Flat Mittag-Leffler modules over countable rings
HTML articles powered by AMS MathViewer
- by Silvana Bazzoni and Jan Šťovíček PDF
- Proc. Amer. Math. Soc. 140 (2012), 1527-1533 Request permission
Abstract:
We show that over any ring, the double Ext-orthogonal class to all flat Mittag-Leffler modules contains all countable direct limits of flat Mittag-Leffler modules. If the ring is countable, then the double orthogonal class consists precisely of all flat modules, and we deduce, using a recent result of Šaroch and Trlifaj, that the class of flat Mittag-Leffler modules is not precovering in $\operatorname {Mod}\text {-}{R}$ unless $R$ is right perfect.References
- Lidia Angeleri Hügel and Dolors Herbera, Mittag-Leffler conditions on modules, Indiana Univ. Math. J. 57 (2008), no. 5, 2459–2517. MR 2463975, DOI 10.1512/iumj.2008.57.3325
- S. Bazzoni, Cotilting modules are pure-injective, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3665–3672. MR 1998172, DOI 10.1090/S0002-9939-03-06938-7
- Vladimir Drinfeld, Infinite-dimensional vector bundles in algebraic geometry: an introduction, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 263–304. MR 2181808, DOI 10.1007/0-8176-4467-9_{7}
- Paul C. Eklof and Alan H. Mekler, Almost free modules, Revised edition, North-Holland Mathematical Library, vol. 65, North-Holland Publishing Co., Amsterdam, 2002. Set-theoretic methods. MR 1914985
- S. Estrada, P. A. Guil Asensio, M. Prest and J. Trlifaj, Model category structures arising from Drinfeld vector bundles, preprint, arXiv:0906.5213v1.
- Rüdiger Göbel and Jan Trlifaj, Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, vol. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006. MR 2251271, DOI 10.1515/9783110199727
- Phillip Griffith, On a subfunctor of $\textrm {Ext}$, Arch. Math. (Basel) 21 (1970), 17–22. MR 262356, DOI 10.1007/BF01220870
- D. Herbera and J. Trlifaj, Almost free modules and Mittag–Leffler conditions, preprint, arXiv:0910.4277v1.
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- 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
- J. Šaroch and J. Trlifaj, Kaplansky classes, finite character, and $\aleph _1$-projectivity, to appear in Forum Math., published online, DOI:10.1515/FORM.2011.101
Additional Information
- Silvana Bazzoni
- Affiliation: Dipartimento di Matematica Pura e Applicata, Universitá di Padova, Via Trieste 63, 35121 Padova, Italy
- MR Author ID: 33015
- Email: bazzoni@math.unipd.it
- Jan Šťovíček
- Affiliation: Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovska 83, 186 75 Praha 8, Czech Republic
- Email: stovicek@karlin.mff.cuni.cz
- Received by editor(s): July 28, 2010
- Received by editor(s) in revised form: January 18, 2011
- Published electronically: September 6, 2011
- Additional Notes: The first author was supported by MIUR, PRIN 2007, project “Rings, algebras, modules and categories” and by Università di Padova (Progetto di Ateneo CPDA071244/07 “Algebras and cluster categories”).
The second author was supported by the Eduard Čech Center for Algebra and Geometry (LC505). - Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1527-1533
- MSC (2010): Primary 16D40; Secondary 16E30, 03E75
- DOI: https://doi.org/10.1090/S0002-9939-2011-11070-0
- MathSciNet review: 2869137