Socle degrees, Resolutions, and Frobenius powers

We first describe a situation in which every graded Betti number in the tail of the resolution of $\frac{R}{J}$ may be read from the socle degrees of $\frac RJ$\mathrm{.}Thenweapplytheaboveresulttotheideals$J$and$J^{[q]}$; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of $R\mathrm{/}{J}^{[q]}$ are equal to the graded Betti numbers in the tail of a shift of the resolution of $R\mathrm{/}J$.