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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( "GradedModules" );
Q := HomalgFieldOfRationalsInSingular();
Q := GradedRing( Q );
R := Q * "x_0..4";
S := Q * "y_0..4";
weights := [ [1,0],[1,0],[1,0],[1,1],[0,1] ];
deg_group := 2 * HOMALG_MATRICES.ZZ;
weights := List( weights, i -> HomalgModuleElement( i, deg_group ) );
SetWeightsOfIndeterminates( R, weights );
SetWeightsOfIndeterminates( S, weights );
T := R * S;
M := 1 * T / GradedLeftSubmodule( "x_0-y_0,x_1-y_1,x_2-y_2,x_3-y_3,x_4-y_4", T );
Resolution( M );