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D.5.6.6 resolutionInLocalization
Procedure from library graal.lib (see graal_lib).
- Usage:
- resolutionInLocalization(I,L); I ideal, L ideal or graalBearer
- Return:
- the resolution of I*A_L, where
A_L is the localization of the current basering (possibly a quotient ring)
at a prime ideal L.
Example:
| LIB "graal.lib";
ring Q = 0,(x,y,z,w),dp;
ideal circle = (x-1)^2+y^2-3,z;
ideal twistedCubic = xz-y2,yw-z2,xw-yz,z;
ideal I = std(intersect(circle,twistedCubic));
// the resolution is more complicated due to the twisted cubic
res(I,0);
==> 1 4 5 2
==> Q <-- Q <-- Q <-- Q
==>
==> 0 1 2 3
==> resolution not minimized yet
==>
// however if we localize outside of the twisted cubic,
// it should become very easy again.
ideal L = std(I+ideal(x-1));
graalBearer Gr = graalMixed(L); Gr;
==> affine coordinate ring:
==> (QQ),(x,y,z,w),(dp(4),C)
==>
==> ideal defining the subvariety:
==> <z,x-1,y2w-3w,y4-3y2>
==>
==> Al:
==> (0,w),(Y(1),Y(2),Y(3),Y(4),x,y,z),(ds(4),c,dp(3))
==> mod <(w)*y^2+(-3*w)-Y(3),x-1-Y(2),z-Y(1),(3*w)*Y(3)+(-w^2)*Y(4)+Y(3)\
^2>
==> graal:
==> (0,w),(Y(1),Y(2),Y(3),Y(4),z),(c,dp(4),dp(1))
==> mod <3*Y(3)+(-w)*Y(4),z^2+10*z-2>
==> where
==> Y(1) represents generator z
==> Y(2) represents generator x-1
==> Y(3) represents generator y2w-3w
==> Y(4) represents generator y4-3y2
==> and x,y,z in Al are mapped to 1,1/3*z+5/3,0 in Graal
==>
markedResolution mr = resolutionInLocalization(I,Gr);
==> // ** full resolution in a qring may be infinite, setting max length to 5
mr;
==> resolution over Al:
==> 1 2 1
==> Al <-- Al <-- Al
==>
==> 0 1 2
==> resolution not minimized yet
==>
==> k=1
==> Y(1),(w^2)*Y(4)+(3*w^2)*Y(2)^2*x+(3*w)*Y(2)*Y(3)-Y(3)^2
==>
==> k=2
==> _[1,1],
==> -Y(1)
==>
==> resolution over Graal:
==> 1 2 1
==> Graal <-- Graal <-- Graal
==>
==> 0 1 2
==>
==> k=1
==> Y(1),(w^2)*Y(4)
==>
==> k=2
==> (w^2)*Y(4),
==> -Y(1)
==>
==>
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