Relative T-injective modules and relative T-flat modules

Abstract

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n(T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.

Keywords

Wakamatsu tilting module / (n, T)-Copure injective module / (n, T)-Copure flat module / T-Projective dimension / T-Injective dimension

References

[1] Angleri-Hugel L., Coelho F. U.. Infinitely generated tilting modules of finite projective dimension. Forum Math., 2001, 13: 239-250

[2] Bazzoni S.. A characterization of n-cotilting and n-tilting modules. J. Algebra, 2005, 273: 359-372

[3] Enochs E. E., Jenda O. M. G.. Relative Homological Algebra, 2000, Berlin, New York: Walter de Gruyter

[4] Green E. L., Reiten I., Solberg O.. Dualities on Generalized Koszul Algebras. Mem. Amer. Math. Soc., 2002, Providence, RI: A. M. S. 754

[5] Nikmehr M. J., Shaveisi F., Nikandish R.. n-Projective modules. Algebras Groups Geom., 2007, 24: 447-454

[6] Nikmehr M. J., Shaveisi F.. T-Dimension and (n + 1/2, T)-projective modules. Southeast Asian Bull. Math., 2011, 35: 1-11

[7] Rotman J. J.. An Introduction to Homological Algebra, 1979, New York: Academic Press

[8] Wakamatsu T.. On modules with trivial self-extension. J. Algebra, 1998, 114: 106-114

[9] Wei J.. n-Star modules and n-tilting modules. J. Algebra, 2005, 283: 711-722