Socle degrees, resolutions, and Frobenius powers

We first describe a situation in which every graded Betti number in the tail of the resolution of may be read from the socle degrees of . Then we apply the above result to the ideals J and J[q]; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of R/J[q] are equal to the graded Betti numbers in the tail of a shift of the resolution of R/J.