descendIdeal -- finds the maximal F-pure Cartier submodule of an ideal viewed as a Cartier module

Synopsis

• Usage:

  descendIdeal(e, tList, fList, J)

• Inputs:
  • e, an integer, the order of the Frobenius root to take
  • fList, a list, consisting of ring elements f_1,\ldots,f_n, for a pair
  • tList, a list, consisting of formal exponents t_1,\ldots,t_n for the elements of fList
  • J, an ideal, the Cartier module to study
• Optional inputs:
  • FrobeniusRootStrategy => a symbol, default value Substitution, selects the strategy for internal frobeniusRoot calls
• Outputs:
  • a sequence, whose first entry is the maximal $F$-pure Cartier submodule of J under the dual-e-iterated Frobenius induced by f_1^{t_1}\ldots f_n^{t_n}, and the second entry is the number of times frobeniusRoot was applied

Description

i1 : R = ZZ/7[x,y,z];

i2 : f = y^2 - x^3;

i3 : descendIdeal(1, {5}, {f}, ideal 1_R) --this computes the non-F-pure ideal of (R, f^{5/6})

o3 = (ideal 1, 0)

o3 : Sequence

i4 : descendIdeal(2, {41}, {f}, ideal 1_R) --this computes the non-F-pure ideal of (R, f^{41/48})

o4 = (ideal (y, x), 1)

o4 : Sequence

i5 : first FPureModule(5/6, f, CanonicalIdeal => ideal 1_R, GeneratorList => {1_R})

o5 = ideal 1

o5 : Ideal of R

i6 : first FPureModule(41/48, f, CanonicalIdeal => ideal 1_R, GeneratorList => {1_R})

o6 = ideal (y, x)

o6 : Ideal of R

See also

• FPureModule -- compute the submodule of the canonical module stable under the image of the trace of Frobenius
• ascendIdeal -- find the smallest ideal containing a given ideal which is compatible with a given Cartier linear map