findFrobeniusLiftConstraints -- Finds the equations satisfied by a delta structure on a ring R

Usage:

J = findFrobeniusLiftConstraints I

J = findFrobeniusLiftConstraints R
Inputs:
- I, an ideal,
- f, a ring element,
- R, a quotient ring,
Optional inputs:
- Homogeneous => ..., default value false
- PerturbationTerm => ..., default value null
Outputs:
- J, an ideal,

Description

Given an ideal I in a characteristic-p polynomial ring S = (ZZ/p)[x_1..x_n], or a quotient ring R = S/I or a generator f of I, this method returns an ideal J in (S/I)[aa_1,...,aa_n]. The generators for J give the equations satisfied by the values of delta(x_i)=aa_i for the resulting Frobenius lift to descend from W_2(k)[x_1..x_n] to W_2(k)[x_1..x_n]/I.

i1 : S = (ZZ/2)[x,y]

o1 = S

o1 : PolynomialRing

i2 : I = ideal(x*y)

o2 = ideal(x*y)

o2 : Ideal of S

i3 : J = findFrobeniusLiftConstraints I

               2       2
o3 = ideal(aa y  + aa x )
             0       1

              ZZ
              --[aa ..aa , x..y]
               2   0    1
o3 : Ideal of ------------------
                      x*y

i4 : J = findFrobeniusLiftConstraints (S/I)

o4 = ideal 0

              ZZ
              --[aa ..aa , x..y]
               2   0    1
o4 : Ideal of ------------------
                      x*y

If the user wants to lift the Frobenius to a different lifting of I to W_2(k)[x_1..x_n], one can use the PerturbationTerm option to specify the coefficients of p in the lift of the defining equations.

i5 : S = (ZZ/2)[x,y,z]

o5 = S

o5 : PolynomialRing

i6 : I = ideal(x*y,z)

o6 = ideal (x*y, z)

o6 : Ideal of S

i7 : J = findFrobeniusLiftConstraints(I,PerturbationTerm=>{1,0})

                     2       2
o7 = ideal (aa , aa y  + aa x  + 1)
              2    0       1

              ZZ
              --[aa ..aa , x..z]
               2   0    2
o7 : Ideal of ------------------
                   (x*y, z)

Ways to use findFrobeniusLiftConstraints:

findFrobeniusLiftConstraints(Ideal) (missing documentation)
findFrobeniusLiftConstraints(Ring) (missing documentation)
findFrobeniusLiftConstraints(RingElement) (missing documentation)

For the programmer

The object findFrobeniusLiftConstraints is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/WittVectors/Documentation.m2:1064:0.