The functions listed below are components of the homalgTable object stored in the ring. They are only indirectly accessible through standard methods that invoke them.
There are matrix methods for which homalg needs a homalgTable entry for non-internal rings, as it cannot provide a suitable fallback. Below is the list of these homalgTable entries.
These are the methods for which it is recommended for performance reasons to have a homalgTable entry for non-internal rings. homalg only provides a generic fallback method.
‣ MonomialMatrix( d, R ) | ( operation ) |
Returns: a homalg matrix
The column matrix of d-th monomials of the homalg graded ring R.
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";; gap> S := GradedRing( R );; gap> m := MonomialMatrix( 2, S ); <A ? x 1 matrix over a graded ring> gap> NumberRows( m ); 6 gap> m; <A 6 x 1 matrix over a graded ring> gap> Display( m ); x^2, x*y, x*z, y^2, y*z, z^2 (over a graded ring)
‣ RandomMatrixBetweenGradedFreeLeftModules( degreesS, degreesT, R ) | ( operation ) |
Returns: a homalg matrix
A random \(r \times c \)-matrix between the graded free left modules \(\textit{R}^{(-\textit{degreesS})} \to \textit{R}^{(-\textit{degreesT})}\), where \(r = \)Length\((\)degreesS\()\) and \(c = \)Length\((\)degreesT\()\).
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";; gap> S := GradedRing( R );; gap> rand := RandomMatrixBetweenGradedFreeLeftModules( [ 2, 3, 4 ], [ 1, 2 ], S ); <A 3 x 2 matrix over a graded ring> gap> #Display( rand ); gap> #a-2*b+2*c, 2, gap> #a^2-a*b+b^2-2*b*c+5*c^2, 3*c, gap> #2*a^3-3*a^2*b+2*a*b^2+3*a^2*c+a*b*c-2*b^2*c-3*b*c^2-2*c^3,a^2-4*a*b-3*a*c-c^2
‣ RandomMatrixBetweenGradedFreeRightModules( degreesT, degreesS, R ) | ( operation ) |
Returns: a homalg matrix
A random \(r \times c \)-matrix between the graded free right modules \(\textit{R}^{(-\textit{degreesS})} \to \textit{R}^{(-\textit{degreesT})}\), where \(r = \)Length\((\)degreesT\()\) and \(c = \)Length\((\)degreesS\()\).
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";; gap> S := GradedRing( R );; gap> rand := RandomMatrixBetweenGradedFreeRightModules( [ 1, 2 ], [ 2, 3, 4 ], S ); <A 2 x 3 matrix over a graded ring> gap> #Display( rand ); gap> #a-2*b-c,a*b+b^2-b*c,2*a^3-a*b^2-4*b^3+4*a^2*c-3*a*b*c-b^2*c+a*c^2+5*b*c^2-2*c^3, gap> #-5, -2*a+c, -2*a^2-a*b-2*b^2-3*a*c
‣ Diff( D, N ) | ( operation ) |
Returns: a homalg matrix
If D is a \(f \times p\)-matrix and N is a \(g \times q\)-matrix then \(H=Diff(\)D,N\()\) is an \(fg \times pq\)-matrix whose entry \(H[g*(i-1)+j,q*(k-1)+l]\) is the result of differentiating N\([j,l]\) by the differential operator corresponding to D\([i,k]\). (Here we follow the Macaulay2 convention.)
gap> S := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" * "x,y,z";; gap> D := HomalgMatrix( "[ \ > x,2*y, \ > y,a-b^2, \ > z,y-b \ > ]", 3, 2, S ); <A 3 x 2 matrix over an external ring> gap> N := HomalgMatrix( "[ \ > x^2-a*y^3,x^3-z^2*y,x*y-b,x*z-c, \ > x, x*y, a-b, x*a*b \ > ]", 2, 4, S ); <A 2 x 4 matrix over an external ring> gap> H := Diff( D, N ); <A 6 x 8 matrix over an external ring> gap> Display( H ); 2*x, 3*x^2, y,z, -6*a*y^2,-2*z^2,2*x,0, 1, y, 0,a*b,0, 2*x, 0, 0, -3*a*y^2,-z^2, x,0, -y^3, 0, 0, 0, 0, x, 0,0, 0, 0, 1, b*x, 0, -2*y*z,0,x, -3*a*y^2,-z^2, x+1,0, 0, 0, 0,0, 0, x, 1, -a*x
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