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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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LoadPackage( "ModulePres" );
LoadPackage( "Homological" );
Q := HomalgFieldOfRationalsInSingular();
R := Q*"x,y";
S := GradedRing( R );
WeightsOfIndeterminates( S );
eins := last[1];
M := HomalgMatrix( "[x^2,xy,y^2]", 3,1, S );
XX := GradedFreeLeftPresentation( 1, S );
S0 := GradedFreeLeftPresentation( 1, S );
S1 := GradedFreeLeftPresentation( 3, S, [ 2*eins, 2*eins, 2*eins ] );
alpha := GradedPresentationMorphism( S1, M, S0 );
M := CokernelObject( alpha );
FFhomo := ResolutionFunctorToComplex( GradedLeftPresentations( S ), CoverByFreeModule );
FF := ResolutionFunctorToCocomplex( GradedLeftPresentations( S ), CoverByFreeModule );
res := ApplyFunctor( FFhomo, M );
SetIsAdditiveCategory( CocomplexCategory( CapCategory( S0 ) ), true );
SetIsAdditiveCategory( ComplexCategory( CapCategory( S0 ) ), true );

M := DirectSum( [ M, S0 ] );

res := FreeResolutionComplexOfModule( M );
res := res[ 1 ];
homres := InternalHomOnComplexWithObject( res, S0 );
CE := CartanEilenbergResolution( homres, FreeResolutionCocomplexOfModule );
homCE := InternalHomOnCocomplexCocomplexWithObject( CE, S0 );