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TestIdeals :: frobeniusPreimage

frobeniusPreimage -- finds the ideal of elements mapped into a given ideal, under all $p^{-e}$-linear maps

Description

Given an ideal Q in a ring R, one frequently considers Ie(Q). This is the ideal of elements x ∈R such that φ(x1/pe) ∈Q for all φ: R1/pe →R. Sometimes this ideal is called the Frobenius pre-image. In a regular ring, it agrees with the frobenius power Q[pe].

i1 : R = ZZ/7[x,y,z]/ideal(x*y-z^2);
i2 : Q = ideal(x, z);

o2 : Ideal of R
i3 : frobeniusPreimage(1, Q)

                3    4
o3 = ideal (0, x z, x )

o3 : Ideal of R

In the previous example I1(Q) agrees with Q(p), the pth symbolic power of Q.

Ways to use frobeniusPreimage :