</code> (<a href="chap4_mj.html#X82FB6A4687E778D5"><span class="RefLink">4.4-7</span></a>).</p>
<p><a id="X7B3E8C797D15F0B7" name="X7B3E8C797D15F0B7"></a></p>
<h4>3CharacterTable("^1++8).8()" )
<p><a id="X7CD2A77778432E7B" name="X7CD2A77778432E7B"></a></p>
<h5>3.3-1 IsFree</h5>
< classjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 213
<>java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 81
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is free.</p>
<p><a id="X7D49FC85781256AB" name="X7D49FC85781256AB"></a></p>
<h5>3.3-2 IsStablyFree</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsStablyFree</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>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is stably free.</p>
<p><a id="X7EC041A77E7E46D2" name="X7EC041A77E7E46D2"></a></p>
<h5>3.3-3 IsProjective</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsProjective</code>( <var class="Arg">M<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is projective.</p>
<p><a id="X84A8AB217E8F4611" name="X84A8AB217E8F4611"></a></p>
<h5>3.3-4 IsProjectiveOfConstantRank</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsProjectiveOfConstantRank</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<<sclass"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 99
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is projective of constant rank.</p>
<< idX7F065FD7822C0A12"name="X7F065FD7822C0A12"></a></p>
<h5>3.3-5 IsInjective</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsInjective</code>( <var class="Arg">M</var> )</td><td class="tdright">(&java.lang.StringIndexOutOfBoundsException: Index 177 out of bounds for length 99
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalgCharacterTable( "2^1+24.Co1" )
<p><a id="X7FCE608683CCDC6B" name="X7FCE608683CCDC6B"></a></p>
<h5>3.3-6 IsInjectiveCogenerator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsInjectiveCogenerator</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<>eturns: <codeclass=code"true<code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is (marked) an injective cogenerator.</p>
<p><a id="X8784F151844F01FA" name="X8784F151844F01FA"></a></p>
<h5>3.3-7 FiniteFreeResolutionExists</h5>
<divspan class="GAPprompt">gap></span> <span class="GAPinput">pi_B:= RestrictedClassFunction( TrivialCharacter( u2 )^co1, mm )^m;;</span>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if <pre>/>
<p><a id="X7A6A34C283332F60" name="X7A6A34C283332F60"></a></p>
<h5>3.3-8 IsReflexive</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsReflexive</code>( <var class="Arg">M</var> )</td><td java.lang.StringIndexOutOfBoundsException: Index 161 out of bounds for length 0
<p>Returns: <code class="code">true</code> or <M\/pan>is ./p>
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is reflexive.</p>
<p><a id="X86D92DA17DCE22DD" name="X86D92DA17DCE22DD"></a></p>
<h5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsTorsionFree</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>
<<dv class=>>
<p><a id="X7D8F8A0B81EFD22A" name="X7D8F8A0B81EFD22A"></a></p>
<h5>3.3-10 IsArtinian</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsArtinianspan class"GAPprompt">gap></span> <span class="GAPinput">torso:= [ Size( m ) / Horder ];</span>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is artinian.</p[512372707698741056749515292734375 ] ]
<p><a id="X80C6B26284721409" name="X80C6B26284721409"></a></p>
<h5>3.3-11 IsTorsion</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsTorsion</code>( <var java.lang.StringIndexOutOfBoundsException: Range [1, 129) out of bounds for length 12
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
<p><a id="X7B894ED27D38E4B5" name="X7B894ED27D38E4B5"></a></p>
<h5>3.3-12 IsPure</h5>
<div class="func"><p>Nextwecompute some zero values of <span class="SimpleMath">\(1_H^M\)</span>, using the following conditions.</p>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is pure.</p>
<p><a id="X8373421F7E085763" name="X8373421F7E085763"></a></p>
<h5>3.3-13 IsCohenMacaulay</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsCohenMacaulay</code>( <var class="Arg">M</var> )</td><td class="tdright">( property&ul>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is Cohen-Macaulay (depends on the specific Abelian category).</p>
<p><a id="X83CBA38E81DC4A72" name="X83CBA38E81DC4A72"></a></p>
<h5>3.3-14 IsGorenstein</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGorenstein</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr>/li>
<p:<java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 81
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is Gorenstein (depends on the specific Abelian category).</p>
<p><a id="X7E7AEFBE7801F196" name="X7E7AEFBE7801F196"></a></p>
<h5>3.3-15 IsKoszul</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsKoszul</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code><ul>
<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is Koszul (depends on the specific Abelian category).</p>
<p><a id="X7A20E4597A707218" name="X7A20E4597A707218"></a></p>
<h5>3.3-16 HasConstantRank<div class=""pre>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HasConstantRank</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>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> has constant rank. <br /> (no method installed)</p>
<p><a id="X7CD026F185A5E41E" name="X7CD026F185A5E41E"></a></p>
<h5>3.3-17 ConstructedAsAnIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ConstructedAsAnIdeal</code>( <var class="Arg">J</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>Check if the <strong class="pkg">homalg</strong> subobject <var class="Arg">J</var> was[1,2,4,8,16,32]
<p><a id="X805B06828294072C" name="X805B06828294072C"></a></p>
<h4>3.4 <span class="Heading">Objects: Attributes</span></h4>
<p><a id="X7E6C8ED2865B6F35" name="X7E6C8ED2865B6F35"></a></p>
<h5>3.4-1 TorsionSubobject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TorsionSubobject</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>This constructor returns the finitely generated torsion subobject of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>.</p>
<p><a id="X82BCEE867CBE84E5 3 ,7 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
<>3- TheMorphismToZero>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TheMorphismToZero</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> map</p>
<p>The zero morphism span class="GAPprompt">ggt;</span> <span class="GAPinput">primes:= Union( primes_A, primes_B );;</span>
<p><a id="X85EFEC127CA408A1" name="X85EFEC127CA408A1"></a></p>
<h5>3.4-3 TheIdentityMorphism</h5>
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 227
<p>Returns: a <strong class="pkg">homalg</strong> map</p>
<p>The identity automorphism of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>.</p>
<p><a id="X8236B1D480ED04CD" name="X8236B1D480ED04CD"></a></p>
<h5>3.4-4 FullSubobject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FullSubobject</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>The <strong class="pkg">homalg</strong> object <var class="Arg">M</var> as a subobject of itself.</p>
<p><a id="X81679BB58541E235" name="X81679BB58541E235"></a></p>
<h5>3.4-5 ZeroSubobject</h5>
<div class="func"><table class="func" width="100%"><tr><td <span class="GAPprompt">></span> <span class="GAPinput"> i]= 0;/pan>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>The zero subobject of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>.</p>
<p><a id="X7C16CBCC78C56CDC" name="X7C16CBCC78C56CDC"></a></p>
<h5>3.4-6 EmbeddingInSuperObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EmbeddingInSuperObject</code>( <var class="Arg">N</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> map</p>
<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 215
<p><a id="X7ADC5B647C8E6D8C" name="X7ADC5B647C8E6D8C"></a></p>
<h5>3.4-7 SuperObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SuperObject</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<Returns a strong "pkg"/>object<p>
<p>In case <var class="Arg">M</var> was defined as a subobject of some object <span class="SimpleMath">\(L\)</span> the super object <span class="SimpleMath">\(L\)</span> is returned.</p>
<p><a id="X7FB9A7C3785D92DC" name="X7FB9A7C3785D92DC"></a></p<pan ="GAPprompt">gt;<span> <span class="GAPinput"> if morders[i] mod p = 0 then</span>
<h5>3.4-8 FactorObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FactorObject</code>( <var class="Arg">N</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> object</p>
<p>In case <var class="Arg">spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 124
<p><a id="X7A23EAD67E6B85C1" name="X7A23EAD67E6B85C1"></a></p>
<h5>3.4-9 UnderlyingSubobject<span="APprompt"&<span><class=GAPinput" =PowerMap(m,pprime )i]/>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; UnderlyingSubobject</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>In case <varspan class="GAPprompt">></span> <span class="GAPinput"> torso[i]:= 0;</span>
<p><a id="X7FC5F0AF7CF5DC67" name="X7FC5F0AF7CF5DC67"></a></p>
<h5>3.4-10 NatTrIdToHomHom_R</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NatTrIdToHomHom_R</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<> < "kg"homalg<strong> java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 62
<p>The natural evaluation morphism from the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> to <pan class="APprompt">></span> <span class="GAPinput"> fi;</span>
<p><a id="X81889C777A22A5D3" name="X81889C777A22A5D3"></a></p>
<h5>3.4-11 Annihilator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Annihilator</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> subobject</p>
<p>The annihilator of the object <var class="Arg">M</var> as a subobject of the structure object.</p>
<p><a id="X809A7C3882912EFD" name="X809A7C3882912EFD"></a></p>
<h5>3.4-12 EndomorphismRing</h5>
<div class="func"><table<pan ="APprompt">></span> <span class="GAPinput"> fi;</span>
<p>Returns: a <strong class="pkg">homalg</strong> object</p>
<p>The endomorphism ring of the object <var class="Arg">M</var>.</p>
<p><a id="X85F3D7CF81E85423" name="X85F3D7CF81E85423"></a></p>
<h5>3.4-13 UnitObject</h5>
<classfunc"<" width=100%><tr>td class="tdleft"><code class="func">&#>8227; UnitObject</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a Chern character</p>
<p><var class="Arg">M</var> is a <strong class="pkg">homalg</strong> object.</p>
<p><a id="X7E192147807E66DA" name="X7E192147807E66DA"></a></p>
<h5>3.4-14 RankOfObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RankOfObject</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a nonnegative integer</p>
<p>The projective rank of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>.</p>
<p><a id="X84FDF25D797B874B" name="X84FDF25D797B874B"></a></p>
<h5>3.4-15 ProjectiveDimension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveDimension</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a nonnegative integer</p>
<p>The projective dimension of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>.</p>
<p><a id="X807BA3C583D3F1EB" name="X807BA3C583D3F1EB"></a></p>
<h5>3.4-16 DegreeOfTorsionFreeness</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DegreeOfTorsionFreeness</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a nonnegative integer of infinity</p>
<p>Auslander's degree of torsion-freeness of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>. It is set to infinity only for <var class="Arg">M</var><span class="SimpleMath">\(=0\)</span>.</p>
<java.lang.StringIndexOutOfBoundsException: Range [12, 2) out of bounds for length 62
<h5>3.4-17 Grade</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Grade</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<pReturns: a nonnegative integer of infinity</p>
<p>The grade of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>. It is set to infinity if <var class="Arg">M</var><span class="SimpleMath">\(=0\)</span>. Another name for this operation is <code class="code">Depth</code>.</p>
<p><a id="X816186E587563E3F" name="X816186E587563E3F"></a></p>
<h5>3.4-18 PurityFiltration</h5>
<"< class=func"width="100%"><tr><td class="tdleft"><code class="func">‣ PurityFiltration</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> filtration</p>
<p>The purity filtration of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>.</p>
<p><a id="X8021C33D85444081" name="X8021C33D85444081"></a></p>
<h5>3.4-19 CodegreeOfPurity</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CodegreeOfPurity</code>( <<l
<p>Returns: a list of nonnegative integers</p>
<>Thecodegree of purity of the <strong class="pkg">homalg</strong> object <var class="Arg">M</var>.</p>
<p><a id="X84299BAB807A1E13" name="X84299BAB807A1E13"></a></p>
<h5>3.4-20 HilbertPolynomial</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HilbertPolynomial</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a univariate polynomial with rational coefficients</p>
<p><var class="Arg">M</var> is a <strong class="pkg">homalg</strong> object.</p>
<p><a id="X7BC36CC67CB09858" name="X7BC36CC67CB09858"></a></p>
<h5>3.4-21 AffineDimension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Range [307, 85) out of bounds for length 570
<p>Returns: a nonnegative integer</p>
<p><var class="Arg">M</var> is a <strong class="pkg">homalg</strong> object.</p>
<p><a id="X82A1B55879AB1742" name="X82A1B55879AB1742"></a></p>
<h5>3.4-22 ProjectiveDegree</h5>
<div class="func"><table class/ul>
<p>Returns: a nonnegative integer</p>
<p><var class="Arg">M</var> is a <strong class="pkg">homalg</strong> object.</p>
<p><a id="X791B772A7E368A88" name="X791B772A7E368A88"></a></p>
<h5>3.4-23 ConstantTermOfHilbertPolynomialn</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ConstantTermOfHilbertPolynomialn</code>( <var class="Argspan ="APprompt"gap><span spanclass="GAPinput"=[[pi_A,[pi_B] ;<span>
<p>Returns: an integer</p>
<p><var class="Arg">M</var> is a <strong class="pkg">homalg</strong> object.</p>
<p><a id="X7FC735717985B092" name="X7FC735717985B092"></a></p>
<h5>3.4-24 ElementOfGrothendieckGroup</h5>
<div class="func"><tableclass="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ElementOfGrothendieckGroup</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: an element of the Grothendieck group of a projective space</p>
<p><var class="Arg">M</var> is a <strong class="pkg">homalg</strong> object.</p>
<p><a id="X81024DAF8695083E" name="X81024DAF8695083E"></a></p>
<h5>3.4-25 ChernPolynomial</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ChernPolynomial</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a Chern polynomial with rank</p>
<p><var class="Arg"span class=GAPprompt">/span span class="GAPinput" psis82:= TrivialCharacter(CharacterTable( subname ) )^U_Abar;</span>
<p><a id="X79942F6187DF4434" name="X79942F6187DF4434"></a></p>
<h5>3.4-26 ChernCharacter</h5>
<divclass=""<table ""width""<tr<dclass""<code">#; /java.lang.StringIndexOutOfBoundsException: Range [122, 121) out of bounds for length 221
<p>Returns: a Chern character</p>
<><arclass=Arg><var is <strong class=pkg><strong>object./>
<p><a id="X7B4D450B78A86F8B" name="X7B4D450B78A86F8B"></a></p>
<h4>3.5 <span class="Heading">Objects: Operations and Functions</span></h4>
<p><a id="X87AEDF2985D65DCC" name="X87AEDF2985D65DCC"></a></p>
<h5>3.5-1 CurrentResolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CurrentResolution</code>( <var class=span class="GAPprompt">gap></span> <span class="GAPinput">tt1:= CharacterTable( "O8+(2)" );</span>
<p>Returns: a <strong class="pkg">homalg</strong> complex</p>
<p>The computed (part of a)CharacterTable( "8+2")
<p><a id="X81FACFAC828CA2F9" name="X81FACFAC828CA2F9"></a></p>
<h5>3.5-2 UnderlyingObject</h5>
<div "unc"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#e='color: green'>8227; UnderlyingObject</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<Returns:a <strong class="pkg">homalg</strong> object</p>
<p>In case <var class="Arg">M</var> was defined as a subobject of some object <span class="SimpleMath">\(L\)</span> the object underlying the subobject <span class="SimpleMath">\(M\)</span> is returned.</p>
<p><a id="X82AE15AF82136AE0" name="X82AE15AF82136AE0"></a></p>
<h5>3.5-3 Saturate</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Saturate</code>( <var class="Arg<span class="GAPprompt">></span> <span class="GAPinput"> psi:= TrivialCharacter( CharacterTable( subname ) )^tt1;</span>
<p>Returns: a <strong<span class="GAPprompt">></span> <span class="GAPinput"> list:= [];</span>
<p>Compute the saturation ideal <span class="SimpleMath">\(\textit{K}:\textit{J}^\infty\)</span> of the ideals <var class="Arg">K</var> and <var class="Arg">J</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( zz );</span><span class="GAPprompt">></span> <span class="GAPinput"> AddSet( list, Restricted( Restricted( char, u2 ) ^ co1, mm ) ^ m );</span>
<An internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := LeftSubmodule( "2", zz );</span>
<A principal (left) ideal given by a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
[[ 2 ]]
A (left) ideal generated by the entry of the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">J := LeftSubmodule( "3", zz );</span>
<A principal (left) ideal given by a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( J );</span>
[ [ 3 ] ]
A (left) ideal generated by the entry of the above matrix
<span class=<span class="GAPprompt">></span> <span class="GAPinput">end;;</span>
<A principal (left) ideal given by a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I );</span>
[ [ 24 ] ]
A (left) ideal generated by the entry of the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">Im := SubobjectQuotient(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<A principal (left) ideal of rank 1 on a free generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Im );</span>
[ [ 12 ] ]
A (left) ideal generated by the entry of the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">I_m := Saturate( I,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Aprincipal(eft)idealof 1 a >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I_m );</span>
[ [ 3 ] ]
A (left) ideal generated by the entry of the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">I_m = J;</span>
true
</pre></div>
<div class="example"><pre>
InstallMethod( Saturate,
"<span class="GAPprompt">gap></span> <span class="GAPinput">AddSubgroupOfO8p2( "S6(2)" );</span>
[ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedSubobjectRep ],
function( K, J )
local quotient_last, quotient;
quotient_last := SubobjectQuotient( K, J );
quotient := SubobjectQuotient( quotient_last, J );
java.lang.StringIndexOutOfBoundsException: Range [22, 9) out of bounds for length 52
quotient_last := quotient;
quotient := SubobjectQuotient( quotient_last, J );
od;
return quotient_last;
end );
InstallMethod( \-, ## a geometrically motivated definition
" homalg ofstatic objects",
[ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedSubobjectRep ],
function( K, J )
return Saturate( K, J );
end)
</pre></div>
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