# 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 => , default value Substitution, selects the strategy for internal frobeniusRoot calls
• Outputs:
• , 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

This command computes the maximal $F$-pure Cartier submodule of an ideal $J$ under the dual-$e$-iterated Frobenius induced by $f_1^{t_1}\ldots f_n^{t_n}$.

The function returns a sequence, where the first entry is the descended ideal, and the second entry is the number of times frobeniusRoot was applied (i.e., the HSL number).

 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

The same two examples could also be accomplished via calls of FPureModule, as illustrated below; however, the descendIdeal construction gives the user more direct control.

 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

The option FrobeniusRootStrategy is passed to internal frobeniusRoot calls.