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# parameterTestIdeal -- compute the parameter test ideal of a Cohen-Macaulay ring

## Synopsis

• Usage:
parameterTestIdeal(R)
• Inputs:
• R, a ring, a Cohen-Macaulay ring
• Optional inputs:
• FrobeniusRootStrategy => , default value Substitution, selects the strategy for internal frobeniusRoot calls
• Outputs:
• an ideal, the parameter test ideal of R

## Description

This function computes the parameter test ideal of a Cohen-Macaulay ring $R$. Technically, it computes \tau($\omega$) : $\omega$ where $\omega$ is a canonical module of $R$, and \tau($\omega$) is the (parameter) test module, as computed by testModule. For example, the ring $R$ in the following example is $F$-rational, and so its parameter test ideal is the unit ideal.

 i1 : T = ZZ/5[x,y]; i2 : S = ZZ/5[a,b,c,d]; i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3}); o3 : RingMap T <--- S i4 : R = S/(ker g); i5 : parameterTestIdeal(R) o5 = ideal 1 o5 : Ideal of R

Consider now a non-$F$-rational Gorenstein ring, whose test ideal and parameter test ideal coincide.

 i6 : R = ZZ/7[x,y,z]/(x^3 + y^3 + z^3); i7 : parameterTestIdeal(R) o7 = ideal (z, y, x) o7 : Ideal of R i8 : testIdeal(R) o8 = ideal (z, y, x) o8 : Ideal of R