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 $R_{\mathfrac p}$

Description

This is the constructor for the type LocalRing.

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

Note that the ideal $P$ is assumed to be prime. Use isWellDefined(LocalRing) to confirm that a local ring is well defined.

The syntax R_f is reserved for localization by inverting a single ring element, but this operation is not currently implemented in this package. Use R_P or localRing(R,P) for localization at a prime ideal P.

Ways to use localRing:

localRing(EngineRing,Ideal)
localRing(Ring,Ideal)

For the programmer

The object localRing is a method function.

The source of this document is in LocalRings/doc.m2:231:0.