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## <#GAPDoc Label="GlobalSectionsAndPurity">
## <Subsection Label="GlobalSectionsAndPurity">
## <Heading>Examples of the ModuleOfGlobalSections Functor and Purity Filtrations</Heading>
## <Example><![CDATA[
## gap> LoadPackage( "GradedRingForHomalg" );;
## gap> Qxyzt := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,t";;
## gap> S := GradedRing( Qxyzt );;
## gap>
## gap> wmat := HomalgMatrix( "[ \
## > x*y, y*z, z*t, 0, 0, 0,\
## > x^3*z,x^2*z^2,0, x*z^2*t, -z^2*t^2, 0,\
## > x^4, x^3*z, 0, x^2*z*t, -x*z*t^2, 0,\
## > 0, 0, x*y, -y^2, x^2-t^2, 0,\
## > 0, 0, x^2*z, -x*y*z, y*z*t, 0,\
## > 0, 0, x^2*y-x^2*t,-x*y^2+x*y*t,y^2*t-y*t^2,0,\
## > 0, 0, 0, 0, -1, 1 \
## > ]", 7, 6, Qxyzt );;
## gap>
## gap> LoadPackage( "GradedModules" );;
## gap> wmor := GradedMap( wmat, "free", "free", "left", S );;
## gap> IsMorphism( wmor );;
## gap> W := LeftPresentationWithDegrees( wmat, S );;
## gap> HW := ModuleOfGlobalSections( W );
## <A graded left module presented by yet unknown relations for 6 generators>
## gap> LinearStrandOfTateResolution( W, 0,4 );
## <A cocomplex containing 4 morphisms of graded left modules at degrees
## [ 0 .. 4 ]>
## gap> purity_iso := IsomorphismOfFiltration( PurityFiltration( W ) );
## <A non-zero isomorphism of graded left modules>
## gap> Hpurity_iso := ModuleOfGlobalSections( purity_iso );
## <An isomorphism of graded left modules>
## gap> ModuleOfGlobalSections( wmor );
## <A homomorphism of graded left modules>
## gap> NaturalMapToModuleOfGlobalSections( W );
## <A homomorphism of graded left modules>
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
LoadPackage( "GradedRingForHomalg", false );;
Qxyzt := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,t";;
S := GradedRing( Qxyzt );;
wmat := HomalgMatrix( "[ \
x*y, y*z, z*t, 0, 0, 0,\
x^3*z,x^2*z^2,0, x*z^2*t, -z^2*t^2, 0,\
x^4, x^3*z, 0, x^2*z*t, -x*z*t^2, 0,\
0, 0, x*y, -y^2, x^2-t^2, 0,\
0, 0, x^2*z, -x*y*z, y*z*t, 0,\
0, 0, x^2*y-x^2*t,-x*y^2+x*y*t,y^2*t-y*t^2,0,\
0, 0, 0, 0, -1, 1 \
]", 7, 6, Qxyzt );;
LoadPackage( "GradedModules", false );;
wmor := GradedMap( wmat, "free", "free", "left", S );;
IsMorphism( wmor );;
W := LeftPresentationWithDegrees( wmat, S );;
HW := ModuleOfGlobalSections( W );
LinearStrandOfTateResolution( W, 0,4 );
purity_iso := IsomorphismOfFiltration( PurityFiltration( W ) );
Hpurity_iso := ModuleOfGlobalSections( purity_iso );
ModuleOfGlobalSections( wmor );
NaturalMapToModuleOfGlobalSections( W );
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