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The Frobenius functor and injective modules

Author: Thomas Marley
Journal: Proc. Amer. Math. Soc. 142 (2014), 1911-1923
MSC (2010): Primary 13H10; Secondary 13D45
Published electronically: March 3, 2014
MathSciNet review: 3182010
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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.

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Thomas Marley
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130

Keywords: Frobenius map, injective module, canonical module
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.