The Frobenius functor and injective modules

Marley, Thomas

doi:10.1090/S0002-9939-2014-11924-1

The Frobenius functor and injective modules
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Proc. Amer. Math. Soc. 142 (2014), 1911-1923 Request permission

Abstract:

We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show that it includes all one-dimensional $F$-pure rings. We also give a characterization of Gorenstein local rings in terms of $\mathrm {Tor}_i^R(R^{f},E)$, where $E$ is the injective hull of the residue field and $R^{f}$ is the ring $R$ whose right $R$-module action is given by the Frobenius map.

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