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<a id="X78B70E1D86624AC1" name="X78B70E1D86624AC1"></a>
<div class="ChapSects"><a href="chap3.html#X78B70E1D86624AC1">3 GradedModules</a>
<div class="ContSect"> <a href="chap3.html#X84BE86BD7CAFCA5F">3.1 GradedModules: Category and Representations</a>

</div>
<div class="ContSect"> <a href="chap3.html#X7B3AF789845366C0">3.2 GradedModules: Constructors</a>

</div>
<div class="ContSect"> <a href="chap3.html#X858BEC417BE013FE">3.3 GradedModules: Properties</a>

</div>
<div class="ContSect"> <a href="chap3.html#X7EEE66BA7E3A4CB8">3.4 GradedModules: Attributes</a>

<div class="ContSSBlock">
 <a href="chap3.html#X78E2B4AD7F671293">3.4-1 BettiTable</a>
 <a href="chap3.html#X854A879B8705130F">3.4-2 CastelnuovoMumfordRegularity</a>
 <a href="chap3.html#X7CB0AA408287A8E2">3.4-3 CastelnuovoMumfordRegularityOfSheafification</a>
</div></div>
<div class="ContSect"> <a href="chap3.html#X795320C4829D4F67">3.5 LISHV: Logical Implications for GradedModules</a>

</div>
<div class="ContSect"> <a href="chap3.html#X877CA99B7CB05AD2">3.6 GradedModules: Operations and Functions</a>

<div class="ContSSBlock">
 <a href="chap3.html#X7E7BA9887C435CD4">3.6-1 MonomialMap</a>
 <a href="chap3.html#X86CB265786A878D8">3.6-2 RandomMatrix</a>
 <a href="chap3.html#X78127AB787A5C681">3.6-3 GeneratorsOfHomogeneousPart</a>
 <a href="chap3.html#X86E9CD307823CC52">3.6-4 SubmoduleGeneratedByHomogeneousPart</a>
 <a href="chap3.html#X870CC71A801346E5">3.6-5 RepresentationMapOfRingElement</a>
 <a href="chap3.html#X797D315F87081C55">3.6-6 RepresentationMatrixOfKoszulId</a>
 <a href="chap3.html#X7DBC9F4F827B4F01">3.6-7 RepresentationMapOfKoszulId</a>
 <a href="chap3.html#X7C2D50247FFA3704">3.6-8 KoszulRightAdjoint</a>
 <a href="chap3.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.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.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.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.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.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_<var class="Arg">d</var> ) -> Hom( A, M_<var class="Arg">d</var>+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_<var class="Arg">d</var> ) -> Hom( A, M_<var class="Arg">d</var>+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.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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