# localRing -- Constructor for local rings

## Synopsis

• Usage:
R_P
R_f
localRing(R, P)
• Inputs:
• R, , the base ring for the localization
• P, an ideal, a prime ideal for the localization
• f, , a ring element to localize (not yet implemented)
• Outputs:
• , 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.

## Ways to use localRing :

• "localRing(EngineRing,Ideal)"
• "localRing(Ring,Ideal)"

## For the programmer

The object localRing is .