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<div class="ChapSects"><a href="chap3_mj.html#X78B70E1D86624AC1">3 GradedModules</a>
<div class="ContSect"> <a href="chap3_mj.html#X84BE86BD7CAFCA5F">3.1 GradedModules: Category and Representations</a>

</div>
<div class="ContSect"> <a href="chap3_mj.html#X7B3AF789845366C0">3.2 GradedModules: Constructors</a>

</div>
<div class="ContSect"> <a href="chap3_mj.html#X858BEC417BE013FE">3.3 GradedModules: Properties</a>

</div>
<div class="ContSect"> <a href="chap3_mj.html#X7EEE66BA7E3A4CB8">3.4 GradedModules: Attributes</a>

<div class="ContSSBlock">
 <a href="chap3_mj.html#X78E2B4AD7F671293">3.4-1 BettiTable</a>
 <a href="chap3_mj.html#X854A879B8705130F">3.4-2 CastelnuovoMumfordRegularity</a>
 <a href="chap3_mj.html#X7CB0AA408287A8E2">3.4-3 CastelnuovoMumfordRegularityOfSheafification</a>
</div></div>
<div class="ContSect"> <a href="chap3_mj.html#X795320C4829D4F67">3.5 LISHV: Logical Implications for GradedModules</a>

</div>
<div class="ContSect"> <a href="chap3_mj.html#X877CA99B7CB05AD2">3.6 GradedModules: Operations and Functions</a>

<div class="ContSSBlock">
 <a href="chap3_mj.html#X7E7BA9887C435CD4">3.6-1 MonomialMap</a>
 <a href="chap3_mj.html#X86CB265786A878D8">3.6-2 RandomMatrix</a>
 <a href="chap3_mj.html#X78127AB787A5C681">3.6-3 GeneratorsOfHomogeneousPart</a>
 <a href="chap3_mj.html#X86E9CD307823CC52">3.6-4 SubmoduleGeneratedByHomogeneousPart</a>
 <a href="chap3_mj.html#X870CC71A801346E5">3.6-5 RepresentationMapOfRingElement</a>
 <a href="chap3_mj.html#X797D315F87081C55">3.6-6 RepresentationMatrixOfKoszulId</a>
 <a href="chap3_mj.html#X7DBC9F4F827B4F01">3.6-7 RepresentationMapOfKoszulId</a>
 <a href="chap3_mj.html#X7C2D50247FFA3704">3.6-8 KoszulRightAdjoint</a>
 <a href="chap3_mj.html#X78E9B52D87FC5F3C">3.6-9 HomogeneousPartOverCoefficientsRing</a>
</div></div>
</div>

<h3>3 GradedModules</h3>

<a id="X84BE86BD7CAFCA5F" name="X84BE86BD7CAFCA5F"></a>

<h4>3.1 GradedModules: Category and Representations</h4>

<a id="X7B3AF789845366C0" name="X7B3AF789845366C0"></a>

<h4>3.2 GradedModules: Constructors</h4>

<a id="X858BEC417BE013FE" name="X858BEC417BE013FE"></a>

<h4>3.3 GradedModules: Properties</h4>

For more properties see the corresponding section <a href="https://homalg-project.github.io/homalg_project/Modules/doc/chap7_mj.html#X83CC1D6079AA2286">Modules: Modules: Properties</a>) in the documentation of the homalg package.

<a id="X7EEE66BA7E3A4CB8" name="X7EEE66BA7E3A4CB8"></a>

<h4>3.4 GradedModules: Attributes</h4>

<a id="X78E2B4AD7F671293" name="X78E2B4AD7F671293"></a>

<h5>3.4-1 BettiTable</h5>

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

The Betti diagram of the homalg graded module <var class="Arg">M</var>.

<a id="X854A879B8705130F" name="X854A879B8705130F"></a>

<h5>3.4-2 CastelnuovoMumfordRegularity</h5>

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

The Castelnuovo-Mumford regularity of the homalg graded module <var class="Arg">M</var>.

<a id="X7CB0AA408287A8E2" name="X7CB0AA408287A8E2"></a>

<h5>3.4-3 CastelnuovoMumfordRegularityOfSheafification</h5>

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

The Castelnuovo-Mumford regularity of the sheafification of homalg graded module <var class="Arg">M</var>.

For more attributes see the corresponding section <a href="https://homalg-project.github.io/homalg_project/Modules/doc/chap7_mj.html#X78A9979B862BD51D">Modules: Modules: Attributes</a>) in the documentation of the homalg package.

<a id="X795320C4829D4F67" name="X795320C4829D4F67"></a>

<h4>3.5 LISHV: Logical Implications for GradedModules</h4>

<a id="X877CA99B7CB05AD2" name="X877CA99B7CB05AD2"></a>

<h4>3.6 GradedModules: Operations and Functions</h4>

<a id="X7E7BA9887C435CD4" name="X7E7BA9887C435CD4"></a>

<h5>3.6-1 MonomialMap</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MonomialMap</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a homalg map

The map from a free graded module onto all degree <var class="Arg">d</var> monomial generators of the finitely generated homalg module <var class="Arg">M</var>.

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
gap> M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );
<A graded non-torsion left module presented by 2 relations for 3 generators>
gap> m := MonomialMap( 1, M );
<A homomorphism of graded left modules>
gap> Display( m );
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>

<a id="X86CB265786A878D8" name="X86CB265786A878D8"></a>

<h5>3.6-2 RandomMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomMatrix</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a homalg matrix

A random matrix between the graded source module <var class="Arg">S</var> and the graded target module <var class="Arg">T</var>.

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;
gap> S := GradedRing( R );;
gap> rand := RandomMatrix( S^1 + S^2, S^2 + S^3 + S^4 );
<A 2 x 3 matrix over a graded ring>
gap> #Display( rand );
gap> #-3*a-b, -1, 
gap> #-a^2+a*b+2*b^2-2*a*c+2*b*c+c^2, -a+c, 
gap> #-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
</pre></div>

<a id="X78127AB787A5C681" name="X78127AB787A5C681"></a>

<h5>3.6-3 GeneratorsOfHomogeneousPart</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfHomogeneousPart</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a homalg matrix

The resulting homalg matrix consists of a generating set (over \(R\)) of the <var class="Arg">d</var>-th homogeneous part of the finitely generated homalg \(S\)-module <var class="Arg">M</var>, where \(R\) is the coefficients ring of the graded ring \(S\) with \(S_0=R\).

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
gap> M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );
<A graded non-torsion left module presented by 2 relations for 3 generators>
gap> m := GeneratorsOfHomogeneousPart( 1, M );
<An unevaluated non-zero 7 x 3 matrix over a graded ring>
gap> Display( m );
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>

Compare with <code class="func">MonomialMap</code> (<a href="chap3_mj.html#X7E7BA9887C435CD4">3.6-1</a>).

<a id="X86E9CD307823CC52" name="X86E9CD307823CC52"></a>

<h5>3.6-4 SubmoduleGeneratedByHomogeneousPart</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubmoduleGeneratedByHomogeneousPart</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a homalg module

The submodule of the homalg 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_mj.html#X7E7BA9887C435CD4">3.6-1</a>)), or equivalently, by the generating set of the <var class="Arg">d</var>-th homogeneous part of <var class="Arg">M</var>.

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
gap> M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );
<A graded non-torsion left module presented by 2 relations for 3 generators>
gap> n := SubmoduleGeneratedByHomogeneousPart( 1, M );
<A graded left submodule given by 7 generators>
gap> Display( M );
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 ])
gap> Display( n );
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 ])
gap> N := UnderlyingObject( n );
<A graded left module presented by yet unknown relations for 7 generators>
gap> Display( N );
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 ])
gap> gens := GeneratorsOfModule( N );
<A set of 7 generators of a homalg left module>
gap> Display( gens );
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>

<a id="X870CC71A801346E5" name="X870CC71A801346E5"></a>

<h5>3.6-5 RepresentationMapOfRingElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
Returns: a homalg matrix

The graded map induced by the homogeneous degree \(1\) ring element <var class="Arg">r</var> (of the underlying homalg graded ring \(S\)) regarded as a \(R\)-linear map between the <var class="Arg">d</var>-th and the \((\)<var class="Arg">d</var>\(+1)\)-st homogeneous part of the graded finitely generated homalg \(S\)-module \(M\), where \(R\) is the coefficients ring of the graded ring \(S\) with \(S_0=R\). The generating set of both modules is given by <code class="func">GeneratorsOfHomogeneousPart</code> (<a href="chap3_mj.html#X78127AB787A5C681">3.6-3</a>). The entries of the matrix presenting the map lie in the coefficients ring \(R\).

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> x := Indeterminate( S, 1 );
x
gap> M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
gap> M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );
<A graded non-torsion left module presented by 2 relations for 3 generators>
gap> m := RepresentationMapOfRingElement( x, M, 0 );
<A "homomorphism" of graded left modules>
gap> Display( m );
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>

<a id="X797D315F87081C55" name="X797D315F87081C55"></a>

<h5>3.6-6 RepresentationMatrixOfKoszulId</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentationMatrixOfKoszulId</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a homalg matrix

It is assumed that all indeterminates of the underlying homalg graded ring \(S\) are of degree \(1\). The output is the homalg matrix of the multiplication map \(Hom( A, M_{\textit{d}} ) \to Hom( A, M_{\textit{d}+1} )\), where \(A\) is the Koszul dual ring of \(S\), defined using the operation <code class="code">KoszulDualRing</code>.

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> A := KoszulDualRing( S, "a,b,c" );;
gap> M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
gap> M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );
<A graded non-torsion left module presented by 2 relations for 3 generators>
gap> m := RepresentationMatrixOfKoszulId( 0, M );
<An unevaluated 3 x 7 matrix over a graded ring>
gap> Display( m );
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>

<a id="X7DBC9F4F827B4F01" name="X7DBC9F4F827B4F01"></a>

<h5>3.6-7 RepresentationMapOfKoszulId</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentationMapOfKoszulId</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a homalg map

It is assumed that all indeterminates of the underlying homalg graded ring \(S\) are of degree \(1\). The output is the the multiplication map \(Hom( A, M_{\textit{d}} ) \to Hom( A, M_{\textit{d}+1} )\), where \(A\) is the Koszul dual ring of \(S\), defined using the operation <code class="code">KoszulDualRing</code>.

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> A := KoszulDualRing( S, "a,b,c" );;
gap> M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
gap> M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ] );
<A graded non-torsion left module presented by 2 relations for 3 generators>
gap> m := RepresentationMapOfKoszulId( 0, M );
<A homomorphism of graded left modules>
gap> Display( m );
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>

<a id="X7C2D50247FFA3704" name="X7C2D50247FFA3704"></a>

<h5>3.6-8 KoszulRightAdjoint</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
Returns: a homalg cocomplex

It is assumed that all indeterminates of the underlying homalg graded ring \(S\) are of degree \(1\). Compute the homalg \(A\)-cocomplex \(C\) of Koszul maps of the homalg \(S\)-module <var class="Arg">M</var> (--> <code class="func">RepresentationMapOfKoszulId</code> (<a href="chap3_mj.html#X7DBC9F4F827B4F01">3.6-7</a>)) in the \([\) <var class="Arg">degree_lowest</var> .. <var class="Arg">degree_highest</var> \(]\). The Castelnuovo-Mumford regularity of <var class="Arg">M</var> is characterized as the highest degree \(d\), such that \(C\) is not exact at \(d\). \(A\) is the Koszul dual ring of \(S\), defined using the operation <code class="code">KoszulDualRing</code>.

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> A := KoszulDualRing( S, "a,b,c" );;
gap> M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
gap> M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ], S );
<A graded non-torsion left module presented by 2 relations for 3 generators>
gap> CastelnuovoMumfordRegularity( M );
1
gap> R := KoszulRightAdjoint( M, -5, 5 );
<A cocomplex containing 10 morphisms of graded left modules at degrees
[ -5 .. 5 ]>
gap> R := KoszulRightAdjoint( M, 1, 5 );
<An acyclic cocomplex containing
4 morphisms of graded left modules at degrees [ 1 .. 5 ]>
gap> R := KoszulRightAdjoint( M, 0, 5 );
<A cocomplex containing 5 morphisms of graded left modules at degrees
[ 0 .. 5 ]>
gap> R := KoszulRightAdjoint( M, -5, 5 );
<A cocomplex containing 10 morphisms of graded left modules at degrees
[ -5 .. 5 ]>
gap> H := Cohomology( R );
<A graded cohomology object consisting of 11 graded left modules at degrees
[ -5 .. 5 ]>
gap> ByASmallerPresentation( H );
<A non-zero graded cohomology object consisting of
11 graded left modules at degrees [ -5 .. 5 ]>
gap> Cohomology( R, -2 );
<A graded zero left module>
gap> Cohomology( R, -3 );
<A graded zero left module>
gap> Cohomology( R, -1 );
<A graded cyclic torsion-free non-free left module presented by 2 relations fo\
r a cyclic generator>
gap> Cohomology( R, 0 );
<A graded non-zero cyclic left module presented by 3 relations for a cyclic ge\
nerator>
gap> Cohomology( R, 1 );
<A graded non-zero cyclic left module presented by 2 relations for a cyclic ge\
nerator>
gap> Cohomology( R, 2 );
<A graded zero left module>
gap> Cohomology( R, 3 );
<A graded zero left module>
gap> Cohomology( R, 4 );
<A graded zero left module>
gap> Display( Cohomology( R, -1 ) );
Q{a,b,c}/< b, a >

(graded, degree of generator: 0)
gap> Display( Cohomology( R, 0 ) );
Q{a,b,c}/< c, b, a >

(graded, degree of generator: 0)
gap> Display( Cohomology( R, 1 ) );
Q{a,b,c}/< b, a >

(graded, degree of generator: 2)
</pre></div>

<a id="X78E9B52D87FC5F3C" name="X78E9B52D87FC5F3C"></a>

<h5>3.6-9 HomogeneousPartOverCoefficientsRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomogeneousPartOverCoefficientsRing</code>( <var class="Arg">d</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a homalg module

The degree \(d\) homogeneous part of the graded \(R\)-module <var class="Arg">M</var> as a module over the coefficient ring or field of \(R\).

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> M := HomalgMatrix( "[ x, y^2, z^3 ]", 3, 1, S );;
gap> M := Subobject( M, ( 1 * S )^0 );
<A graded torsion-free (left) ideal given by 3 generators>
gap> CastelnuovoMumfordRegularity( M );
4
gap> M1 := HomogeneousPartOverCoefficientsRing( 1, M );
<A graded left vector space of dimension 1 on a free generator>
gap> gen1 := GeneratorsOfModule( M1 );
<A set consisting of a single generator of a homalg left module>
gap> Display( M1 );
Q^(1 x 1)

(graded, degree of generator: 1)
gap> M2 := HomogeneousPartOverCoefficientsRing( 2, M );
<A graded left vector space of dimension 4 on free generators>
gap> Display( M2 );
Q^(1 x 4)

(graded, degrees of generators: [ 2, 2, 2, 2 ])
gap> gen2 := GeneratorsOfModule( M2 );
<A set of 4 generators of a homalg left module>
gap> M3 := HomogeneousPartOverCoefficientsRing( 3, M );
<A graded left vector space of dimension 9 on free generators>
gap> Display( M3 );
Q^(1 x 9)

(graded, degrees of generators: [ 3, 3, 3, 3, 3, 3, 3, 3, 3 ])
gap> gen3 := GeneratorsOfModule( M3 );
<A set of 9 generators of a homalg left module>
gap> Display( gen1 );
x

a set consisting of a single generator given by (the row of) the above matrix
gap> Display( gen2 );
x^2,
x*y,
x*z,
y^2

a set of 4 generators given by the rows of the above matrix
gap> Display( gen3 );
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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