Characterizing Cohen-Macaulay local rings by Frobenius maps

Let R be a commutative noetherian local ring of prime characteristic. Denote by ^eR the ring R regarded as an R-algebra through e-times composition of the Frobenius map. Suppose that R is F-finite, i.e., ¹R is a finitely generated R-module. We prove that R is Cohen-Macaulay if and only if the R-modules ^eR have finite Cohen-Macaulay dimensions for infinitely many integers e.