Abstract
It is proved that a module M over a commutative noetherian ring R is injective if \(\mathrm {Ext}_{R}^{i}((R/{\mathfrak p})_{\mathfrak p},M)=0\) holds for every \(i\geqslant 1\) and every prime ideal \(\mathfrak {p}\) in R. This leads to the following characterization of injective modules: If F is faithfully flat, then a module M such that \({\text {Hom}}_R(F,M)\) is injective and \({\text {Ext}}^i_R(F,M)=0\) for all \(i\geqslant 1\) is injective. A limited version of this characterization is also proved for certain non-noetherian rings.
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Acknowledgments
L. W. C. thanks Fatih Köksal for conversations related to this work; the joint paper [7] provided much inspiration. S. B. I. thanks Dave Benson, Henning Krause, and Julia Pevtsova for discussions related to this work. The statement of Theorem 1.1 emerged out of an on-going collaboration with them. We also thank Tom Marley for pointing out an error in an earlier version of Remark 2.3.
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We thank the Centre de Recerca Matemàtica, Barcelona, for hospitality during visits in Spring 2015, when part of the work reported in this article was done. L. W. C. was partly supported by NSA Grant H98230-14-0140, and S. B. I. was partly supported by NSF Grant DMS-1503044.
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Christensen, L.W., Iyengar, S.B. Tests for injectivity of modules over commutative rings. Collect. Math. 68, 243–250 (2017). https://doi.org/10.1007/s13348-016-0176-0
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DOI: https://doi.org/10.1007/s13348-016-0176-0