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<p id="mathjaxlink" class="pcenter"><a href="chap3_mj.html">[MathJax on]</a></p>
<p><a id="X78497E777FB3E402" name="X78497E777FB3E402"></a></p>
<div class="ChapSects"><a href="chap3.html#X78497E777FB3E402">3 <span class="Heading">Objects</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E3651DF87064E72">3.1 <span class="Heading">Objects: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E610FA77A49B9EC">3.1-1 IsHomalgObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79FC4A848517AF55">3.1-2 IsHomalgStaticObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F1BC3F77949E779">3.1-3 IsFinitelyPresentedObjectRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79ED26577A1C2E09">3.1-4 IsStaticFinitelyPresentedObjectOrSubobjectRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B645ADA876153F2">3.1-5 IsStaticFinitelyPresentedObjectRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X837C31E38502E580">3.1-6 IsStaticFinitelyPresentedSubobjectRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7BD901538362C36E">3.2 <span class="Heading">Objects: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X810D3BFB7D9FE47E">3.2-1 Subobject</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B3E8C797D15F0B7">3.3 <span class="Heading">Objects: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CD2A77778432E7B">3.3-1 IsFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D49FC85781256AB">3.3-2 IsStablyFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7EC041A77E7E46D2">3.3-3 IsProjective</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84A8AB217E8F4611">3.3-4 IsProjectiveOfConstantRank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F065FD7822C0A12">3.3-5 IsInjective</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FCE608683CCDC6B">3.3-6 IsInjectiveCogenerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8784F151844F01FA">3.3-7 FiniteFreeResolutionExists</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A6A34C283332F60">3.3-8 IsReflexive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86D92DA17DCE22DD">3.3-9 IsTorsionFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D8F8A0B81EFD22A">3.3-10 IsArtinian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80C6B26284721409">3.3-11 IsTorsion</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B894ED27D38E4B5">3.3-12 IsPure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8373421F7E085763">3.3-13 IsCohenMacaulay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83CBA38E81DC4A72">3.3-14 IsGorenstein</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E7AEFBE7801F196">3.3-15 IsKoszul</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A20E4597A707218">3.3-16 HasConstantRank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CD026F185A5E41E">3.3-17 ConstructedAsAnIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X805B06828294072C">3.4 <span class="Heading">Objects: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E6C8ED2865B6F35">3.4-1 TorsionSubobject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82BCEE867CBE84E5">3.4-2 TheMorphismToZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85EFEC127CA408A1">3.4-3 TheIdentityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8236B1D480ED04CD">3.4-4 FullSubobject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81679BB58541E235">3.4-5 ZeroSubobject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C16CBCC78C56CDC">3.4-6 EmbeddingInSuperObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7ADC5B647C8E6D8C">3.4-7 SuperObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FB9A7C3785D92DC">3.4-8 FactorObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A23EAD67E6B85C1">3.4-9 UnderlyingSubobject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FC5F0AF7CF5DC67">3.4-10 NatTrIdToHomHom_R</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81889C777A22A5D3">3.4-11 Annihilator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X809A7C3882912EFD">3.4-12 EndomorphismRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85F3D7CF81E85423">3.4-13 UnitObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E192147807E66DA">3.4-14 RankOfObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84FDF25D797B874B">3.4-15 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X807BA3C583D3F1EB">3.4-16 DegreeOfTorsionFreeness</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E32A9FC81E0E101">3.4-17 Grade</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X816186E587563E3F">3.4-18 PurityFiltration</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8021C33D85444081">3.4-19 CodegreeOfPurity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84299BAB807A1E13">3.4-20 HilbertPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BC36CC67CB09858">3.4-21 AffineDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82A1B55879AB1742">3.4-22 ProjectiveDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X791B772A7E368A88">3.4-23 ConstantTermOfHilbertPolynomialn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FC735717985B092">3.4-24 ElementOfGrothendieckGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81024DAF8695083E">3.4-25 ChernPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79942F6187DF4434">3.4-26 ChernCharacter</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B4D450B78A86F8B">3.5 <span class="Heading">Objects: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87AEDF2985D65DCC">3.5-1 CurrentResolution</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81FACFAC828CA2F9">3.5-2 UnderlyingObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.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">‣ IsHomalgObject</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="func">IsHomalgStaticObject</code> (<a href="chap3.html#X79FC4A848517AF55"><span class="RefLink">3.1-2</span></a>), <code class="func">IsHomalgComplex</code> (<a href="chap6.html#X8166F9FD7BFDA207"><span class="RefLink">6.1-1</span></a>), <code class="func">IsHomalgBicomplex</code> (<a href="chap8.html#X80B7C45A850F4C3E"><span class="RefLink">8.1-1</span></a>), <code class="func">IsHomalgBigradedObject</code> (<a href="chap9.html#X795C082E83748032"><span class="RefLink">9.1-1</span></a>), and <code class="func">IsHomalgSpectralSequence</code> (<a href="chap10.html#X795DCCD88630BA47"><span class="RefLink">10.1-1</span></a>). We need this <strong class="pkg">GAP</strong>-category to be able to build complexes with *objects* being objects of <strong class="pkg">homalg</strong> categories or again complexes.</p>


<div class="example"><pre>
DeclareCategory( "IsHomalgObject",
        IsHomalgObjectOrMorphism and
        IsStructureObjectOrObject and
        IsAdditiveElementWithZero );
</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">‣ 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">‣ 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.html#X7E610FA77A49B9EC"><span class="RefLink">3.1-1</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representations <code class="code">IsStructureObjectOrFinitelyPresentedObjectRep</code>.)</p>


<div class="example"><pre>
DeclareRepresentation( "IsFinitelyPresentedObjectRep",
        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><td class="tdleft"><code class="func">‣ IsStaticFinitelyPresentedObjectOrSubobjectRep</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> static objects.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgStaticObject</code> (<a href="chap3.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">‣ IsStaticFinitelyPresentedObjectRep</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> static objects.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgStaticObject</code> (<a href="chap3.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",
        IsStaticFinitelyPresentedObjectOrSubobjectRep and
        IsFinitelyPresentedObjectRep,
        [ ] );
</pre></div>

<p><a id="X837C31E38502E580" name="X837C31E38502E580"></a></p>

<h5>3.1-6 IsStaticFinitelyPresentedSubobjectRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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.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( "IsStaticFinitelyPresentedSubobjectRep",
        IsStaticFinitelyPresentedObjectOrSubobjectRep and
        IsFinitelyPresentedObjectRep,
        [ ] );
</pre></div>

<p><a id="X7BD901538362C36E" name="X7BD901538362C36E"></a></p>

<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">‣ 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.html#X82FB6A4687E778D5"><span class="RefLink">4.4-7</span></a>).</p>

<p><a id="X7B3E8C797D15F0B7" name="X7B3E8C797D15F0B7"></a></p>

<h4>3.3 <span class="Heading">Objects: Properties</span></h4>

<p><a id="X7CD2A77778432E7B" name="X7CD2A77778432E7B"></a></p>

<h5>3.3-1 IsFree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFree</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 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">‣ 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">‣ IsProjective</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 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">‣ IsProjectiveOfConstantRank</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 projective of constant rank.</p>

<p><a id="X7F065FD7822C0A12" 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">‣ IsInjective</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 (marked) injective.</p>

<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">‣ IsInjectiveCogenerator</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 (marked) an injective cogenerator.</p>

<p><a id="X8784F151844F01FA" name="X8784F151844F01FA"></a></p>

<h5>3.3-7 FiniteFreeResolutionExists</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FiniteFreeResolutionExists</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> allows a finite free resolution. <br /> (no method installed)</p>

<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">‣ IsReflexive</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 reflexive.</p>

<p><a id="X86D92DA17DCE22DD" name="X86D92DA17DCE22DD"></a></p>

<h5>3.3-9 IsTorsionFree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>

<p>Check if the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> is torsion-free.</p>

<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">‣ IsArtinian</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 artinian.</p>

<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">‣ IsTorsion</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 torsion.</p>

<p><a id="X7B894ED27D38E4B5" name="X7B894ED27D38E4B5"></a></p>

<h5>3.3-12 IsPure</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPure</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 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">‣ IsCohenMacaulay</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 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">‣ IsGorenstein</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 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">‣ IsKoszul</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 Koszul (depends on the specific Abelian category).</p>

<p><a id="X7A20E4597A707218" name="X7A20E4597A707218"></a></p>

<h5>3.3-16 HasConstantRank</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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 constructed as an ideal. <br /> (no method installed)</p>

<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">‣ 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" name="X82BCEE867CBE84E5"></a></p>

<h5>3.4-2 TheMorphismToZero</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 from the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> to zero.</p>

<p><a id="X85EFEC127CA408A1" name="X85EFEC127CA408A1"></a></p>

<h5>3.4-3 TheIdentityMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TheIdentityMorphism</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 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">‣ 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 class="tdleft"><code class="func">‣ ZeroSubobject</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 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">‣ 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>

<p>In case <var class="Arg">N</var> was defined as a subobject of some object <span class="SimpleMath">L</span> the embedding of <var class="Arg">N</var> in <span class="SimpleMath">L</span> is returned.</p>

<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">‣ SuperObject</code>( <var class="Arg">M</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">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>

<h5>3.4-8 FactorObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">N</var> was defined as a subobject of some object <span class="SimpleMath">L</span> the factor object <span class="SimpleMath">L/</span><var class="Arg">N</var> is returned.</p>

<p><a id="X7A23EAD67E6B85C1" name="X7A23EAD67E6B85C1"></a></p>

<h5>3.4-9 UnderlyingSubobject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 <var class="Arg">M</var> was defined as the object underlying a subobject <span class="SimpleMath">L</span> then <span class="SimpleMath">L</span> is returned. <br /> (no method installed)</p>

<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">‣ NatTrIdToHomHom_R</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> morphism</p>

<p>The natural evaluation morphism from the <strong class="pkg">homalg</strong> object <var class="Arg">M</var> to its double dual <code class="code">HomHom</code><span class="SimpleMath">(</span><var class="Arg">M</var><span class="SimpleMath">)</span>.</p>

<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">‣ 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 class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EndomorphismRing</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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">‣ 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">‣ 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 homalg object M. It is set to infinity only for M=0.



<p><a id="X7E32A9FC81E0E101" name="X7E32A9FC81E0E101"></a></p>

<h5>3.4-17 Grade</h5>

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

<div class="func"><table 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">‣ CodegreeOfPurity</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of nonnegative integers</p>

<p>The codegree 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">‣ 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 class="func">‣ AffineDimension</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveDegree</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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">‣ ConstantTermOfHilbertPolynomialn</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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">M</var> is a <strong class="pkg">homalg</strong> object.</p>

<p><a id="X79942F6187DF4434" name="X79942F6187DF4434"></a></p>

<h5>3.4-26 ChernCharacter</h5>

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

<p>The computed (part of a) resolution of the static object <var class="Arg">M</var>.</p>

<p><a id="X81FACFAC828CA2F9" name="X81FACFAC828CA2F9"></a></p>

<h5>3.5-2 UnderlyingObject</h5>

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

<p>Compute the saturation ideal <span class="SimpleMath"><var class="Arg">K</var>:<var class="Arg">J</var>^∞</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>
<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="GAPprompt">gap></span> <span class="GAPinput">I := Intersect( J, m^3 );</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( I, m );</span>
<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, m );</span>
<A principal (left) ideal of rank 1 on a free generator>
<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,
        "for homalg subobjects of static objects",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedSubobjectRep ],
        
  function( K, J )
    local quotient_last, quotient;
    
    quotient_last := SubobjectQuotient( K, J );
    
    quotient := SubobjectQuotient( quotient_last, J );
    
    while not IsSubset( quotient_last, quotient ) do
        quotient_last := quotient;
        quotient := SubobjectQuotient( quotient_last, J );
    od;
    
    return quotient_last;
    
end );


InstallMethod( \-, ## a geometrically motivated definition
        "for homalg subobjects of static objects",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedSubobjectRep ],
        
  function( K, J )
    
    return Saturate( K, J );
    
end );
</pre></div>


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