Abstract
We describe how some aspects of abstract localization on module categories have applications to the study of injective comodules over some special types of corings. We specialize the general results to the case of Doi-Koppinen modules, generalizing previous results in this setting.
Professor Robert Wisbauer gewidmet
This research is supported by the grants MTM2004-01406 and MTM2007-61673 from the Ministerio de Educación y Ciencia of Spain, and P06-FQM-01889 from the Consejería de Innovación, Ciencia y Empresa of Andalucía, with funds from FEDER (Unión Europea).
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References
G.D. Abrams, Morita equivalence for rings with local units, Comm. Algebra 11 (1983), 801–837.
J.Y. Abuhlail, Rational modules for corings, Comm. Algebra 31 (2003), no. 12, 5793–5840.
J.Y. Abuhlail, J. Gómez-Torrecillas, and F.J. Lobillo, Duality and rational modules in Hopf algebras over commutative rings, J. Algebra 240 (2001), 165–184.
T. Brzeziński, A note on coring extensions, Ann. Univ. Ferrara, Sez. VII, Sc. Mat. LI(III) (2005), 15–27.
T. Brzeziński and R. Wisbauer, Corings and comodules, LMS, vol. 309, Cambridge University Press, 2003.
T. Brzeziński, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Alg. Rep. Theory 5 (2002), 389–410.
T. Brzeziński and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467–492.
C. Cai and H. Chen, Coactions, smash product, and Hopf modules, J. Algebra 167 (1994), 85–99.
S. Dăscălescu, C. Năstăsescu, A. del Rio, and F. van Oystaeyen, Gradings of finite support. Applications to injective objects, J. Pure Appl. Algebra 107 (1996), 193–206.
S. Dăscălescu, C. Năstăsescu, and B. Torrecillas, Co-Frobenius Hopf algebras: Integrals, Doi-Koppinen modules and injective objects, J. Algebra 220 (1999), 542–560.
Y. Doi, Unifying Hopf modules, J. Algebra. 153 (1992), 373–385.
L. El Kaoutit, J. Gómez-Torrecillas, and F.J. Lobillo, Semisimple corings, Algebra Colloq. 11 (2004), 427–442.
C. Faith, Algebra: rings, modules and categories, I, Grundlehren Math. Wiss., 190, Springer, New York, 1973.
P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.
J. Gómez-Torrecillas, Coalgebras and comodules over commutative rings, Rev. Roumaine Math. Pures Appl. 43 (1998), no. 5–6, 591–603.
J. Gómez-Torrecillas, Separable functors in corings, Int. J. Math. Math. Sci. 30 (2002), no. 4, 203–225.
J. Gómez-Torrecillas and C. Năstăsescu, Quasi-co-Fronenius coalgebras, J. Algebra, 174 (1995), 909–923.
M. Koppinen, Variations on the smash product with applications to group-graded rings, J. Pure Appl. Algebra 104 (1995), 61–80.
M. Sweedler, The predual theorem to the Jacobson-Bourbaki theorem, Trans. Amer. Math. Soc. 213 (1975), 391–406.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading-Paris, 1991.
B. Zimmermann-Huisgen, Pure submodules of direct products of free modules, Math. Ann. 224 (1976), 233–245.
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El Kaoutit, L., Gómez-Torrecillas, J. (2008). Corings with Exact Rational Functors and Injective Objects. In: Brzeziński, T., Gómez Pardo, J.L., Shestakov, I., Smith, P.F. (eds) Modules and Comodules. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8742-6_12
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DOI: https://doi.org/10.1007/978-3-7643-8742-6_12
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