Frobenius criteria of freeness and Gorensteinness

摘要

Let F ~n(-) be the Frobenius functor of Peskine and Szpiro. In this note, we show that the maximal Cohen-Macaulayness of F ~n(M) forces M to be free, provided M has a rank. We apply this result to obtain several Frobenius-related criteria for the Gorensteinness of a local ring R, one of which improves a previous characterization due to Hanes and Huneke. We also establish a special class of finite length modules over Cohen-Macaulay rings, which are rigid against Frobenius.