Frobenius criteria of freeness and Gorensteinness

Details Frobenius criteria of freeness and Gorensteinness Frobenius criteria of freeness and Gorensteinness

2012-04-11

arxiv.org/abs/1204.2456v1

Mathematics

Commutative Algebra

Scientific paper

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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