All 𝑛-cotilting modules are pure-injective

Šťovíček, Jan

doi:10.1090/S0002-9939-06-08256-6

All $n$-cotilting modules are pure-injective
HTML articles powered by AMS MathViewer

by Jan Šťovíček

Proc. Amer. Math. Soc. 134 (2006), 1891-1897

DOI: https://doi.org/10.1090/S0002-9939-06-08256-6

Published electronically: January 17, 2006

PDF | Request permission

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.

References

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 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 10.1016/S0021-8693(03)00432-0
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
Silvana Bazzoni, $n$-cotilting modules and pure-injectivity, Bull. London Math. Soc. 36 (2004), no. 5, 599–612. MR 2070436, DOI 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 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 10.1006/jabr.1999.8103
Phillip Griffith, On a subfunctor of $\textrm {Ext}$, Arch. Math. (Basel) 21 (1970), 17–22. MR 262356, DOI 10.1007/BF01220870
Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR 933092, DOI 10.1017/CBO9780511600562
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, DOI 10.1007/BFb0094173