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<p><a id="X782EF48B7D997E9E" name="X782EF48B7D997E9E"></a></p>
<div class="ChapSects"><a href="chap7_mj.html#X782EF48B7D997E9E">7 <span class="Heading">Chain Morphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X8703B8017F55336F">7.1 <span class="Heading">ChainMorphisms: Categories and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7CB62E188027B7C5">7.1-1 IsHomalgChainMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X853BD37084BFC602">7.1-2 IsHomalgChainEndomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7C35D69F7B09BD47">7.1-3 IsChainMorphismOfFinitelyPresentedObjectsRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7DF3EA1D817266C1">7.1-4 IsCochainMorphismOfFinitelyPresentedObjectsRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X83637FBE86C5DDF1">7.2 <span class="Heading">Chain Morphisms: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X853361547FB213CA">7.2-1 HomalgChainMorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X789E2EC07C041D78">7.3 <span class="Heading">Chain Morphisms: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X798B6A897FE4FF12">7.3-1 IsMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8194427F8423EB00">7.3-2 IsGeneralizedMorphismWithFullDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X84FE6CFD85AB7B73">7.3-3 IsGeneralizedEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7C7A07FD795C903E">7.3-4 IsGeneralizedMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7D686DF9832AE258">7.3-5 IsGeneralizedIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X790FC54F7DF8B5B1">7.3-6 IsOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8709A2597FE67C7F">7.3-7 IsMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7C8E0B1A7A8EE198">7.3-8 IsEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8724A5E77FD88D49">7.3-9 IsSplitMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X87508506872F4FC3">7.3-10 IsSplitEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X85180A1E83C01BAA">7.3-11 IsIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X856D1F5C7E289064">7.3-12 IsAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X81B2B7BC7B27A1F4">7.3-13 IsGradedMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7B5C2D788794699E">7.3-14 IsQuasiIsomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X83FBA43B7E5833F0">7.4 <span class="Heading">Chain Morphisms: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X81A0D7187D28BA34">7.4-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X842454D5851D0C79">7.4-2 Range</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7DD92C727DD630DA">7.5 <span class="Heading">Chain Morphisms: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X875F27D07EB78998">7.5-1 ByASmallerPresentation</a></span>
</div></div>
</div>

<h3>7 <span class="Heading">Chain Morphisms</span></h3>

<p><a id="X8703B8017F55336F" name="X8703B8017F55336F"></a></p>

<h4>7.1 <span class="Heading">ChainMorphisms: Categories and Representations</span></h4>

<p><a id="X7CB62E188027B7C5" name="X7CB62E188027B7C5"></a></p>

<h5>7.1-1 IsHomalgChainMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomalgChainMorphism</code>( <var class="Arg">cm</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>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> (co)chain morphisms.</p>

<p>(It is a subcategory of the <strong class="pkg">GAP</strong> category <code class="code">IsHomalgMorphism</code>.)</p>

<p><a id="X853BD37084BFC602" name="X853BD37084BFC602"></a></p>

<h5>7.1-2 IsHomalgChainEndomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomalgChainEndomorphism</code>( <var class="Arg">cm</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>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> (co)chain endomorphisms.</p>

<p>(It is a subcategory of the <strong class="pkg">GAP</strong> categories <code class="code">IsHomalgChainMorphism</code> and <code class="code">IsHomalgEndomorphism</code>.)</p>

<p><a id="X7C35D69F7B09BD47" name="X7C35D69F7B09BD47"></a></p>

<h5>7.1-3 IsChainMorphismOfFinitelyPresentedObjectsRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsChainMorphismOfFinitelyPresentedObjectsRep</code>( <var class="Arg">c</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 chain morphisms of finitely presented <strong class="pkg">homalg</strong> objects.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgChainMorphism</code> (<a href="chap7_mj.html#X7CB62E188027B7C5"><span class="RefLink">7.1-1</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representation <code class="code">IsMorphismOfFinitelyGeneratedObjectsRep</code>.)</p>

<p><a id="X7DF3EA1D817266C1" name="X7DF3EA1D817266C1"></a></p>

<h5>7.1-4 IsCochainMorphismOfFinitelyPresentedObjectsRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCochainMorphismOfFinitelyPresentedObjectsRep</code>( <var class="Arg">c</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 cochain morphisms of finitely presented <strong class="pkg">homalg</strong> objects.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgChainMorphism</code> (<a href="chap7_mj.html#X7CB62E188027B7C5"><span class="RefLink">7.1-1</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representation <code class="code">IsMorphismOfFinitelyGeneratedObjectsRep</code>.)</p>

<p><a id="X83637FBE86C5DDF1" name="X83637FBE86C5DDF1"></a></p>

<h4>7.2 <span class="Heading">Chain Morphisms: Constructors</span></h4>

<p><a id="X853361547FB213CA" name="X853361547FB213CA"></a></p>

<h5>7.2-1 HomalgChainMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomalgChainMorphism</code>( <var class="Arg">phi</var>[, <var class="Arg">C</var>][, <var class="Arg">D</var>][, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> chain morphism</p>

<p>The constructor creates a (co)chain morphism given a source <strong class="pkg">homalg</strong> (co)chain complex <var class="Arg">C</var>, a target <strong class="pkg">homalg</strong> (co)chain complex <var class="Arg">D</var>, and a <strong class="pkg">homalg</strong> morphism <var class="Arg">phi</var> at (co)homological degree <var class="Arg">d</var>. The returned (co)chain morphism will cautiously be indicated using parenthesis: "chain morphism". To verify if the result is indeed a (co)chain morphism use <code class="func">IsMorphism</code> (<a href="chap7_mj.html#X798B6A897FE4FF12"><span class="RefLink">7.3-1</span></a>). If source and target are identical objects, and only then, the (co)chain morphism is created as a (co)chain endomorphism.</p>

<p>The following examples shows a chain morphism that induces the zero morphism on homology, but is itself <em>not</em> zero in the derived category:</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">M := 1 * zz;</span>
<The free left module of rank 1 on a free generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M );</span>
Z^(1 x 1)
<span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[3]", 1, 1, zz );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := LeftPresentation( N );</span>
<A cyclic torsion left module presented by 1 relation for
 a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( N );</span>
Z/< 3 >
<span class="GAPprompt">gap></span> <span class="GAPinput">a := HomalgMap( HomalgMatrix( "[2]", 1, 1, zz ), M, M );</span>
<An endomorphism of a left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">c := HomalgMap( HomalgMatrix( "[2]", 1, 1, zz ), M, N );</span>
<A homomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">b := HomalgMap( HomalgMatrix( "[1]", 1, 1, zz ), M, M );</span>
<An endomorphism of a left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">d := HomalgMap( HomalgMatrix( "[1]", 1, 1, zz ), M, N );</span>
<A homomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">C1 := HomalgComplex( a );</span>
<A non-zero acyclic complex containing a single morphism of left modules at de\
grees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">C2 := HomalgComplex( c );</span>
<A non-zero acyclic complex containing a single morphism of left modules at de\
grees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">cm := HomalgChainMorphism( d, C1, C2 );</span>
<A "chain morphism" containing a single left morphism at degree 0>
<span class="GAPprompt">gap></span> <span class="GAPinput">Add( cm, b );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( cm );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">cm;</span>
<A chain morphism containing 2 morphisms of left modules at degrees
[ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">hcm := DefectOfExactness( cm );</span>
<A chain morphism of graded objects containing
2 morphisms of left modules at degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsZero( hcm );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsZero( Source( hcm ) );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsZero( Range( hcm ) );</span>
false
</pre></div>

<p><a id="X789E2EC07C041D78" name="X789E2EC07C041D78"></a></p>

<h4>7.3 <span class="Heading">Chain Morphisms: Properties</span></h4>

<p><a id="X798B6A897FE4FF12" name="X798B6A897FE4FF12"></a></p>

<h5>7.3-1 IsMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMorphism</code>( <var class="Arg">cm</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 <var class="Arg">cm</var> is a well-defined chain morphism, i.e. independent of all involved presentations.</p>

<p><a id="X8194427F8423EB00" name="X8194427F8423EB00"></a></p>

<h5>7.3-2 IsGeneralizedMorphismWithFullDomain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedMorphismWithFullDomain</code>( <var class="Arg">cm</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 <var class="Arg">cm</var> is a generalized morphism.</p>

<p><a id="X84FE6CFD85AB7B73" name="X84FE6CFD85AB7B73"></a></p>

<h5>7.3-3 IsGeneralizedEpimorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedEpimorphism</code>( <var class="Arg">cm</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 <var class="Arg">cm</var> is a generalized epimorphism.</p>

<p><a id="X7C7A07FD795C903E" name="X7C7A07FD795C903E"></a></p>

<h5>7.3-4 IsGeneralizedMonomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedMonomorphism</code>( <var class="Arg">cm</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 <var class="Arg">cm</var> is a generalized monomorphism.</p>

<p><a id="X7D686DF9832AE258" name="X7D686DF9832AE258"></a></p>

<h5>7.3-5 IsGeneralizedIsomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedIsomorphism</code>( <var class="Arg">cm</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 <var class="Arg">cm</var> is a generalized isomorphism.</p>

<p><a id="X790FC54F7DF8B5B1" name="X790FC54F7DF8B5B1"></a></p>

<h5>7.3-6 IsOne</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOne</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is the identity chain morphism.</p>

<p><a id="X8709A2597FE67C7F" name="X8709A2597FE67C7F"></a></p>

<h5>7.3-7 IsMonomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonomorphism</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is a monomorphism.</p>

<p><a id="X7C8E0B1A7A8EE198" name="X7C8E0B1A7A8EE198"></a></p>

<h5>7.3-8 IsEpimorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEpimorphism</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is an epimorphism.</p>

<p><a id="X8724A5E77FD88D49" name="X8724A5E77FD88D49"></a></p>

<h5>7.3-9 IsSplitMonomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSplitMonomorphism</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is a split monomorphism. <br /></p>

<p><a id="X87508506872F4FC3" name="X87508506872F4FC3"></a></p>

<h5>7.3-10 IsSplitEpimorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSplitEpimorphism</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is a split epimorphism. <br /></p>

<p><a id="X85180A1E83C01BAA" name="X85180A1E83C01BAA"></a></p>

<h5>7.3-11 IsIsomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIsomorphism</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is an isomorphism.</p>

<p><a id="X856D1F5C7E289064" name="X856D1F5C7E289064"></a></p>

<h5>7.3-12 IsAutomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAutomorphism</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is an automorphism.</p>

<p><a id="X81B2B7BC7B27A1F4" name="X81B2B7BC7B27A1F4"></a></p>

<h5>7.3-13 IsGradedMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGradedMorphism</code>( <var class="Arg">cm</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 source and target complex of the <strong class="pkg">homalg</strong> chain morphism <var class="Arg">cm</var> are graded objects, i.e. if all their morphisms vanish.</p>

<p><a id="X7B5C2D788794699E" name="X7B5C2D788794699E"></a></p>

<h5>7.3-14 IsQuasiIsomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsQuasiIsomorphism</code>( <var class="Arg">cm</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> chain morphism <var class="Arg">cm</var> is a quasi-isomorphism.</p>

<p><a id="X83FBA43B7E5833F0" name="X83FBA43B7E5833F0"></a></p>

<h4>7.4 <span class="Heading">Chain Morphisms: Attributes</span></h4>

<p><a id="X81A0D7187D28BA34" name="X81A0D7187D28BA34"></a></p>

<h5>7.4-1 Source</h5>

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

<p>The source of the <strong class="pkg">homalg</strong> chain morphism <var class="Arg">cm</var>.</p>

<p><a id="X842454D5851D0C79" name="X842454D5851D0C79"></a></p>

<h5>7.4-2 Range</h5>

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

<p>The target (range) of the <strong class="pkg">homalg</strong> chain morphism <var class="Arg">cm</var>.</p>

<p><a id="X7DD92C727DD630DA" name="X7DD92C727DD630DA"></a></p>

<h4>7.5 <span class="Heading">Chain Morphisms: Operations and Functions</span></h4>

<p><a id="X875F27D07EB78998" name="X875F27D07EB78998"></a></p>

<h5>7.5-1 ByASmallerPresentation</h5>

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

<p>See <code class="func">ByASmallerPresentation</code> (<a href="chap6_mj.html#X79677A407C9EF3A0"><span class="RefLink">6.5-2</span></a>) on complexes.</p>


<div class="example"><pre>
InstallMethod( ByASmallerPresentation,
        "for homalg chain morphisms",
        [ IsHomalgChainMorphism ],
        
  function( cm )
    
    ByASmallerPresentation( Source( cm ) );
    ByASmallerPresentation( Range( cm ) );
    
    List( MorphismsOfChainMorphism( cm ), DecideZero );
    
    return cm;
    
end );
</pre></div>

<p>This method performs side effects on its argument <var class="Arg">cm</var> and returns it.</p>


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