## <#GAPDoc Label="KoszulRightAdjoint:example">
## <Example><![CDATA[
## 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)
## ]]></Example>
## <#/GAPDoc>
LoadPackage( "GradedModules", false );
R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
S := GradedRing( R );;
A := KoszulDualRing( S, "a,b,c" );;
M := HomalgMatrix( "[ x^3, y^2, z, z, 0, 0 ]", 2, 3, S );;
M := LeftPresentationWithDegrees( M, [ -1, 0, 1 ], S );
CastelnuovoMumfordRegularity( M );
R := KoszulRightAdjoint( M, -5, 5 );
R := KoszulRightAdjoint( M, 1, 5 );
R := KoszulRightAdjoint( M, 0, 5 );
R := KoszulRightAdjoint( M, -5, 5 );
H := Cohomology( R );
ByASmallerPresentation( H );
Cohomology( R, -2 );
Cohomology( R, -3 );
Cohomology( R, -1 );
Cohomology( R, 0 );
Cohomology( R, 1 );
Cohomology( R, 2 );
Cohomology( R, 3 );
Cohomology( R, 4 );
Display( Cohomology( R, -1 ) );
Display( Cohomology( R, 0 ) );
Display( Cohomology( R, 1 ) );
