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

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

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 NonPrincipalTestIdeals.m2:1147:0.