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LoadPackage( "RingsForHomalg" );
R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
var := Indeterminates( R );
LoadPackage( "Modules" );
M0 := 1 * R;
I1 := LeftSubmodule( var{[1]}, R );
M1 := FactorObject( I1 );
Assert( 0, Grade( I1 ) = 1 );
Assert( 0, Grade( M1 ) = 1 );
I2 := LeftSubmodule( var{[1 .. 2]}, R );
M2 := FactorObject( I2 );
Assert( 0, Grade( I2 ) = 2 );
Assert( 0, Grade( M2 ) = 2 );
I3 := LeftSubmodule( var{[1 .. 3]}, R );
M3 := FactorObject( I3 );
Assert( 0, Grade( I3 ) = 3 );
Assert( 0, Grade( M3 ) = 3 );
m := I3;
## Grade is the official name for Depth in homalg
Assert( 0, Grade( m, M0 ) + ProjectiveDimension( M0 ) = Grade( m ) );
Assert( 0, Grade( m, M1 ) + ProjectiveDimension( M1 ) = Grade( m ) );
Assert( 0, Grade( m, M2 ) + ProjectiveDimension( M2 ) = Grade( m ) );
Assert( 0, Grade( m, M3 ) + ProjectiveDimension( M3 ) = Grade( m ) );
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