Spracherkennung für: .g vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
## <#GAPDoc Label="Intersection">
## <Section Label="Intersection">
## <Heading>Testing the Intersection Formula</Heading>
## We want to check Serre's intersection formula
## <M>i(I_1, I_2; 0)=\sum_i(-1)^i length(Tor^{R_0}_i(R_0/I_1,R_0/I_2))</M>
## on an easy affine example.
## <Example>
## <![CDATA[
## gap> LoadPackage( "RingsForHomalg" );;
## gap> R := HomalgFieldOfRationalsInSingular() * "w,x,y,z";;
## gap> LoadPackage( "LocalizeRingForHomalg" );;
## gap> R0 := LocalizePolynomialRingAtZeroWithMora( R );;
## gap> M1 := HomalgMatrix( "[\
## > (w-x^2)*y, \
## > (w-x^2)*z, \
## > (x-w^2)*y, \
## > (x-w^2)*z \
## > ]", 4, 1, R );;
## gap> M2 := HomalgMatrix( "[\
## > (w-x^2)-y, \
## > (x-w^2)-z \
## > ]", 2, 1, R );;
## gap> LoadPackage( "Modules" );;
## gap> RmodI1 := LeftPresentation( M1 );;
## gap> RmodI2 := LeftPresentation( M2 );;
## gap> T:=Tor( RmodI1, RmodI2 );
## <A graded homology object consisting of 4 left modules at degrees [ 0 .. 3 ]>
## gap> List( ObjectsOfComplex( T ), AffineDegree );
## [ 12, 4, 0, 0 ]
## ]]></Example>
## We read, that the intersection multiplicity is 12-4=8 globally.
## <Example><![CDATA[
## gap> M10 := R0 * M1;
## <A 4 x 1 matrix over a local (Mora) ring>
## gap> M20 := R0 * M2;
## <A 2 x 1 matrix over a local (Mora) ring>
## gap> R0modI10 := LeftPresentation( M10 );;
## gap> R0modI20 := LeftPresentation( M20 );;
## gap> T0 := Tor( R0modI10, R0modI20 );
## <A graded homology object consisting of 4 left modules at degrees [ 0 .. 3 ]>
## gap> List( ObjectsOfComplex( T0 ), AffineDegree );
## [ 3, 1, 0, 0 ]
## ]]></Example>
## The intersection multiplicity at zero is 3-1=2.
## </Section>
## <#/GAPDoc>
LoadPackage( "RingsForHomalg" );;
R := HomalgFieldOfRationalsInSingular() * "w,x,y,z";;
LoadPackage( "LocalizeRingForHomalg" );;
R0 := LocalizePolynomialRingAtZeroWithMora( R );;
M1 := HomalgMatrix( "[\
(w-x^2)*y, \
(w-x^2)*z, \
(x-w^2)*y, \
(x-w^2)*z \
]", 4, 1, R );;
M2 := HomalgMatrix( "[\
(w-x^2)-y, \
(x-w^2)-z \
]", 2, 1, R );;
LoadPackage( "Modules" );;
RmodI1 := LeftPresentation( M1 );;
RmodI2 := LeftPresentation( M2 );;
T:=Tor( RmodI1, RmodI2 );
Assert( 0, List( ObjectsOfComplex( T ), AffineDegree ) = [ 12, 4, 0, 0 ] );
#We read, that the intersection multiplicity is 12-4=8 globally.
M10 := R0 * M1;
M20 := R0 * M2;
R0modI10 := LeftPresentation( M10 );;
R0modI20 := LeftPresentation( M20 );;
T0 := Tor( R0modI10, R0modI20 );
Assert( 0, List( ObjectsOfComplex( T0 ), AffineDegree ) = [ 3, 1, 0, 0 ] );
#The intersection multiplicity at zero is 3-1=2.
