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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
## MatrixOverGradedRing.gi GradedRingForHomalg package
##
## Copyright 2009-2010, Mohamed Barakat, University of Kaiserslautern
## Markus Lange-Hegermann, RWTH-Aachen University
##
## Implementations for matrices over graded rings.
##
#############################################################################
####################################
#
# global variables:
#
####################################
if IsBound( HOMALG_MATRICES ) and IsBound( HOMALG_MATRICES.colored_info ) then
HOMALG_MATRICES.colored_info.LinearSyzygiesGeneratorsOfRows := [ 2, HOMALG_MATRICES.colors.BOH ];
HOMALG_MATRICES.colored_info.LinearSyzygiesGeneratorsOfColumns := [ 2, HOMALG_MATRICES.colors.BOH ];
fi;
####################################
#
# representations:
#
####################################
## <#GAPDoc Label="IsHomalgMatrixOverGradedRingRep">
## <ManSection>
## <Filt Type="Representation" Arg="A" Name="IsHomalgMatrixOverGradedRingRep"/>
## <Returns>true or false</Returns>
## <Description>
## The representation of &homalg; matrices with entries in a &homalg; graded ring. <P/>
## (It is a representation of the &GAP; category <C>IsMatrixOverGradedRing</C>.)
## <Listing Type="Code"><![CDATA[
DeclareRepresentation( "IsHomalgMatrixOverGradedRingRep",
IsMatrixOverGradedRing,
[ ] );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
####################################
#
# families and types:
#
####################################
BindGlobal( "TheTypeHomalgMatrixOverGradedRing",
NewType( TheFamilyOfHomalgMatrices,
IsHomalgMatrixOverGradedRingRep ) );
####################################
#
# methods for operations:
#
####################################
##
InstallMethod( UnderlyingMatrixOverNonGradedRing,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep and IsEmptyMatrix ],
function( A )
local B;
if not HasEval( A ) then
B := HomalgZeroMatrix( NrRows( A ), NrColumns( A ), UnderlyingNonGradedRing( HomalgRing( A ) ) );
SetEval( A, B );
fi;
return Eval( A );
end );
##
InstallMethod( UnderlyingMatrixOverNonGradedRing,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep ],
Eval );
## <#GAPDoc Label="UnderlyingNonGradedRing:matrix">
## <ManSection>
## <Oper Arg="mat" Name="UnderlyingNonGradedRing" Label="for matrices over graded rings"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## The nongraded ring underlying <C>HomalgRing</C>(<A>mat</A>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( UnderlyingNonGradedRing,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep ],
function( A )
return UnderlyingNonGradedRing( HomalgRing(A) );
end );
##
InstallMethod( BlindlyCopyMatrixPropertiesToMatrixOverGradedRing, ## under construction
"for matrices over graded rings",
[ IsHomalgMatrix, IsHomalgMatrixOverGradedRingRep ],
function( S, T )
local c;
for c in [ NrRows, NrColumns ] do
if Tester( c )( S ) then
Setter( c )( T, c( S ) );
fi;
od;
for c in [ IsZero, IsOne, IsDiagonalMatrix ] do
if Tester( c )( S ) and c( S ) then
Setter( c )( T, c( S ) );
fi;
od;
end );
## <#GAPDoc Label="SetMatElm">
## <ManSection>
## <Oper Arg="mat, i, j, r, R" Name="SetMatElm" Label="for matrices over graded rings"/>
## <Description>
## Changes the entry (<A>i,j</A>) of the matrix <A>mat</A> to the value <A>r</A>. Here <A>R</A> is the graded &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( SetMatElm,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep and IsMutable, IsPosInt, IsPosInt, IsHomalgGradedRingElementRep, IsHomalgGradedRingRep ],
function( M, r, c, s, R )
SetMatElm( UnderlyingMatrixOverNonGradedRing( M ), r, c, UnderlyingNonGradedRingElement( s ), UnderlyingNonGradedRing( R ) );
end );
## <#GAPDoc Label="AddToMatElm">
## <ManSection>
## <Oper Arg="mat, i, j, r, R" Name="AddToMatElm" Label="for matrices over graded rings"/>
## <Description>
## Changes the entry (<A>i,j</A>) of the matrix <A>mat</A> by adding the value <A>r</A> to it. Here <A>R</A> is the (graded) &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AddToMatElm,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep and IsMutable, IsPosInt, IsPosInt, IsHomalgGradedRingElementRep, IsHomalgGradedRingRep ],
function( M, r, c, s, R )
AddToMatElm( UnderlyingMatrixOverNonGradedRing( M ), r, c, UnderlyingNonGradedRingElement( s ), UnderlyingNonGradedRing( R ) );
end );
## <#GAPDoc Label="MatElmAsString">
## <ManSection>
## <Oper Arg="mat, i, j, R" Name="MatElmAsString" Label="for matrices over graded rings"/>
## <Returns>a string</Returns>
## <Description>
## Returns the entry (<A>i,j</A>) of the matrix <A>mat</A> as a string. Here <A>R</A> is the (graded) &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( MatElmAsString,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep, IsPosInt, IsPosInt, IsHomalgGradedRingRep ],
function( M, r, c, R )
return MatElmAsString( UnderlyingMatrixOverNonGradedRing( M ), r, c, UnderlyingNonGradedRing( R ) );
end );
## <#GAPDoc Label="MatElm">
## <ManSection>
## <Oper Arg="mat, i, j, R" Name="MatElm" Label="for matrices over graded rings"/>
## <Returns>a graded ring element</Returns>
## <Description>
## Returns the entry (<A>i,j</A>) of the matrix <A>mat</A>. Here <A>R</A> is the (graded) &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( MatElm,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep, IsPosInt, IsPosInt, IsHomalgGradedRingRep ],
function( M, r, c, R )
return GradedRingElement( MatElm( UnderlyingMatrixOverNonGradedRing( M ), r, c, UnderlyingNonGradedRing( R ) ), R );
end );
##
InstallMethod( SaveHomalgMatrixToFile,
"for matrices over graded rings",
[ IsString, IsHomalgMatrixOverGradedRingRep, IsHomalgGradedRingRep ],
function( filename, M, R )
return SaveHomalgMatrixToFile( filename, UnderlyingMatrixOverNonGradedRing( M ), UnderlyingNonGradedRing( R ) );
end );
##
InstallMethod( LoadHomalgMatrixFromFile,
"for matrices over graded rings",
[ IsString, IsInt, IsInt, IsHomalgGradedRingRep ],
function( filename, r, c, R )
return MatrixOverGradedRing( LoadHomalgMatrixFromFile( filename, r, c, UnderlyingNonGradedRing( R ) ), R );
end );
##
InstallMethod( MonomialMatrixWeighted,
"for homalg rings",
[ IsInt, IsHomalgRing and IsHomalgResidueClassRingRep, IsList ],
function( d, R, weights )
return R * MonomialMatrixWeighted( d, AmbientRing( R ), weights );
end );
## <#GAPDoc Label="MonomialMatrix">
## <ManSection>
## <Oper Arg="d, R" Name="MonomialMatrix"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The column matrix of <A>d</A>-th monomials of the &homalg; graded ring <A>R</A>.
## <Example><![CDATA[
## gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
## gap> S := GradedRing( R );;
## gap> m := MonomialMatrix( 2, S );
## <A ? x 1 matrix over a graded ring>
## gap> NrRows( 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)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( MonomialMatrix,
"for homalg rings",
[ IsInt, IsHomalgGradedRing ],
function( d, S )
local weights, weightlist;
weights := WeightsOfIndeterminates( S );
if IsHomalgElement( weights[ 1 ] ) then
weightlist := List( weights, UnderlyingListOfRingElements );
if Length( weightlist[ 1 ] ) = 1 then
weightlist := List( weightlist, i -> i[ 1 ] );
fi;
fi;
return MatrixOverGradedRing(
MonomialMatrixWeighted(
d, UnderlyingNonGradedRing( S ), weightlist ),
S );
end );
##
InstallMethod( MonomialMatrix,
"for homalg rings",
[ IsList, IsHomalgGradedRing ],
function( d, S )
local weights, weightlist;
weights := WeightsOfIndeterminates( S );
if IsHomalgElement( weights[ 1 ] ) then
weightlist := List( weights, UnderlyingListOfRingElements );
if Length( weightlist[ 1 ] ) = 1 then
weightlist := List( weightlist, i -> i[ 1 ] );
fi;
fi;
return MatrixOverGradedRing(
MonomialMatrixWeighted(
d, UnderlyingNonGradedRing( S ), weightlist ),
S );
end );
## <#GAPDoc Label="RandomMatrixBetweenGradedFreeLeftModules">
## <ManSection>
## <Oper Arg="degreesS,degreesT,R" Name="RandomMatrixBetweenGradedFreeLeftModules"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A random <M>r \times c </M>-matrix between the graded free <E>left</E> modules
## <M><A>R</A>^{(-<A>degreesS</A>)} \to <A>R</A>^{(-<A>degreesT</A>)}</M>,
## where <M>r = </M><C>Length</C><M>(</M><A>degreesS</A><M>)</M> and
## <M>c = </M><C>Length</C><M>(</M><A>degreesT</A><M>)</M>.
## <Example><![CDATA[
## 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
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( RandomMatrixBetweenGradedFreeLeftModules,
"for homalg graded rings",
[ IsList, IsList, IsHomalgGradedRingRep ],
function( degreesS, degreesT, S )
local weights, weightlist;
weights := WeightsOfIndeterminates( S );
if IsHomalgElement( weights[ 1 ] ) then
weightlist := List( weights, UnderlyingListOfRingElements );
if Length( weightlist[ 1 ] ) = 1 then
weightlist := List( weightlist, i -> i[ 1 ] );
fi;
fi;
if ForAny( degreesS, IsHomalgElement ) then
degreesS := List( degreesS, function( i )
if IsHomalgElement( i ) then
return UnderlyingListOfRingElements( i );
else
return i;
fi;
end );
if ForAny( degreesS, IsList ) then
degreesS := List( degreesS, function( i )
if IsList( i ) and Length( i ) = 1 then
return i[ 1 ];
else
return i;
fi;
end );
fi;
fi;
if IsHomalgElement( degreesT[ 1 ] ) then
degreesT := List( degreesT, function( i )
if IsHomalgElement( i ) then
return UnderlyingListOfRingElements( i );
else
return i;
fi;
end );
if ForAny( degreesT, IsList ) then
degreesT := List( degreesT, function( i )
if IsList( i ) and Length( i ) = 1 then
return i[ 1 ];
else
return i;
fi;
end );
fi;
fi;
return MatrixOverGradedRing(
RandomMatrixBetweenGradedFreeLeftModulesWeighted(
degreesS, degreesT,
UnderlyingNonGradedRing( S ), weightlist ),
S );
end );
## <#GAPDoc Label="RandomMatrixBetweenGradedFreeRightModules">
## <ManSection>
## <Oper Arg="degreesT,degreesS,R" Name="RandomMatrixBetweenGradedFreeRightModules"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A random <M>r \times c </M>-matrix between the graded free <E>right</E> modules
## <M><A>R</A>^{(-<A>degreesS</A>)} \to <A>R</A>^{(-<A>degreesT</A>)}</M>,
## where <M>r = </M><C>Length</C><M>(</M><A>degreesT</A><M>)</M> and
## <M>c = </M><C>Length</C><M>(</M><A>degreesS</A><M>)</M>.
## <Example><![CDATA[
## 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
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( RandomMatrixBetweenGradedFreeRightModules,
"for homalg graded rings",
[ IsList, IsList, IsHomalgGradedRingRep ],
function( degreesS, degreesT, S )
local weights, weightlist;
weights := WeightsOfIndeterminates( S );
if IsHomalgElement( weights[ 1 ] ) then
weightlist := List( weights, UnderlyingListOfRingElements );
if Length( weightlist[ 1 ] ) = 1 then
weightlist := List( weightlist, i -> i[ 1 ] );
fi;
fi;
if IsHomalgElement( degreesS[ 1 ] ) then
degreesS := List( degreesS, UnderlyingListOfRingElements );
if Length( degreesS[ 1 ] ) = 1 then
degreesS := List( degreesS, i -> i[ 1 ] );
fi;
fi;
if IsHomalgElement( degreesT[ 1 ] ) then
degreesT := List( degreesT, UnderlyingListOfRingElements );
if Length( degreesT[ 1 ] ) = 1 then
degreesT := List( degreesT, i -> i[ 1 ] );
fi;
fi;
return MatrixOverGradedRing(
RandomMatrixBetweenGradedFreeRightModulesWeighted(
degreesS, degreesT,
UnderlyingNonGradedRing( S ), weightlist ),
S );
end );
##
InstallMethod( DegreesOfEntries,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgGradedRing ],
function( C, S )
if IsZero( C ) then
return ListWithIdenticalEntries( NrRows( C ),
ListWithIdenticalEntries( NrColumns( C ),
DegreeOfRingElement( Zero( S ) ) ) );
fi;
return DegreesOfEntriesFunction( S )( C );
end );
##
InstallMethod( NonTrivialDegreePerRow,
"for a homalg matrix and a graded ring",
[ IsHomalgMatrix, IsHomalgGradedRing ],
function( C, S )
local degrees;
if IsOne( C ) then
return ListWithIdenticalEntries( NrRows( C ), DegreeOfRingElement( One( S ) ) );
elif IsZero( C ) then
return ListWithIdenticalEntries( NrRows( C ), DegreeOfRingElement( One( S ) ) ); ## One( S ) is not a mistake
fi;
## CASHING ##
if IsBound( C!.NonTrivialDegreePerRow ) then
degrees := _ElmPObj_ForHomalg( C!.NonTrivialDegreePerRow, S, fail );
if degrees <> fail then
return degrees;
fi;
else
C!.NonTrivialDegreePerRow :=
ContainerForPointers(
TheTypeContainerForPointersOnComputedValues,
[ "operation", "NonTrivialDegreePerRow" ] );
fi;
## ENDCASHING ##
degrees := NonTrivialDegreePerRowWithColPositionFunction( S )( C );
## CASHING ##
if not IsMutable( C ) then
_AddTwoElmPObj_ForHomalg( C!.NonTrivialDegreePerRow, S, degrees );
fi;
## ENDCASHING ##
return degrees;
end );
##
InstallMethod( NonTrivialDegreePerRow,
"for a homalg matrix and a graded ring",
[ IsHomalgMatrix, IsHomalgGradedRing, IsList ],
function( C, S, col_degrees )
local degs, col_pos, f;
if Length( col_degrees ) <> NrColumns( C ) then
Error( "the number of entries in the list of column degrees does not match the number of columns of the matrix\n" );
fi;
if IsOne( C ) then
return col_degrees;
elif IsEmptyMatrix( C ) then
return ListWithIdenticalEntries( NrRows( C ), DegreeOfRingElement( One( S ) ) ); ## One( S ) is not a mistake
elif IsZero( C ) then
return ListWithIdenticalEntries( NrRows( C ), col_degrees[1] ); ## this is not a mistake
fi;
degs := NonTrivialDegreePerRow( C, S );
col_pos := PositionOfFirstNonZeroEntryPerRow( C );
f := function( i )
local c;
c := col_pos[i];
if c = 0 then
return col_degrees[1];
else
return degs[i] + col_degrees[c];
fi;
end;
return List( [ 1 .. NrRows( C ) ], f );
end );
##
InstallMethod( NonTrivialDegreePerColumn,
"for a homalg matrix and a graded ring",
[ IsHomalgMatrix, IsHomalgGradedRing ],
function( C, S )
local degrees;
if IsOne( C ) then
return ListWithIdenticalEntries( NrColumns( C ), DegreeOfRingElement( One( S ) ) );
elif IsZero( C ) then
return ListWithIdenticalEntries( NrColumns( C ), DegreeOfRingElement( One( S ) ) ); ## One( S ) is not a mistake
fi;
if IsBound( C!.NonTrivialDegreePerColumn ) then
degrees := _ElmPObj_ForHomalg( C!.NonTrivialDegreePerColumn, S, fail );
if degrees <> fail then
return degrees;
fi;
else
C!.NonTrivialDegreePerColumn :=
ContainerForPointers(
TheTypeContainerForPointersOnComputedValues,
[ "operation", "NonTrivialDegreePerColumn" ] );
fi;
degrees := NonTrivialDegreePerColumnWithRowPositionFunction( S )( C );
if not IsMutable( C ) then
_AddTwoElmPObj_ForHomalg( C!.NonTrivialDegreePerColumn, S, degrees );
fi;
return degrees;
end );
##
InstallMethod( NonTrivialDegreePerColumn,
"for a homalg matrix and a graded ring",
[ IsHomalgMatrix, IsHomalgGradedRing, IsList ],
function( C, S, row_degrees )
local degs, row_pos, f;
if Length( row_degrees ) <> NrRows( C ) then
Error( "the number of entries in the list of row degrees does not match the number of rows of the matrix\n" );
fi;
if IsOne( C ) then
return row_degrees;
elif IsEmptyMatrix( C ) then
return ListWithIdenticalEntries( NrColumns( C ), DegreeOfRingElement( One( S ) ) ); ## One( S ) is not a mistake
elif IsZero( C ) then
return ListWithIdenticalEntries( NrColumns( C ), row_degrees[1] ); ## this is not a mistake
fi;
degs := NonTrivialDegreePerColumn( C, S );
row_pos := PositionOfFirstNonZeroEntryPerColumn( C );
f := function( j )
local r;
r := row_pos[j];
if r = 0 then
return row_degrees[1];
else
return degs[j] + row_degrees[r];
fi;
end;
return List( [ 1 .. NrColumns( C ) ], f );
end );
####################################
#
# constructor functions and methods:
#
####################################
## <#GAPDoc Label="MatrixOverGradedRing">
## <ManSection>
## <Func Arg="mat, S" Name="MatrixOverGradedRing" Label="constructor for matrices over graded rings"/>
## <Returns>a matrix over a graded ring</Returns>
## <Description>
## Creates a matrix for the graded ring <A>S</A>, where <A>mat</A> is a matrix over <C>UnderlyingNonGradedRing</C>(<A>S</A>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( MatrixOverGradedRing,
"constructor for matrices over graded rings",
[ IsHomalgMatrix, IsHomalgGradedRingRep ],
function( A, R )
local G, type, matrix, ComputationRing, rr, AA;
if IsHomalgMatrixOverGradedRingRep( A ) then
return A;
fi;
G := HomalgRing( A );
ComputationRing := UnderlyingNonGradedRing( R );
if not IsIdenticalObj( ComputationRing , G ) then
Error( "underlying rings do not match" );
fi;
matrix := rec(
ring := R,
);
ObjectifyWithAttributes(
matrix, TheTypeHomalgMatrixOverGradedRing,
Eval, A
);
BlindlyCopyMatrixPropertiesToMatrixOverGradedRing( A, matrix );
return matrix;
end );
InstallMethod( MatrixOverGradedRing,
"constructor for matrices over graded rings",
[ IsList, IsInt, IsInt, IsHomalgGradedRingRep ],
function( A, r, c, R )
return MatrixOverGradedRing( HomalgMatrix( A, r, c, UnderlyingNonGradedRing( R ) ), R );
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgGradedRingRep, IsHomalgMatrix ],
function( R, m )
return MatrixOverGradedRing( UnderlyingNonGradedRing( R ) * m, R );
end );
##
InstallMethod( \*,
"for matrices over graded rings",
[ IsHomalgRing, IsHomalgMatrixOverGradedRingRep ],
function( R, m )
return R * UnderlyingMatrixOverNonGradedRing( m );
end );
##
InstallMethod( PostMakeImmutable,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep and HasEval ],
function( A )
MakeImmutable( UnderlyingMatrixOverNonGradedRing( A ) );
end );
##
InstallMethod( SetIsMutableMatrix,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep, IsBool ],
function( A, b )
if b = true then
SetFilterObj( A, IsMutable );
else
ResetFilterObj( A, IsMutable );
fi;
SetIsMutableMatrix( UnderlyingMatrixOverNonGradedRing( A ), b );
end );
##
InstallMethod( SaveHomalgMatrixToFile,
"for matrices over graded rings",
[ IsString, IsHomalgMatrixOverGradedRingRep ],
function( filename, M )
return SaveHomalgMatrixToFile( filename, UnderlyingMatrixOverNonGradedRing( M ) );
end );
##
InstallMethod( LoadHomalgMatrixFromFile,
"for a string and a graded ring",
[ IsString, IsHomalgGradedRingRep ],
function( filename, S )
return LoadHomalgMatrixFromFile( filename, UnderlyingNonGradedRing( S ) );
end );
##
InstallMethod( LoadHomalgMatrixFromFile,
"for a string, two integers, and a graded ring",
[ IsString, IsInt, IsInt, IsHomalgGradedRingRep ],
function( filename, r, c, S )
return LoadHomalgMatrixFromFile( filename, r, c, UnderlyingNonGradedRing( S ) );
end );
####################################
#
# View, Print, and Display methods:
#
####################################
##
InstallMethod( Display,
"for matrices over graded rings",
[ IsHomalgMatrixOverGradedRingRep ],
function( A )
Display( UnderlyingMatrixOverNonGradedRing( A ) );
Print( "(over a graded ring)\n" );
end );