localRing -- Constructor for local rings at prime ideals

• Usage:

  R_P

  localRing(R, P)

• Inputs:
  • R, a polynomial ring, the base ring for the localization
  • P, an ideal, a prime ideal for the localization
  • f, a ring element, a ring element; localization by a single element is not implemented
• Outputs:
  • a local ring, the local ring $R_{\mathfrac p}$

i1 : R = QQ[x,y,z,w];

i2 : P = ideal"xz-y2,yw-z2,xw-yz"; -- The twisted cubic curve

o2 : Ideal of R

i3 : I = ideal"xz-y2,z(yw-z2)-w(xw-yz)";

o3 : Ideal of R

i4 : RP = R_P

o4 = RP

                                  2           2
o4 : LocalRing, maximal ideal (- y  + x*z, - z  + y*w, - y*z + x*w)

i5 : M = RP^1/promote(I, RP)

o5 = cokernel | -y2+xz -z3+2yzw-xw2 |

                              1
o5 : RP-module, quotient of RP

i6 : length M

o6 = 2