Positive_Characteristic_Berkeley2014

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title: Positive Characteristic Berkeley2014 permalink: wiki/Positive_Characteristic_Berkeley2014/ layout: wiki

Goals

We would like to build upon, clean up, and unify existing positive characteristic tools related to singularities and Frobenius. For example, test ideals, Hilbert-Kunz multiplicity, etc.

Specific target projects

• Parameter test modules and parameter test ideals
• F-injectivity and F-rationality tests
• Test ideals of non-principal ideals
• Integrate more of Katzman's code into the project
• Better isolated graded FPT and F-jumping number computation (Hernandez-Witt?)
• ???

Description of methods and functions implemented: ethRoot(Ideal I,ZZ e):

compute smallest ideal J whose p^eth Frobenius power contains J.

ethRoot(RingElement f, Ideal I, ZZ a, ZZ e):

compute (f^a*I)^{[1/p^e]} in such a way that we don't blow exponent buffers

ethRoot (Ideal I, ZZ m , ZZ e) := (I,m,e)

compute (I^m)^{[1/p^e]} in such a way that we don't blow exponent buffers

ethRoot (Matrix A, ZZ e):

find smallest submodule of free module whose p^eth Frobenius power contains the image of A

minimalCompatible(Ideal I,RingElement u,ZZ e):

find the smallest ideal J containing I which satisfies uJ\subset J^{[p^e]} 

minimalCompatible(Ideal,RingElement,ZZ,ZZ) := (Jk, hk, ak, ek) -> ascendIdealSafe (Jk, hk, ak, ek)

a "safe" version

minimalCompatible(Matrix A,Matrix U,ZZ e):

find the smallest submodule B of a free module containing containing the image of A which satisfies U B\subset B^{[p^e]} 

findAllCompatibleIdeals = (RingElement u):

FIND IDEALS COMPATIBLE WITH A GIVEN NEAR-SPLITTING

    "findGeneratingMorphisms",     --MK
    "findHSLloci",                 --MK
    "canonicalIdeal",
    "frobeniusPower",
   "HSL",
   "nonFInjectiveLocus",   --MK
    "paraTestModule",
    "paraTestModuleAmbient",

--- to be conntinued