This is Example B.2 in [Bar09].
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z"; Q[x,y,z] gap> wmat := HomalgMatrix( "[ \ > x*y, y*z, z, 0, 0, \ > x^3*z,x^2*z^2,0, x*z^2, -z^2, \ > x^4, x^3*z, 0, x^2*z, -x*z, \ > 0, 0, x*y, -y^2, x^2-1,\ > 0, 0, x^2*z, -x*y*z, y*z, \ > 0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \ > ]", 6, 5, Qxyz ); <A 6 x 5 matrix over an external ring> gap> W := LeftPresentation( wmat ); <A left module presented by 6 relations for 5 generators> gap> Y := Hom( Qxyz, W ); <A right module on 5 generators satisfying yet unknown relations> gap> SetInfoLevel( InfoWarning, 0 ); gap> F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, Y, "TensorY" ); <The functor TensorY for f.p. modules and their maps over computable rings> gap> SetInfoLevel( InfoWarning, 1 ); gap> G := LeftDualizingFunctor( Qxyz );; gap> II_E := GrothendieckSpectralSequence( F, G, W ); <A stable homological spectral sequence with sheets at levels [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x [ 0 .. 3 ]> gap> Display( II_E ); The associated transposed spectral sequence: a homological spectral sequence at bidegrees [ [ 0 .. 3 ], [ -3 .. 0 ] ] --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s s s s . . . . . . . . . . . . Now the spectral sequence of the bicomplex: a homological spectral sequence at bidegrees [ [ -3 .. 0 ], [ 0 .. 3 ] ] --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * * * * * . * * * . . . * --------- Level 2: * * s s * * * * . * * * . . . * --------- Level 3: * s s s * s s s . . s * . . . * --------- Level 4: s s s s . s s s . . s s . . . s gap> filt := FiltrationBySpectralSequence( II_E, 0 ); <An ascending filtration with degrees [ -3 .. 0 ] and graded parts: 0: <A non-zero left module presented by yet unknown relations for 23 generator\ s> -1: <A non-zero left module presented by 37 relations for 22 generators> -2: <A non-zero left module presented by 31 relations for 10 generators> -3: <A non-zero left module presented by 33 relations for 5 generators> of <A non-zero left module presented by 102 relations for 37 generators>> gap> ByASmallerPresentation( filt ); <An ascending filtration with degrees [ -3 .. 0 ] and graded parts: 0: <A non-zero left module presented by 26 relations for 16 generators> -1: <A non-zero left module presented by 30 relations for 14 generators> -2: <A non-zero left module presented by 18 relations for 7 generators> -3: <A non-zero left module presented by 12 relations for 4 generators> of <A non-zero left module presented by 48 relations for 20 generators>> gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules>
This is Example B.3 in [Bar09].
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z"; Q[x,y,z] gap> wmat := HomalgMatrix( "[ \ > x*y, y*z, z, 0, 0, \ > x^3*z,x^2*z^2,0, x*z^2, -z^2, \ > x^4, x^3*z, 0, x^2*z, -x*z, \ > 0, 0, x*y, -y^2, x^2-1,\ > 0, 0, x^2*z, -x*y*z, y*z, \ > 0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \ > ]", 6, 5, Qxyz ); <A 6 x 5 matrix over an external ring> gap> W := LeftPresentation( wmat ); <A left module presented by 6 relations for 5 generators> gap> filt := PurityFiltration( W ); <The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts: 0: <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\ generators> -1: <A codegree-1-pure grade 1 left module presented by 4 relations for 3 gene\ rators> -2: <A cyclic reflexively pure grade 2 left module presented by 2 relations fo\ r a cyclic generator> -3: <A cyclic reflexively pure grade 3 left module presented by 3 relations fo\ r a cyclic generator> of <A non-pure rank 2 left module presented by 6 relations for 5 generators>> gap> W; <A non-pure rank 2 left module presented by 6 relations for 5 generators> gap> II_E := SpectralSequence( filt ); <A stable homological spectral sequence with sheets at levels [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x [ 0 .. 3 ]> gap> Display( II_E ); The associated transposed spectral sequence: a homological spectral sequence at bidegrees [ [ 0 .. 3 ], [ -3 .. 0 ] ] --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s . . . . . . . . . . . . . . . Now the spectral sequence of the bicomplex: a homological spectral sequence at bidegrees [ [ -3 .. 0 ], [ 0 .. 3 ] ] --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * * * * * . * * * . . . * --------- Level 2: s . . . * s . . . * * . . . . * --------- Level 3: s . . . * s . . . . s . . . . * --------- Level 4: s . . . . s . . . . s . . . . s gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules> gap> IsIdenticalObj( Range( m ), W ); true gap> Source( m ); <A left module presented by 12 relations for 9 generators (locked)> gap> Display( last ); 0, 0,x, -y,0,1, 0, 0, 0, x*y,0,-z,0, 0,0, 0, 0, 0, x^2,0,0, -z,1,0, 0, 0, 0, 0, 0,0, 0, y,-z,0, 0, 0, 0, 0,0, 0, 0,x, -y, -1, 0, 0, 0,0, 0, x,0, -z, 0, -1, 0, 0,0, 0, 0,-y,x^2-1,0, 0, 0, 0,0, 0, 0,0, 0, z, 0, 0, 0,0, 0, 0,0, 0, y-1,0, 0, 0,0, 0, 0,0, 0, 0, z, 0, 0,0, 0, 0,0, 0, 0, y, 0, 0,0, 0, 0,0, 0, 0, x Cokernel of the map Q[x,y,z]^(1x12) --> Q[x,y,z]^(1x9), currently represented by the above matrix gap> Display( filt ); Degree 0: 0, 0,x, -y, x*y,0,-z,0, x^2,0,0, -z Cokernel of the map Q[x,y,z]^(1x3) --> Q[x,y,z]^(1x4), currently represented by the above matrix ---------- Degree -1: y,-z,0, 0,x, -y, x,0, -z, 0,-y,x^2-1 Cokernel of the map Q[x,y,z]^(1x4) --> Q[x,y,z]^(1x3), currently represented by the above matrix ---------- Degree -2: Q[x,y,z]/< z, y-1 > ---------- Degree -3: Q[x,y,z]/< z, y, x > gap> Display( m ); 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -y, -1, 0, 0, 0, -x, 0, -1, 0, -x^2,-x*z, 0, -z, 0, 0, 0, x, -y, 0, 0, 0, 0, 0, -1, 0, 0, x^2,-x*y,y, -x^3,-x^2*z,0, -x*z,z the map is currently represented by the above 9 x 5 matrix
This is Example B.4 in [Bar09].
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z"; Q[x,y,z] gap> A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" ); Q[x,y,z]<Dx,Dy,Dz> gap> nmat := HomalgMatrix( "[ \ > 3*Dy*Dz-Dz^2+Dx+3*Dy-Dz, 3*Dy*Dz-Dz^2, \ > Dx*Dz+Dz^2+Dz, Dx*Dz+Dz^2, \ > Dx*Dy, 0, \ > Dz^2-Dx+Dz, 3*Dx*Dy+Dz^2, \ > Dx^2, 0, \ > -Dz^2+Dx-Dz, 3*Dx^2-Dz^2, \ > Dz^3-Dx*Dz+Dz^2, Dz^3, \ > 2*x*Dz^2-2*x*Dx+2*x*Dz+3*Dx+3*Dz+3,2*x*Dz^2+3*Dx+3*Dz\ > ]", 8, 2, A3 ); <A 8 x 2 matrix over an external ring> gap> N := LeftPresentation( nmat ); <A left module presented by 8 relations for 2 generators> gap> filt := PurityFiltration( N ); <The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts: 0: <A zero left module> -1: <A cyclic reflexively pure grade 1 left module presented by 1 relation for\ a cyclic generator> -2: <A cyclic reflexively pure grade 2 left module presented by 2 relations fo\ r a cyclic generator> -3: <A cyclic reflexively pure grade 3 left module presented by 3 relations fo\ r a cyclic generator> of <A non-pure grade 1 left module presented by 8 relations for 2 generators>> gap> II_E := SpectralSequence( filt ); <A stable homological spectral sequence with sheets at levels [ 0 .. 2 ] each consisting of left modules at bidegrees [ -4 .. 0 ]x [ 0 .. 3 ]> gap> Display( II_E ); The associated transposed spectral sequence: a homological spectral sequence at bidegrees [ [ 0 .. 3 ], [ -4 .. 0 ] ] --------- Level 0: * * * * . * * * . . * * . . . * . . . * --------- Level 1: * * * * . . . . . . . . . . . . . . . . --------- Level 2: s . . . . . . . . . . . . . . . . . . . Now the spectral sequence of the bicomplex: a homological spectral sequence at bidegrees [ [ -4 .. 0 ], [ 0 .. 3 ] ] --------- Level 0: * * * * * . . * * * . . . * * . . . . * --------- Level 1: * * * * * . . * * * . . . * * . . . . . --------- Level 2: . s . . . . . s . . . . . s . . . . . . gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules> gap> IsIdenticalObj( Range( m ), N ); true gap> Source( m ); <A left module presented by 6 relations for 3 generators (locked)> gap> Display( last ); Dx,1/3,1/216*x, 0, Dy, -1/144, 0, Dx, 1/48, 0, 0, Dz, 0, 0, Dy, 0, 0, Dx Cokernel of the map R^(1x6) --> R^(1x3), ( for R := Q[x,y,z]<Dx,Dy,Dz> ) currently represented by the above matrix gap> Display( filt ); Degree 0: 0 ---------- Degree -1: Q[x,y,z]<Dx,Dy,Dz>/< Dx > ---------- Degree -2: Q[x,y,z]<Dx,Dy,Dz>/< Dy, Dx > ---------- Degree -3: Q[x,y,z]<Dx,Dy,Dz>/< Dz, Dy, Dx > gap> Display( m ); 1, 1, 3*Dz+3, 3*Dz, 144*Dz^2-144*Dx+144*Dz,144*Dz^2 the map is currently represented by the above 3 x 2 matrix
This is Example B.5 in [Bar09].
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z"; Q[x,y,z] gap> wmat := HomalgMatrix( "[ \ > x*y, y*z, z, 0, 0, \ > x^3*z,x^2*z^2,0, x*z^2, -z^2, \ > x^4, x^3*z, 0, x^2*z, -x*z, \ > 0, 0, x*y, -y^2, x^2-1,\ > 0, 0, x^2*z, -x*y*z, y*z, \ > 0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \ > ]", 6, 5, Qxyz ); <A 6 x 5 matrix over an external ring> gap> W := LeftPresentation( wmat ); <A left module presented by 6 relations for 5 generators> gap> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, W, "TensorW" ); <The functor TensorW for f.p. modules and their maps over computable rings> gap> G := LeftDualizingFunctor( Qxyz );; gap> II_E := GrothendieckSpectralSequence( F, G, W ); <A stable cohomological spectral sequence with sheets at levels [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x [ 0 .. 3 ]> gap> Display( II_E ); The associated transposed spectral sequence: a cohomological spectral sequence at bidegrees [ [ 0 .. 3 ], [ -3 .. 0 ] ] --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s s s s . . . . . . . . . . . . Now the spectral sequence of the bicomplex: a cohomological spectral sequence at bidegrees [ [ -3 .. 0 ], [ 0 .. 3 ] ] --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * * * * * . * * * . . . * --------- Level 2: * * s s * * * * . * * * . . . * --------- Level 3: * s s s . s s s . . s * . . . s --------- Level 4: s s s s . s s s . . s s . . . s gap> filt := FiltrationBySpectralSequence( II_E, 0 ); <A descending filtration with degrees [ -3 .. 0 ] and graded parts: -3: <A non-zero cyclic torsion left module presented by yet unknown relations \ for a cyclic generator> -2: <A non-zero left module presented by 17 relations for 6 generators> -1: <A non-zero left module presented by 27 relations for 12 generators> 0: <A non-zero left module presented by 13 relations for 10 generators> of <A left module presented by yet unknown relations for 49 generators>> gap> ByASmallerPresentation( filt ); <A descending filtration with degrees [ -3 .. 0 ] and graded parts: -3: <A non-zero cyclic torsion left module presented by 3 relations for a cycl\ ic generator> -2: <A non-zero left module presented by 12 relations for 4 generators> -1: <A non-zero left module presented by 21 relations for 8 generators> 0: <A non-zero left module presented by 11 relations for 10 generators> of <A non-zero left module presented by 27 relations for 14 generators>> gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules>
This is Example B.6 in [Bar09].
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z"; Q[x,y,z] gap> wmat := HomalgMatrix( "[ \ > x*y, y*z, z, 0, 0, \ > x^3*z,x^2*z^2,0, x*z^2, -z^2, \ > x^4, x^3*z, 0, x^2*z, -x*z, \ > 0, 0, x*y, -y^2, x^2-1,\ > 0, 0, x^2*z, -x*y*z, y*z, \ > 0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \ > ]", 6, 5, Qxyz ); <A 6 x 5 matrix over an external ring> gap> W := LeftPresentation( wmat ); <A left module presented by 6 relations for 5 generators> gap> P := Resolution( W ); <A right acyclic complex containing 3 morphisms of left modules at degrees [ 0 .. 3 ]> gap> GP := Hom( P ); <A cocomplex containing 3 morphisms of right modules at degrees [ 0 .. 3 ]> gap> FGP := GP * P; <A cocomplex containing 3 morphisms of left complexes at degrees [ 0 .. 3 ]> gap> BC := HomalgBicomplex( FGP ); <A bicocomplex containing left modules at bidegrees [ 0 .. 3 ]x[ -3 .. 0 ]> gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1]; [ 0 .. 3 ] gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees ); <A stable cohomological spectral sequence with sheets at levels [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x [ 0 .. 3 ]> gap> Display( II_E ); The associated transposed spectral sequence: a cohomological spectral sequence at bidegrees [ [ 0 .. 3 ], [ -3 .. 0 ] ] --------- Level 0: * * * * * * * * * * * * * * * * --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s s s s . . . . . . . . . . . . Now the spectral sequence of the bicomplex: a cohomological spectral sequence at bidegrees [ [ -3 .. 0 ], [ 0 .. 3 ] ] --------- Level 0: * * * * * * * * * * * * * * * * --------- Level 1: * * * * * * * * * * * * * * * * --------- Level 2: * * s s * * * * . * * * . . . * --------- Level 3: * s s s . s s s . . s * . . . s --------- Level 4: s s s s . s s s . . s s . . . s gap> filt := FiltrationBySpectralSequence( II_E, 0 ); <A descending filtration with degrees [ -3 .. 0 ] and graded parts: -3: <A non-zero cyclic torsion left module presented by yet unknown relations \ for a cyclic generator> -2: <A non-zero left module presented by 15 relations for 6 generators> -1: <A non-zero left module presented by 27 relations for 13 generators> 0: <A non-zero left module presented by 13 relations for 10 generators> of <A left module presented by yet unknown relations for 31 generators>> gap> ByASmallerPresentation( filt ); <A descending filtration with degrees [ -3 .. 0 ] and graded parts: -3: <A non-zero cyclic torsion left module presented by 3 relations for a cycl\ ic generator> -2: <A non-zero left module presented by 11 relations for 4 generators> -1: <A non-zero left module presented by 23 relations for 9 generators> 0: <A non-zero left module presented by 11 relations for 10 generators> of <A non-zero left module presented by 24 relations for 12 generators>> gap> m := IsomorphismOfFiltration( filt ); <A non-zero isomorphism of left modules>
This is Example B.7 in [Bar09].
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z"; Q[x,y,z] gap> vmat := HomalgMatrix( "[ \ > 0, 0, x,-z, \ > x*z,z^2,y,0, \ > x^2,x*z,0,y \ > ]", 3, 4, Qxyz ); <A 3 x 4 matrix over an external ring> gap> V := LeftPresentation( vmat ); <A non-torsion left module presented by 3 relations for 4 generators> gap> wmat := HomalgMatrix( "[ \ > 0, 0, x,-y, \ > x*y,y*z,z,0, \ > x^2,x*z,0,z \ > ]", 3, 4, Qxyz ); <A 3 x 4 matrix over an external ring> gap> W := LeftPresentation( wmat ); <A non-torsion left module presented by 3 relations for 4 generators> gap> Rank( V ); 2 gap> Rank( W ); 2 gap> ProjectiveDimension( V ); 2 gap> ProjectiveDimension( W ); 2 gap> DegreeOfTorsionFreeness( V ); 1 gap> DegreeOfTorsionFreeness( W ); 1 gap> CodegreeOfPurity( V ); [ 2 ] gap> CodegreeOfPurity( W ); [ 1, 1 ] gap> filtV := PurityFiltration( V ); <The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts: 0: <A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 ge\ nerators> -1: <A zero left module> -2: <A zero left module> of <A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 gener\ ators>> gap> filtW := PurityFiltration( W ); <The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts: 0: <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\ generators> -1: <A zero left module> -2: <A zero left module> of <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4 ge\ nerators>> gap> II_EV := SpectralSequence( filtV ); <A stable homological spectral sequence with sheets at levels [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x [ 0 .. 2 ]> gap> Display( II_EV ); The associated transposed spectral sequence: a homological spectral sequence at bidegrees [ [ 0 .. 2 ], [ -3 .. 0 ] ] --------- Level 0: * * * * * * * * * . * * --------- Level 1: * * * . . . . . . . . . --------- Level 2: s . . . . . . . . . . . Now the spectral sequence of the bicomplex: a homological spectral sequence at bidegrees [ [ -3 .. 0 ], [ 0 .. 2 ] ] --------- Level 0: * * * * * * * * . * * * --------- Level 1: * * * * * * * * . . * * --------- Level 2: * . . . * . . . . . * * --------- Level 3: * . . . . . . . . . . * --------- Level 4: . . . . . . . . . . . s gap> II_EW := SpectralSequence( filtW ); <A stable homological spectral sequence with sheets at levels [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x [ 0 .. 2 ]> gap> Display( II_EW ); The associated transposed spectral sequence: a homological spectral sequence at bidegrees [ [ 0 .. 2 ], [ -3 .. 0 ] ] --------- Level 0: * * * * * * . * * . . * --------- Level 1: * * * . . . . . . . . . --------- Level 2: s . . . . . . . . . . . Now the spectral sequence of the bicomplex: a homological spectral sequence at bidegrees [ [ -3 .. 0 ], [ 0 .. 2 ] ] --------- Level 0: * * * * . * * * . . * * --------- Level 1: * * * * . * * * . . . * --------- Level 2: * . . . . * . . . . . * --------- Level 3: * . . . . . . . . . . * --------- Level 4: . . . . . . . . . . . s
This corresponds to the example of Section 2 in [BR06].
gap> R := HomalgRingOfIntegersInExternalGAP( ) / 2^8; Z/( 256 ) gap> Display( R ); <A residue class ring> gap> M := LeftPresentation( [ 2^5 ], R ); <A cyclic left module presented by 1 relation for a cyclic generator> gap> Display( M ); Z/( 256 )/< |[ 32 ]| > gap> M; <A cyclic left module presented by 1 relation for a cyclic generator> gap> _M := LeftPresentation( [ 2^3 ], R ); <A cyclic left module presented by 1 relation for a cyclic generator> gap> Display( _M ); Z/( 256 )/< |[ 8 ]| > gap> _M; <A cyclic left module presented by 1 relation for a cyclic generator> gap> alpha2 := HomalgMap( [ 1 ], M, _M ); <A "homomorphism" of left modules> gap> IsMorphism( alpha2 ); true gap> alpha2; <A homomorphism of left modules> gap> Display( alpha2 ); [ [ 1 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix gap> M_ := Kernel( alpha2 ); <A cyclic left module presented by yet unknown relations for a cyclic generato\ r> gap> alpha1 := KernelEmb( alpha2 ); <A monomorphism of left modules> gap> seq := HomalgComplex( alpha2 ); <An acyclic complex containing a single morphism of left modules at degrees [ 0 .. 1 ]> gap> Add( seq, alpha1 ); gap> seq; <A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]> gap> IsShortExactSequence( seq ); true gap> seq; <A short exact sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]> gap> Display( seq ); ------------------------- at homology degree: 2 Z/( 256 )/< |[ 4 ]| > ------------------------- [ [ 8 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 1 Z/( 256 )/< |[ 32 ]| > ------------------------- [ [ 1 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 0 Z/( 256 )/< |[ 8 ]| > ------------------------- gap> K := LeftPresentation( [ 2^7 ], R ); <A cyclic left module presented by 1 relation for a cyclic generator> gap> L := RightPresentation( [ 2^4 ], R ); <A cyclic right module on a cyclic generator satisfying 1 relation> gap> triangle := LHomHom( 4, seq, K, L, "t" ); <An exact triangle containing 3 morphisms of left complexes at degrees [ 1, 2, 3, 1 ]> gap> lehs := LongSequence( triangle ); <A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]> gap> ByASmallerPresentation( lehs ); <A non-zero sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]> gap> IsExactSequence( lehs ); false gap> lehs; <A non-zero left acyclic complex containing 14 morphisms of left modules at degrees [ 0 .. 14 ]> gap> Assert( 0, IsLeftAcyclic( lehs ) ); gap> Display( lehs ); ------------------------- at homology degree: 14 Z/( 256 )/< |[ 4 ]| > ------------------------- [ [ 4 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 13 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 12 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 11 Z/( 256 )/< |[ 4 ]| > ------------------------- [ [ 4 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 10 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 9 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 8 Z/( 256 )/< |[ 4 ]| > ------------------------- [ [ 4 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 7 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 6 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 5 Z/( 256 )/< |[ 4 ]| > ------------------------- [ [ 4 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 4 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 3 Z/( 256 )/< |[ 8 ]| > ------------------------- [ [ 2 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 2 Z/( 256 )/< |[ 4 ]| > ------------------------- [ [ 8 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 1 Z/( 256 )/< |[ 16 ]| > ------------------------- [ [ 1 ] ] modulo [ 256 ] the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 0 Z/( 256 )/< |[ 8 ]| > -------------------------
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,l,m"; Q[x,y,z,l,m] gap> var := Indeterminates( R ); [ x, y, z, l, m ] gap> x := var[1];; y := var[2];; z := var[3];; l := var[4];; m := var[5];; gap> L := [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ]; [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ] gap> e := Eliminate( L, [ l, m ] ); <A non-zero right regular 3 x 1 matrix over an external ring> gap> Display( e ); 4*y+z, 4*x-5*z, 21*z^2-8 gap> I := LeftSubmodule( e ); <A torsion-free (left) ideal given by 3 generators> gap> Display( I ); 4*y+z, 4*x-5*z, 21*z^2-8 A (left) ideal generated by the 3 entries of the above matrix gap> J := LeftSubmodule( "x+y-z, -2*z-3*y+x, x^2+y^2+z^2-1", R ); <A torsion-free (left) ideal given by 3 generators> gap> I = J; true
generated by GAPDoc2HTML