Injective modules under faithfully flat ring extensions
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- by Lars Winther Christensen and Fatih Köksal PDF
- Proc. Amer. Math. Soc. 144 (2016), 1015-1020 Request permission
Abstract:
Let $R$ be a commutative ring and let $S$ be an $R$-algebra. It is well-known that if $N$ is an injective $R$-module, then $\operatorname {Hom}_R(S,N)$ is an injective $S$-module. The converse is not true, not even if $R$ is a commutative noetherian local ring and $S$ is its completion, but it is close: It is a special case of our main theorem that, in this setting, an $R$-module $N$ with $\operatorname {Ext}^{>0}_R(S,N) =0$ is injective if $\operatorname {Hom}_R(S,N)$ is an injective $S$-module.References
- Stephen T. Aldrich, Edgar E. Enochs, and Juan A. Lopez Ramos, Derived functors of Hom relative to flat covers, Math. Nachr. 242 (2002), 17–26. MR 1916846, DOI 10.1002/1522-2616(200207)242:1<17::AID-MANA17>3.0.CO;2-F
- Luchezar L. Avramov and Hans-Bjørn Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), no. 2-3, 129–155. MR 1117631, DOI 10.1016/0022-4049(91)90144-Q
- Dave Benson, Srikanth B. Iyengar, and Henning Krause, Stratifying triangulated categories, J. Topol. 4 (2011), no. 3, 641–666. MR 2832572, DOI 10.1112/jtopol/jtr017
- David J. Benson, Srikanth B. Iyengar, and Henning Krause, Colocalizing subcategories and cosupport, J. Reine Angew. Math. 673 (2012), 161–207. MR 2999131, DOI 10.1515/crelle.2011.180
- Lars Winther Christensen and Sean Sather-Wagstaff, Transfer of Gorenstein dimensions along ring homomorphisms, J. Pure Appl. Algebra 214 (2010), no. 6, 982–989. MR 2580673, DOI 10.1016/j.jpaa.2009.09.007
- Hans-Bjørn Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra 15 (1979), no. 2, 149–172. MR 535182, DOI 10.1016/0022-4049(79)90030-6
- L. Gruson and C. U. Jensen, Dimensions cohomologiques reliées aux foncteurs $\underleftarrow {\mmlToken {mi}{lim}}^{(i)}$, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980) Lecture Notes in Math., vol. 867, Springer, Berlin-New York, 1981, pp. 234–294 (French). MR 633523
- C. U. Jensen, On the vanishing of $\underleftarrow {\mmlToken {mi}{lim}}^{(i)}$, J. Algebra 15 (1970), 151–166. MR 260839, DOI 10.1016/0021-8693(70)90071-2
- T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
- Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 308104, DOI 10.1007/BF01390094
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Additional Information
- Lars Winther Christensen
- Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409
- MR Author ID: 671759
- ORCID: 0000-0002-9360-123X
- Email: lars.w.christensen@ttu.edu
- Fatih Köksal
- Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409
- Address at time of publication: Department of Mathematics and Computer Science, Lewis University, One Univeristy Parkway, Romeoville, Illinois 60446-2200
- Email: koksalfa@lewisu.edu
- Received by editor(s): September 29, 2014
- Received by editor(s) in revised form: March 24, 2015
- Published electronically: July 30, 2015
- Additional Notes: This research was partly supported by a Simons Foundation Collaboration Grant for Mathematicians, award no. 281886, and by grant no. H98230-14-0140 from the National Security Agency.
- Communicated by: Irena Peeva
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1015-1020
- MSC (2010): Primary 13C11; Secondary 13D05
- DOI: https://doi.org/10.1090/proc/12791
- MathSciNet review: 3447655