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testModuleMinusEpsilon -- computes the (parameter) test module of a pair for values arbitrarily close to, but below, t

Description

This function computes $\tau(\omega_R, J^{t-\epsilon})$, the test module of the pair $(R, J^{t-\epsilon})$ for $1 \gg \epsilon > 0$. It returns a list with two entries. The first is the ideal of $R$ isomorphic viewed as an ideal of $R$ isomorphic to $\tau(\omega_R, J^t)$. The second is an ideal $R$ isomorphic to $\omega_R$, in which the first module sits as a subideal.

i1 : R = ZZ/5[x,y,z];
 
i2 : J = ideal(x^2,y^2,z^2);

o2 : Ideal of R
 
i3 : testModuleMinusEpsilon(3/2,J)

o3 = (ideal (0, 1, y*z, x*z, x*y), ideal 1)

o3 : Sequence
 
i4 : apply(oo, JJ -> trim JJ)

o4 = (ideal 1, ideal 1)

o4 : Sequence
 
i5 : testModule(3/2, J)

o5 = (ideal (z, y, x), ideal 1)

o5 : Sequence

In the above example, one can see that 3/2 is the F-pure threshold of the given ideal $J$, as expected.

Caveat

There is not a guarantee different calls to the function will always produce the same ambient canonical module.

See also

Ways to use testModuleMinusEpsilon:

  • testModuleMinusEpsilon(QQ,Ideal)
  • testModuleMinusEpsilon(ZZ,Ideal)

For the programmer

The object testModuleMinusEpsilon is a method function with options.

The source of this document is in NonPrincipalTestIdeals.m2:1104:0.