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.