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## <#GAPDoc Label="ResidueClass">
## <Section Label="ResidueClass">
## <Heading>ResidueClass</Heading>
## We want to show, how localization can work together with residue class rings.
## <Example><![CDATA[
## gap> LoadPackage( "RingsForHomalg", ">= 2020.04.17" );;
## gap> Qxy := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y";
## Q[x,y]
## gap> wmat := HomalgMatrix(
## > "[ y^3-y^2 , x^3-x^2 , y^3+y^2 , x^3+x^2 ]",
## > 2, 2, Qxy );
## <A 2 x 2 matrix over an external ring>
## gap> ec := HomalgRingElement( "-x^3-x^2+2*y^2", Qxy );
## -x^3-x^2+2*y^2
## ]]></Example>
## Compute globally:
## <Example><![CDATA[
## gap> LoadPackage( "Modules" );;
## gap> W := LeftPresentation( wmat );
## <A left module presented by 2 relations for 2 generators>
## gap> Res := Resolution( 2 , W );
## <A right acyclic complex containing 2 morphisms of left modules at degrees
## [ 0 .. 2 ]>
## gap> Display( Res );
## -------------------------
## at homology degree: 2
## 0
## -------------------------
## (an empty 0 x 2 matrix)
##
## the map is currently represented by the above 0 x 2 matrix
## ------------v------------
## at homology degree: 1
## Q[x,y]^(1 x 2)
## -------------------------
## y^2, x^2,
## x*y^2-y^3,0
##
## the map is currently represented by the above 2 x 2 matrix
## ------------v------------
## at homology degree: 0
## Q[x,y]^(1 x 2)
## -------------------------
## ]]></Example>
## Try a localization of a residue class ring:
## <Example><![CDATA[
## gap> R1 := Qxy / ec;
## Q[x,y]/( -x^3-x^2+2*y^2 )
## gap> Display( R1 );
## <A residue class ring>
## gap> wmat1 := R1 * wmat;
## <A 2 x 2 matrix over a residue class ring>
## gap> LoadPackage( "LocalizeRingForHomalg" );;
## gap> R10 := LocalizeAt( R1 ,
## > [ HomalgRingElement( "x", R1 ),
## > HomalgRingElement( "y", R1 ) ]
## > );
## Q[x,y]/( x^3+x^2-2*y^2 )_< |[ x ]|, |[ y ]| >
## gap> Display( R10 );
## <A local ring>
## gap> wmat10 := HomalgLocalMatrix( wmat, R10 );
## <A 2 x 2 matrix over a local ring>
## gap> W10 := LeftPresentation( wmat10 );
## <A left module presented by 2 relations for 2 generators>
## gap> Res10 := Resolution( 2 , W10 );
## <A right acyclic complex containing 2 morphisms of left modules at degrees
## [ 0 .. 2 ]>
## gap> Display( Res10 );
## -------------------------
## at homology degree: 2
## 0
## -------------------------
## (an empty 0 x 2 matrix)
##
## the map is currently represented by the above 0 x 2 matrix
## ------------v------------
## at homology degree: 1
## Q[x,y]/( x^3+x^2-2*y^2 )_< |[ x ]|, |[ y ]| >^(1 x 2)
## -------------------------
## x*y^2+y^2,2*y^2,
## y^2, y^4-2*y^3+2*y^2
##
## modulo [ x^3+x^2-2*y^2 ]
## / |[ 1 ]|
##
## the map is currently represented by the above 2 x 2 matrix
## ------------v------------
## at homology degree: 0
## Q[x,y]/( x^3+x^2-2*y^2 )_< |[ x ]|, |[ y ]| >^(1 x 2)
## -------------------------
## ]]></Example>
## Try a residue class ring of a localization:
## <Example><![CDATA[
## gap> R0 := LocalizeAtZero( Qxy );
## Q[x,y]_< x, y >
## gap> Display( R0 );
## <A local ring>
## gap> wmat0 := R0 * wmat;
## <A 2 x 2 matrix over a local ring>
## gap> R01 := R0 / ( ec / R0 );
## Q[x,y]_< x, y >/( (-x^3-x^2+2*y^2)/1 )
## gap> Display( R01 );
## <A residue class ring>
## gap> wmat01 := R01 * wmat0;
## <A 2 x 2 matrix over a residue class ring>
## gap> W01 := LeftPresentation( wmat01 );
## <A left module presented by 2 relations for 2 generators>
## gap> Res01 := Resolution( 2 , W01 );
## <A right acyclic complex containing 2 morphisms of left modules at degrees
## [ 0 .. 2 ]>
## gap> Display( Res01 );
## -------------------------
## at homology degree: 2
## 0
## -------------------------
## (an empty 0 x 2 matrix)
##
## the map is currently represented by the above 0 x 2 matrix
## ------------v------------
## at homology degree: 1
## Q[x,y]_< x, y >/( (x^3+x^2-2*y^2)/1 )^(1 x 2)
## -------------------------
## y^3+y^2,2*y^2,
## 0, x*y^2-y^3
## / 1
##
## modulo [ (x^3+x^2-2*y^2)/1 ]
##
## the map is currently represented by the above 2 x 2 matrix
## ------------v------------
## at homology degree: 0
## Q[x,y]_< x, y >/( (x^3+x^2-2*y^2)/1 )^(1 x 2)
## -------------------------
## ]]></Example>
## </Section>
## <#/GAPDoc>
LoadPackage( "RingsForHomalg", ">= 2020.04.17" );;
Qxy := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y";;
wmat := HomalgMatrix(
"[ y^3-y^2 , x^3-x^2 , y^3+y^2 , x^3+x^2 ]",
2, 2, Qxy );
ec := HomalgRingElement( "-x^3-x^2+2*y^2", Qxy );
LoadPackage( "Modules" );
#Compute globally:
W := LeftPresentation( wmat );
Res := Resolution( 2 , W );
Display(Res);
#Try a localization of a residue class ring:
R1 := Qxy / ec;
wmat1 := R1 * wmat;
LoadPackage( "LocalizeRingForHomalg" );;
R10 := LocalizeAt( R1 ,
[ HomalgRingElement( "x", R1 ),
HomalgRingElement( "y", R1 ) ]
);
wmat10 := HomalgLocalMatrix( wmat, R10 );
W10 := LeftPresentation( wmat10 );
Res10 := Resolution( 2 , W10 );
Display(Res10);
#Try a residue class ring of a localization:
R0 := LocalizeAtZero( Qxy );
wmat0 := R0 * wmat;
R01 := R0 / ( ec / R0 );
wmat01 := R01 * wmat0;
W01 := LeftPresentation( wmat01 );
Res01 := Resolution( 2 , W01 );
Display(Res01);
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