The Frobenius functor and injective modules
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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.References
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Additional Information
- Thomas Marley
- Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130
- MR Author ID: 263869
- Email: tmarley1@unl.edu
- Received by editor(s): January 5, 2012
- Received by editor(s) in revised form: February 28, 2012, June 13, 2012, and July 6, 2012
- Published electronically: March 3, 2014
- Communicated by: Irena Peeva
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1911-1923
- MSC (2010): Primary 13H10; Secondary 13D45
- DOI: https://doi.org/10.1090/S0002-9939-2014-11924-1
- MathSciNet review: 3182010