For more properties see the corresponding section Modules: Modules: Properties) in the documentation of the homalg package.
‣ BettiTable( M ) | ( attribute ) |
Returns: a homalg diagram
The Betti diagram of the homalg graded module M.
‣ CastelnuovoMumfordRegularity( M ) | ( attribute ) |
Returns: an integer
The Castelnuovo-Mumford regularity of the homalg graded module M.
‣ 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.
‣ 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 ])
‣ 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
‣ 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 (3.6-1).
‣ 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 (3.6-1)), 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 ); 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
‣ 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 (3.6-3). 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 ])
‣ 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_d ) -> Hom( A, M_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)
‣ 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_d ) -> Hom( A, M_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 (3.6-7)) 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)
‣ 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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