gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );;
gap> M1 := HomalgMatrix( "[ \
> Dx  \
> ]", 1, 1, A3 );;
gap> M2 := HomalgMatrix( "[ \
> Dx, \
> Dy  \
> ]", 2, 1, A3 );;
gap> M3 := HomalgMatrix( "[ \
> Dx, \
> Dy, \
> Dz  \
> ]", 3, 1, A3 );;
gap> M := DiagMat( [ M1, M2, M3 ] );;
gap> M := ShallowCopy( M );;
gap> SetIsMutableMatrix( M, true );;
gap> M[ 1, 2 ] := "1";;
gap> M[ 2, 3 ] := "1";;
gap> M[ 3, 3 ] := "1";;
gap> MakeImmutable( M );;
gap> tau1 := HomalgMatrix( "[ \
> 1, Dx, Dz, \
> 0,  0,  1, \
> 0,  1, Dy  \
> ]", 3, 3, A3 );;
gap> tau2 := HomalgMatrix( "[ \
> 0,  1, Dz+x*y, \
> 0,  0,      1, \
> 1, Dz,    x-y  \
> ]", 3, 3, A3 );;
gap> tau3 := HomalgMatrix( "[ \
> 1,  0, 0, \
> 1,  1, 0, \
> 0, -1, 1  \
> ]", 3, 3, A3 );;
gap> tau := tau1 * tau2 * tau3;;
gap> M := M * tau;;
gap> rows := CategoryOfRows( A3 );;
gap> alpha := AsCategoryOfRowsMorphism( M, rows );;
gap> Mrows := FreydCategoryObject( alpha );;
gap> Srows := SimplifyObject( Mrows, infinity );;
gap> RankOfObject( Source( RelationMorphism( Srows ) ) );
4
gap> RankOfObject( Range( RelationMorphism( Srows ) ) );
2
gap> IsIsomorphism( SimplifyObject_IsoFromInputObject( Mrows, infinity ) );
true
gap> IsIsomorphism( SimplifyObject_IsoToInputObject( Mrows, infinity ) );
true

gap> mu1 := GradeFiltrationNthMonomorphism( Mrows, 1 );;
gap> IsZero( mu1 );
false
gap> IsMonomorphism( mu1 );
true
gap> mu2 := GradeFiltrationNthMonomorphism( Mrows, 2 );;
gap> IsZero( mu2 );
false
gap> IsMonomorphism( mu2 );
true
gap> mu3 := GradeFiltrationNthMonomorphism( Mrows, 3 );;
gap> IsZero( mu3 );
false
gap> IsMonomorphism( mu3 );
true
gap> mu4 := GradeFiltrationNthMonomorphism( Mrows, 4 );;
gap> IsZero( mu4 );
true

gap> cols := CategoryOfColumns( A3 );;
gap> alpha := AsCategoryOfColumnsMorphism( M, cols );;
gap> Mcols := FreydCategoryObject( alpha );;
gap> Scols := SimplifyObject( Mcols, infinity );;
gap> RankOfObject( Source( RelationMorphism( Scols ) ) );
1
gap> RankOfObject( Range( RelationMorphism( Scols ) ) );
4
gap> IsIsomorphism( SimplifyObject_IsoFromInputObject( Mcols, infinity ) );
true
gap> IsIsomorphism( SimplifyObject_IsoToInputObject( Mcols, infinity ) );
true