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<div class="ChapSects"><a href="chap3.html#X78B70E1D86624AC1">3 <span class="Heading">GradedModules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X84BE86BD7CAFCA5F">3.1 <span class="Heading">GradedModules: Category and Representations</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B3AF789845366C0">3.2 <span class="Heading">GradedModules: Constructors</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X858BEC417BE013FE">3.3 <span class="Heading">GradedModules: Properties</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7EEE66BA7E3A4CB8">3.4 <span class="Heading">GradedModules: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78E2B4AD7F671293">3.4-1 BettiTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X854A879B8705130F">3.4-2 CastelnuovoMumfordRegularity</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7CB0AA408287A8E2">3.4-3 CastelnuovoMumfordRegularityOfSheafification</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X795320C4829D4F67">3.5 <span class="Heading"><strong class="pkg">LISHV</strong>: Logical Implications for GradedModules</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X877CA99B7CB05AD2">3.6 <span class="Heading">GradedModules: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E7BA9887C435CD4">3.6-1 MonomialMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86CB265786A878D8">3.6-2 RandomMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78127AB787A5C681">3.6-3 GeneratorsOfHomogeneousPart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86E9CD307823CC52">3.6-4 SubmoduleGeneratedByHomogeneousPart</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X870CC71A801346E5">3.6-5 RepresentationMapOfRingElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X797D315F87081C55">3.6-6 RepresentationMatrixOfKoszulId</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DBC9F4F827B4F01">3.6-7 RepresentationMapOfKoszulId</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C2D50247FFA3704">3.6-8 KoszulRightAdjoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78E9B52D87FC5F3C">3.6-9 HomogeneousPartOverCoefficientsRing</a></span>
</div></div>
</div>
<h3>3 <span class="Heading">GradedModules</span></h3>
<p><a id="X84BE86BD7CAFCA5F" name="X84BE86BD7CAFCA5F"></a></p>
<h4>3.1 <span class="Heading">GradedModules: Category and Representations</span></h4>
<p><a id="X7B3AF789845366C0" name="X7B3AF789845366C0"></a></p>
<h4>3.2 <span class="Heading">GradedModules: Constructors</span></h4>
<p><a id="X858BEC417BE013FE" name="X858BEC417BE013FE"></a></p>
<h4>3.3 <span class="Heading">GradedModules: Properties</span></h4>
<p>For more properties see the corresponding section <a href="https://homalg-project.github.io/homalg_project/Modules/doc/chap7.html#X83CC1D6079AA2286"><span class="RefLink">Modules: Modules: Properties</span></a>) in the documentation of the <strong class="pkg">homalg</strong> package.</p>
<p><a id="X7EEE66BA7E3A4CB8" name="X7EEE66BA7E3A4CB8"></a></p>
<h4>3.4 <span class="Heading">GradedModules: Attributes</span></h4>
<p><a id="X78E2B4AD7F671293" name="X78E2B4AD7F671293"></a></p>
<h5>3.4-1 BettiTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BettiTable</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> diagram</p>
<p>The Betti diagram of the <strong class="pkg">homalg</strong> graded module <var class="Arg">M</var>.</p>
<p><a id="X854A879B8705130F" name="X854A879B8705130F"></a></p>
<h5>3.4-2 CastelnuovoMumfordRegularity</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CastelnuovoMumfordRegularity</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an integer</p>
<p>The Castelnuovo-Mumford regularity of the <strong class="pkg">homalg</strong> graded module <var class="Arg">M</var>.</p>
<p><a id="X7CB0AA408287A8E2" name="X7CB0AA408287A8E2"></a></p>
<h5>3.4-3 CastelnuovoMumfordRegularityOfSheafification</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CastelnuovoMumfordRegularityOfSheafification</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an integer</p>
<p>The Castelnuovo-Mumford regularity of the sheafification of <strong class="pkg">homalg</strong> graded module <var class="Arg">M</var>.</p>
<p>For more attributes see the corresponding section <a href="https://homalg-project.github.io/homalg_project/Modules/doc/chap7.html#X78A9979B862BD51D"><span class="RefLink">Modules: Modules: Attributes</span></a>) in the documentation of the <strong class="pkg">homalg</strong> package.</p>
<p><a id="X795320C4829D4F67" name="X795320C4829D4F67"></a></p>
<h4>3.5 <span class="Heading"><strong class="pkg">LISHV</strong>: Logical Implications for GradedModules</span></h4>
<p><a id="X877CA99B7CB05AD2" name="X877CA99B7CB05AD2"></a></p>
<h4>3.6 <span class="Heading">GradedModules: Operations and Functions</span></h4>
<p><a id="X7E7BA9887C435CD4" name="X7E7BA9887C435CD4"></a></p>
<h5>3.6-1 MonomialMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MonomialMap</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> map</p>
<p>The map from a free graded module onto all degree <var class="Arg">d</var> monomial generators of the finitely generated <strong class="pkg">homalg</strong> module <var class="Arg">M</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );</span>
<A graded non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := MonomialMap( 1, M );</span>
<A homomorphism of graded left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
x^2,0,0,
x*y,0,0,
x*z,0,0,
y^2,0,0,
y*z,0,0,
z^2,0,0,
0, x,0,
0, y,0,
0, z,0,
0, 0,1
the graded map is currently represented by the above 10 x 3 matrix
(degrees of generators of target: [ -1, 0, 1 ])
</pre></div>
<p><a id="X86CB265786A878D8" name="X86CB265786A878D8"></a></p>
<h5>3.6-2 RandomMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomMatrix</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>
<p>A random matrix between the graded source module <var class="Arg">S</var> and the graded target module <var class="Arg">T</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rand := RandomMatrix( S^1 + S^2, S^2 + S^3 + S^4 );</span>
<A 2 x 3 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">#Display( rand );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#-3*a-b, -1, </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#-a^2+a*b+2*b^2-2*a*c+2*b*c+c^2, -a+c, </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#-2*a^3+5*a^2*b-3*b^3+3*a*b*c+3*b^2*c+2*a*c^2+2*b*c^2+c^3,-3*b^2-2*a*c-2*b*c+c^2</span>
</pre></div>
<p><a id="X78127AB787A5C681" name="X78127AB787A5C681"></a></p>
<h5>3.6-3 GeneratorsOfHomogeneousPart</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GeneratorsOfHomogeneousPart</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>
<p>The resulting <strong class="pkg">homalg</strong> matrix consists of a generating set (over <span class="SimpleMath">R</span>) of the <var class="Arg">d</var>-th homogeneous part of the finitely generated <strong class="pkg">homalg</strong> <span class="SimpleMath">S</span>-module <var class="Arg">M</var>, where <span class="SimpleMath">R</span> is the coefficients ring of the graded ring <span class="SimpleMath">S</span> with <span class="SimpleMath">S_0=R</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );</span>
<A graded non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := GeneratorsOfHomogeneousPart( 1, M );</span>
<An unevaluated non-zero 7 x 3 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
x^2,0,0,
x*y,0,0,
y^2,0,0,
0, x,0,
0, y,0,
0, z,0,
0, 0,1
(over a graded ring)
</pre></div>
<p>Compare with <code class="func">MonomialMap</code> (<a href="chap3.html#X7E7BA9887C435CD4"><span class="RefLink">3.6-1</span></a>).</p>
<p><a id="X86E9CD307823CC52" name="X86E9CD307823CC52"></a></p>
<h5>3.6-4 SubmoduleGeneratedByHomogeneousPart</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SubmoduleGeneratedByHomogeneousPart</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> module</p>
<p>The submodule of the <strong class="pkg">homalg</strong> module <var class="Arg">M</var> generated by the image of the <var class="Arg">d</var>-th monomial map (--> <code class="func">MonomialMap</code> (<a href="chap3.html#X7E7BA9887C435CD4"><span class="RefLink">3.6-1</span></a>)), or equivalently, by the generating set of the <var class="Arg">d</var>-th homogeneous part of <var class="Arg">M</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );</span>
<A graded non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">n := SubmoduleGeneratedByHomogeneousPart( 1, M );</span>
<A graded left submodule given by 7 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M );</span>
z, 0, 0,
0, y^2*z,z^2,
x^3,y^2, z
Cokernel of the map
Q[x,y,z]^(1x3) --> Q[x,y,z]^(1x3),
currently represented by the above matrix
(graded, degrees of generators: [ -1, 0, 1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( n );</span>
x^2,0,0,
x*y,0,0,
y^2,0,0,
0, x,0,
0, y,0,
0, z,0,
0, 0,1
A left submodule generated by the 7 rows of the above matrix
(graded, degrees of generators: [ 1, 1, 1, 1, 1, 1, 1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">N := UnderlyingObject( n );</span>
<A graded left module presented by yet unknown relations for 7 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( N );</span>
0, 0, z,0, 0, 0,0,
0, z, 0,0, 0, 0,0,
z, 0, 0,0, 0, 0,0,
0, 0, 0,0, -z, y,0,
0, 0, 0,-z,0, x,0,
0, 0, 0,-y,x, 0,0,
0, -y,x,0, 0, 0,0,
-y,x, 0,0, 0, 0,0,
x, 0, 0,0, y, 0,z,
0, 0, 0,0, y*z,0,z^2
Cokernel of the map
Q[x,y,z]^(1x10) --> Q[x,y,z]^(1x7),
currently represented by the above matrix
(graded, degrees of generators: [ 1, 1, 1, 1, 1, 1, 1 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratorsOfModule( N );</span>
<A set of 7 generators of a homalg left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( gens );</span>
x^2,0,0,
x*y,0,0,
y^2,0,0,
0, x,0,
0, y,0,
0, z,0,
0, 0,1
a set of 7 generators given by the rows of the above matrix
</pre></div>
<p><a id="X870CC71A801346E5" name="X870CC71A801346E5"></a></p>
<h5>3.6-5 RepresentationMapOfRingElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RepresentationMapOfRingElement</code>( <var class="Arg">r</var>, <var class="Arg">M</var>, <var class="Arg">d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>
<p>The graded map induced by the homogeneous degree <em><span class="SimpleMath">1</span></em> ring element <var class="Arg">r</var> (of the underlying <strong class="pkg">homalg</strong> graded ring <span class="SimpleMath">S</span>) regarded as a <span class="SimpleMath">R</span>-linear map between the <var class="Arg">d</var>-th and the <span class="SimpleMath">(</span><var class="Arg">d</var><span class="SimpleMath">+1)</span>-st homogeneous part of the graded finitely generated <strong class="pkg">homalg</strong> <span class="SimpleMath">S</span>-module <span class="SimpleMath">M</span>, where <span class="SimpleMath">R</span> is the coefficients ring of the graded ring <span class="SimpleMath">S</span> with <span class="SimpleMath">S_0=R</span>. The generating set of both modules is given by <code class="func">GeneratorsOfHomogeneousPart</code> (<a href="chap3.html#X78127AB787A5C681"><span class="RefLink">3.6-3</span></a>). The entries of the matrix presenting the map lie in the coefficients ring <span class="SimpleMath">R</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Indeterminate( S, 1 );</span>
x
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );</span>
<A graded non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := RepresentationMapOfRingElement( x, M, 0 );</span>
<A "homomorphism" of graded left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
1,0,0,0,0,0,0,
0,1,0,0,0,0,0,
0,0,0,1,0,0,0
the graded map is currently represented by the above 3 x 7 matrix
(degrees of generators of target: [ 1, 1, 1, 1, 1, 1, 1 ])
</pre></div>
<p><a id="X797D315F87081C55" name="X797D315F87081C55"></a></p>
<h5>3.6-6 RepresentationMatrixOfKoszulId</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RepresentationMatrixOfKoszulId</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>
<p>It is assumed that all indeterminates of the underlying <strong class="pkg">homalg</strong> graded ring <span class="SimpleMath">S</span> are of degree <span class="SimpleMath">1</span>. The output is the <strong class="pkg">homalg</strong> matrix of the multiplication map <span class="SimpleMath">Hom( A, M_<var class="Arg">d</var> ) -> Hom( A, M_<var class="Arg">d</var>+1 )</span>, where <span class="SimpleMath">A</span> is the Koszul dual ring of <span class="SimpleMath">S</span>, defined using the operation <code class="code">KoszulDualRing</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := KoszulDualRing( S, "a,b,c" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );</span>
<A graded non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := RepresentationMatrixOfKoszulId( 0, M );</span>
<An unevaluated 3 x 7 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
a,b,0,0,0,0,0,
0,a,b,0,0,0,0,
0,0,0,a,b,c,0
(over a graded ring)
</pre></div>
<p><a id="X7DBC9F4F827B4F01" name="X7DBC9F4F827B4F01"></a></p>
<h5>3.6-7 RepresentationMapOfKoszulId</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RepresentationMapOfKoszulId</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> map</p>
<p>It is assumed that all indeterminates of the underlying <strong class="pkg">homalg</strong> graded ring <span class="SimpleMath">S</span> are of degree <span class="SimpleMath">1</span>. The output is the the multiplication map <span class="SimpleMath">Hom( A, M_<var class="Arg">d</var> ) -> Hom( A, M_<var class="Arg">d</var>+1 )</span>, where <span class="SimpleMath">A</span> is the Koszul dual ring of <span class="SimpleMath">S</span>, defined using the operation <code class="code">KoszulDualRing</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := KoszulDualRing( S, "a,b,c" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );</span>
<A graded non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := RepresentationMapOfKoszulId( 0, M );</span>
<A homomorphism of graded left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
a,b,0,0,0,0,0,
0,a,b,0,0,0,0,
0,0,0,a,b,c,0
the graded map is currently represented by the above 3 x 7 matrix
(degrees of generators of target: [ 4, 4, 4, 4, 4, 4, 4 ])
</pre></div>
<p><a id="X7C2D50247FFA3704" name="X7C2D50247FFA3704"></a></p>
<h5>3.6-8 KoszulRightAdjoint</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; KoszulRightAdjoint</code>( <var class="Arg">M</var>, <var class="Arg">degree_lowest</var>, <var class="Arg">degree_highest</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> cocomplex</p>
<p>It is assumed that all indeterminates of the underlying <strong class="pkg">homalg</strong> graded ring <span class="SimpleMath">S</span> are of degree <span class="SimpleMath">1</span>. Compute the <strong class="pkg">homalg</strong> <span class="SimpleMath">A</span>-cocomplex <span class="SimpleMath">C</span> of Koszul maps of the <strong class="pkg">homalg</strong> <span class="SimpleMath">S</span>-module <var class="Arg">M</var> (--> <code class="func">RepresentationMapOfKoszulId</code> (<a href="chap3.html#X7DBC9F4F827B4F01"><span class="RefLink">3.6-7</span></a>)) in the <span class="SimpleMath">[</span> <var class="Arg">degree_lowest</var> .. <var class="Arg">degree_highest</var> <span class="SimpleMath">]</span>. The Castelnuovo-Mumford regularity of <var class="Arg">M</var> is characterized as the highest degree <span class="SimpleMath">d</span>, such that <span class="SimpleMath">C</span> is not exact at <span class="SimpleMath">d</span>. <span class="SimpleMath">A</span> is the Koszul dual ring of <span class="SimpleMath">S</span>, defined using the operation <code class="code">KoszulDualRing</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := KoszulDualRing( S, "a,b,c" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ], S );</span>
<A graded non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">CastelnuovoMumfordRegularity( M );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">R := KoszulRightAdjoint( M, -5, 5 );</span>
<A cocomplex containing 10 morphisms of graded left modules at degrees
[ -5 .. 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := KoszulRightAdjoint( M, 1, 5 );</span>
<An acyclic cocomplex containing
4 morphisms of graded left modules at degrees [ 1 .. 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := KoszulRightAdjoint( M, 0, 5 );</span>
<A cocomplex containing 5 morphisms of graded left modules at degrees
[ 0 .. 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := KoszulRightAdjoint( M, -5, 5 );</span>
<A cocomplex containing 10 morphisms of graded left modules at degrees
[ -5 .. 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := Cohomology( R );</span>
<A graded cohomology object consisting of 11 graded left modules at degrees
[ -5 .. 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( H );</span>
<A non-zero graded cohomology object consisting of
11 graded left modules at degrees [ -5 .. 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, -2 );</span>
<A graded zero left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, -3 );</span>
<A graded zero left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, -1 );</span>
<A graded cyclic torsion-free non-free left module presented by 2 relations fo\
r a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, 0 );</span>
<A graded non-zero cyclic left module presented by 3 relations for a cyclic ge\
nerator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, 1 );</span>
<A graded non-zero cyclic left module presented by 2 relations for a cyclic ge\
nerator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, 2 );</span>
<A graded zero left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, 3 );</span>
<A graded zero left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology( R, 4 );</span>
<A graded zero left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Cohomology( R, -1 ) );</span>
Q{a,b,c}/< b, a >
(graded, degree of generator: 0)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Cohomology( R, 0 ) );</span>
Q{a,b,c}/< c, b, a >
(graded, degree of generator: 0)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Cohomology( R, 1 ) );</span>
Q{a,b,c}/< b, a >
(graded, degree of generator: 2)
</pre></div>
<p><a id="X78E9B52D87FC5F3C" name="X78E9B52D87FC5F3C"></a></p>
<h5>3.6-9 HomogeneousPartOverCoefficientsRing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HomogeneousPartOverCoefficientsRing</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> module</p>
<p>The degree <span class="SimpleMath">d</span> homogeneous part of the graded <span class="SimpleMath">R</span>-module <var class="Arg">M</var> as a module over the coefficient ring or field of <span class="SimpleMath">R</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ x, y^2, z^3 ]", 3, 1, S );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := Subobject( M, ( 1 * S )^0 );</span>
<A graded torsion-free (left) ideal given by 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">CastelnuovoMumfordRegularity( M );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">M1 := HomogeneousPartOverCoefficientsRing( 1, M );</span>
<A graded left vector space of dimension 1 on a free generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">gen1 := GeneratorsOfModule( M1 );</span>
<A set consisting of a single generator of a homalg left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M1 );</span>
Q^(1 x 1)
(graded, degree of generator: 1)
<span class="GAPprompt">gap></span> <span class="GAPinput">M2 := HomogeneousPartOverCoefficientsRing( 2, M );</span>
<A graded left vector space of dimension 4 on free generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M2 );</span>
Q^(1 x 4)
(graded, degrees of generators: [ 2, 2, 2, 2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">gen2 := GeneratorsOfModule( M2 );</span>
<A set of 4 generators of a homalg left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">M3 := HomogeneousPartOverCoefficientsRing( 3, M );</span>
<A graded left vector space of dimension 9 on free generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M3 );</span>
Q^(1 x 9)
(graded, degrees of generators: [ 3, 3, 3, 3, 3, 3, 3, 3, 3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">gen3 := GeneratorsOfModule( M3 );</span>
<A set of 9 generators of a homalg left module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( gen1 );</span>
x
a set consisting of a single generator given by (the row of) the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( gen2 );</span>
x^2,
x*y,
x*z,
y^2
a set of 4 generators given by the rows of the above matrix
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( gen3 );</span>
x^3,
x^2*y,
x^2*z,
x*y*z,
x*z^2,
x*y^2,
y^3,
y^2*z,
z^3
a set of 9 generators given by the rows of the above matrix
</pre></div>
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