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# isFJumpingExponentModule -- decides if a rational number is a jumping exponent of a generalized parameter test module

- Usage:
    `b = isFJumpingExponentModule(t, J)`
- Inputs:
  - t, a rational number, the exponent
  - J, an ideal, the ideal of the pair
- Optional inputs:
  - AtOrigin => ..., default value false
  - FrobeniusRootStrategy => ..., default value Substitution
  - Verbose => ..., default value false
- Outputs:
  - b, a Boolean value,

## Description

Given an ideal $J$ in a ring $R$ and a rational number $t$, this function determines if $\tau(\omega_R, J^t) \neq \tau(\omega_R, J^{t-\epsilon})$ for all $1 \gg \epsilon > 0$. If so, it returns true, that is $t$ is a jumping exponent of $\tau(\omega_R, J^*)$, and otherwise it returns false.

    i1 : R = ZZ/3[x,y,z];

    i2 : J = ideal(x^3,y^4,z^5);

    o2 : Ideal of R

    i3 : isFJumpingExponent(1/3+1/4+1/5, J) -- should be true, this is the fpt = lct

    o3 = true

    i4 : isFJumpingExponent(1/3+1/4, J) -- should be false

    o4 = false

The option AtOrigin (default) checks whether the jump happens is done at the origin (as opposed to anywhere).

    i5 : R = ZZ/3[x,y];

    i6 : J = ideal(x-1,y);

    o6 : Ideal of R

    i7 : isFJumpingExponentModule(2, J, AtOrigin=>false)

    o7 = true

    i8 : isFJumpingExponentModule(2, J, AtOrigin=>true)

    o8 = false

The formulation isFJumpingExponent(t, J) also computes whether the test ideal jumps at $t$. However, it only works if $R$ is a quasi-Gorenstein normal domain as in that case the jumping numbers of $\tau(R, J^t)$ agree with those of $\tau(\omega_R, J^t)$.

## See also

- isFJumpingExponent -- whether a given number is an F-jumping exponent
- isFPT(QQ,Ideal)

## Ways to use isFJumpingExponentModule:

- `isFJumpingExponentModule(QQ,Ideal)`
- `isFJumpingExponentModule(ZZ,Ideal)`

## For the programmer

The object isFJumpingExponentModule is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NonPrincipalTestIdeals.m2:1147:0.