Abstract
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.
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