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 => a symbol, 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

Ways to use parameterTestIdeal :

parameterTestIdeal(Ring)

For the programmer

The object parameterTestIdeal is a method function with options.

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

Synopsis

Description

See also

Ways to use parameterTestIdeal :

For the programmer