#### 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)
    ==>
    ==>

User manual for Singular version 4.4.1, 2025, generated by texi2html.