torsionOrder -- computes the local torsion order of an ideal in a normal domain

Usage:

(m,J) = torsionOrder(n, I)
Inputs:
- n, an integer, the maximum order to check
- I, an ideal, the ideal you want to check
Outputs:
- m, an integer, the local torsion order of the ideal in the class group, or 0 if not found
- J, an ideal, the reflexive hull of $I^n$

Description

Given an ideal $I$ in a normal (or G1 + S2) domain, this computes the local torsion order of the class of $I$. In other words, this computes the smallest integer $m$ such that $I^{m}$ is locally principal up to reflexification (that is, $(I^{m})^{**}$ is locally principal).

This function returns a pair $(m, (I^{m})^{**})$ if such an $m$ is found. If no $m$ is found, this returns $(0, ideal 0)$.

i1 : R = QQ[x,y,z]/ideal(x*y+z^7);

i2 : I = ideal(x,z);

o2 : Ideal of R

i3 : torsionOrder(10, I) --the order should be 7

o3 = (7, ideal x)

o3 : Sequence

i4 : torsionOrder(5, I) --we only check up to 5, so this should return 0

o4 = (0, ideal 0)

o4 : Sequence

As we are computing the reflexive hull, this function does not distinguish between the ideal $I$ and the ideal made up of the height-one components of a primary decomposition of $I$.

i5 : R = ZZ/5[a,b,c]/ideal(a*b-c^8)

o5 = R

o5 : QuotientRing

i6 : torsionOrder(10, (ideal(a,b,c))^2*(ideal(a,c)))

o6 = (8, ideal a)

o6 : Sequence

Ways to use torsionOrder:

torsionOrder(ZZ,Ideal)

For the programmer

The object torsionOrder is a method function with options.

The source of this document is in NonPrincipalTestIdeals.m2:1243:0.