Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Flat Mittag-Leffler modules over countable rings

Authors: Silvana Bazzoni and Jan Šťovíček
Journal: Proc. Amer. Math. Soc. 140 (2012), 1527-1533
MSC (2010): Primary 16D40; Secondary 16E30, 03E75
Posted: September 6, 2011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $\operatorname{Mod}$ - unless is right perfect.

References

Bibliography

1. Lidia Angeleri Hügel and Dolors Herbera, Mittag-Leffler conditions on modules, Indiana Univ. Math. J. 57 (2008), no. 5, 2459–2517. MR 2463975 (2009i:16002), http://dx.doi.org/10.1512/iumj.2008.57.3325
2. S. Bazzoni, Cotilting modules are pure-injective, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3665–3672 (electronic). MR 1998172 (2004f:16049), http://dx.doi.org/10.1090/S0002-9939-03-06938-7
3. 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 (2007d:14038)
4. 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 (2003e:20002)
5. S. Estrada, P. A. Guil Asensio, M. Prest and J. Trlifaj, Model category structures arising from Drinfeld vector bundles, preprint, arXiv:0906.5213v1.
6. 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 (2007m:16007)
7. Phillip Griffith, On a subfunctor of 𝐸𝑥𝑡, Arch. Math. (Basel) 21 (1970), 17–22. MR 0262356 (41 #6964)
8. D. Herbera and J. Trlifaj, Almost free modules and Mittag-Leffler conditions, preprint, arXiv:0910.4277v1.
9. Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. Pure and Applied Mathematics. MR 506523 (80a:03062)
10. 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 0308104 (46 #7219)
11. 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16D40, 16E30, 03E75

Retrieve articles in all journals with MSC (2010): 16D40, 16E30, 03E75

Additional Information

Silvana Bazzoni
Affiliation: Dipartimento di Matematica Pura e Applicata, Universitá di Padova, Via Trieste 63, 35121 Padova, Italy
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11070-0
PII: S 0002-9939(2011)11070-0
Keywords: Flat Mittag-Leffler modules, precovers, Ext-orthogonal classes
Received by editor(s): July 28, 2010
Received by editor(s) in revised form: January 18, 2011
Posted: 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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|