All $n$-cotilting modules are pure-injective
Author:
Jan Šťovíček
Journal:
Proc. Amer. Math. Soc. 134 (2006), 1891-1897
MSC (2000):
Primary 16D90; Secondary 16E30, 03E75
DOI:
https://doi.org/10.1090/S0002-9939-06-08256-6
Published electronically:
January 17, 2006
MathSciNet review:
2215116
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Abstract | References | Similar Articles | Additional Information
Abstract: We prove that all $n$-cotilting $R$-modules are pure-injective for any ring $R$ and any $n \ge 0$. To achieve this, we prove that ${^{\perp _1} U}$ is a covering class whenever $U$ is an $R$-module such that ${^{\perp _1} U}$ is closed under products and pure submodules.
- Lidia Angeleri Hügel and Flávio Ulhoa Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math. 13 (2001), no. 2, 239–250. MR 1813669, DOI https://doi.org/10.1515/form.2001.006
- L. Angeleri Hügel, D. Herbera and J.Trlifaj, Tilting modules and Gorenstein rings, to appear in Forum. Math.
- Silvana Bazzoni, A characterization of $n$-cotilting and $n$-tilting modules, J. Algebra 273 (2004), no. 1, 359–372. MR 2032465, DOI https://doi.org/10.1016/S0021-8693%2803%2900432-0
- S. Bazzoni, Cotilting modules are pure-injective, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3665–3672. MR 1998172, DOI https://doi.org/10.1090/S0002-9939-03-06938-7
- Silvana Bazzoni, $n$-cotilting modules and pure-injectivity, Bull. London Math. Soc. 36 (2004), no. 5, 599–612. MR 2070436, DOI https://doi.org/10.1112/S0024609304003352
- S. Bazzoni, Tilting and cotilting modules over Prüfer domains, preprint.
- S. Bazzoni, R. Göbel and L. Strüngmann, Pure injectivity of $n$-cotilting modules: the Prüfer and the countable case, to appear in Archiv der Mathematik.
- Paul C. Eklof and Alan H. Mekler, Almost free modules, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR 1055083
- Paul C. Eklof and Jan Trlifaj, Covers induced by Ext, J. Algebra 231 (2000), no. 2, 640–651. MR 1778163, DOI https://doi.org/10.1006/jabr.2000.8343
- Rüdiger Göbel and Jan Trlifaj, Cotilting and a hierarchy of almost cotorsion groups, J. Algebra 224 (2000), no. 1, 110–122. MR 1736696, DOI https://doi.org/10.1006/jabr.1999.8103
- Phillip Griffith, On a subfunctor of ${\rm Ext}$, Arch. Math. (Basel) 21 (1970), 17–22. MR 262356, DOI https://doi.org/10.1007/BF01220870
- Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR 933092
- Luigi Salce, Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977) Academic Press, London-New York, 1979, pp. 11–32. MR 565595
- J. Šaroch and J. Trlifaj, Completeness of cotorsion pairs, preprint.
- J. Trlifaj, Infinite dimensional tilting modules and cotorsion pairs, to appear in the Handbook of Tilting Theory (London Math. Soc. Lect. Notes).
- Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. MR 1438789
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Additional Information
Jan Šťovíček
Affiliation:
Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
Address at time of publication:
Institutt for Matematiske FAG, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Email:
stovicek@karlin.mff.cuni.cz, stovicek@math.ntnu.no
Received by editor(s):
February 22, 2005
Published electronically:
January 17, 2006
Additional Notes:
This research was supported by a grant of the Industrie Club Duesseldorf and by GAČR 201/05/H005.
Communicated by:
Martin Lorenz
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
