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#### 5.3 GradedModules: Properties

For more properties see the corresponding section Modules: Modules: Properties) in the documentation of the homalg package.

#### 5.4 GradedModules: Attributes

##### 5.4-1 BettiTable
 ‣ BettiTable( M ) ( attribute )

Returns: a homalg diagram

The Betti diagram of the homalg graded module M.

##### 5.4-2 CastelnuovoMumfordRegularity
 ‣ CastelnuovoMumfordRegularity( M ) ( attribute )

Returns: an integer

The Castelnuovo-Mumford regularity of the homalg graded module M.

##### 5.4-3 CastelnuovoMumfordRegularityOfSheafification
 ‣ CastelnuovoMumfordRegularityOfSheafification( M ) ( attribute )

Returns: an integer

The Castelnuovo-Mumford regularity of the sheafification of homalg graded module M.

For more attributes see the corresponding section Modules: Modules: Attributes) in the documentation of the homalg package.

#### 5.6 GradedModules: Operations and Functions

##### 5.6-1 MonomialMap
 ‣ MonomialMap( d, M ) ( operation )

Returns: a homalg map

The map from a free graded module onto all degree d monomial generators of the finitely generated homalg module M.

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 ])


##### 5.6-2 RandomMatrix
 ‣ RandomMatrix( S, T ) ( operation )

Returns: a homalg matrix

A random matrix between the graded source module S and the graded target module T.

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


##### 5.6-3 GeneratorsOfHomogeneousPart
 ‣ GeneratorsOfHomogeneousPart( d, M ) ( operation )

Returns: a homalg matrix

The resulting homalg matrix consists of a generating set (over $$R$$) of the d-th homogeneous part of the finitely generated homalg $$S$$-module M, where $$R$$ is the coefficients ring of the graded ring $$S$$ with $$S_0=R$$.

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)


Compare with MonomialMap (5.6-1).

##### 5.6-4 SubmoduleGeneratedByHomogeneousPart
 ‣ SubmoduleGeneratedByHomogeneousPart( d, M ) ( operation )

Returns: a homalg module

The submodule of the homalg module M generated by the image of the d-th monomial map (--> MonomialMap (5.6-1)), or equivalently, by the generating set of the d-th homogeneous part of M.

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 );
z, 0, 0,0,    0,  0,0,
0, z, 0,0,    0,  0,0,
0, 0, z,0,    0,  0,0,
0, 0, 0,0,    -z, y,0,
x, 0, 0,0,    y,  0,z,
-y,x, 0,0,    0,  0,0,
0, -y,x,0,    0,  0,0,
0, 0, 0,-y,   x,  0,0,
0, 0, 0,-z,   0,  x,0,
0, 0, 0,0,    y*z,0,z^2,
0, 0, 0,y^2*z,0,  0,x*z^2

Cokernel of the map

Q[x,y,z]^(1x11) --> 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


##### 5.6-5 RepresentationMapOfRingElement
 ‣ RepresentationMapOfRingElement( r, M, d ) ( operation )

Returns: a homalg matrix

The graded map induced by the homogeneous degree $$1$$ ring element r (of the underlying homalg graded ring $$S$$) regarded as a $$R$$-linear map between the d-th and the $$($$d$$+1)$$-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 GeneratorsOfHomogeneousPart (5.6-3). The entries of the matrix presenting the map lie in the coefficients ring $$R$$.

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 ])


##### 5.6-6 RepresentationMatrixOfKoszulId
 ‣ RepresentationMatrixOfKoszulId( d, M ) ( operation )

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 KoszulDualRing.

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)


##### 5.6-7 RepresentationMapOfKoszulId
 ‣ RepresentationMapOfKoszulId( d, M ) ( operation )

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 KoszulDualRing.

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 ])


 ‣ KoszulRightAdjoint( M, degree_lowest, degree_highest ) ( operation )

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 M (--> RepresentationMapOfKoszulId (5.6-7)) in the $$[$$ degree_lowest .. degree_highest $$]$$. The Castelnuovo-Mumford regularity of M 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 KoszulDualRing.

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)


##### 5.6-9 HomogeneousPartOverCoefficientsRing
 ‣ HomogeneousPartOverCoefficientsRing( d, M ) ( operation )

Returns: a homalg module

The degree $$d$$ homogeneous part of the graded $$R$$-module M as a module over the coefficient ring or field of $$R$$.

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

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