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## <#GAPDoc Label="DE-2.2">
## <Subsection Label="DE-2.2">
## <Heading>DE-2.2</Heading>
## <Example><![CDATA[
## gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0,x1,x2";;
## gap> S := GradedRing( R );;
## gap> mat := HomalgMatrix( "[ x0^2, x1^2, x2^2 ]", 1, 3, S );
## <A 1 x 3 matrix over a graded ring>
## gap> M := RightPresentationWithDegrees( mat, S );
## <A graded cyclic right module on a cyclic generator satisfying 3 relations>
## gap> M := RightPresentationWithDegrees( mat );
## <A graded cyclic right module on a cyclic generator satisfying 3 relations>
## gap> d := Resolution( M );
## <A right acyclic complex containing
## 3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
## gap> betti := BettiTable( d );
## <A Betti diagram of <A right acyclic complex containing
## 3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
## gap> Display( betti );
## total: 1 3 3 1
## ----------------
## 0: 1 . . .
## 1: . 3 . .
## 2: . . 3 .
## 3: . . . 1
## ----------------
## degree: 0 1 2 3
## gap> ## we are still below the Castelnuovo-Mumford regularity, which is 3:
## gap> M2 := SubmoduleGeneratedByHomogeneousPart( 2, M );
## <A graded torsion right submodule given by 3 generators>
## gap> d2 := Resolution( M2 );
## <A right acyclic complex containing
## 3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
## gap> betti2 := BettiTable( d2 );
## <A Betti diagram of <A right acyclic complex containing
## 3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
## gap> Display( betti2 );
## total: 3 8 6 1
## ----------------
## 2: 3 8 6 .
## 3: . . . 1
## ----------------
## degree: 0 1 2 3
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
LoadPackage( "GradedModules", false );
R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0,x1,x2";
S := GradedRing( R );
mat := HomalgMatrix( "[ x0^2, x1^2, x2^2 ]", 1, 3, S );
M := RightPresentationWithDegrees( mat, S );
d := Resolution( M );
betti := BettiTable( d );
## we are still below the Castelnuovo-Mumford regularity, which is 3:
M2 := SubmoduleGeneratedByHomogeneousPart( 2, M );
d2 := Resolution( M2 );
betti2 := BettiTable( d2 );
Display( betti2 );
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