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Cohomology for Bicomodules. Separable and Maschke functors

Published online by Cambridge University Press:  30 November 2007

L. El Kaoutit  and
J. Vercruysse
L. El Kaoutit
Affiliation:
Departamento de Álgebra, Facultad de Educación y Humanidades de Ceuta, Universidad de Granada, El Greco N. 10. E-51002 Ceuta, Spain, kaoutit@ugr.es.
J. Vercruysse
Affiliation:
Department of Mathematics, Faculty of Engineering, Vrije Universiteit Brussel (VUB), B-1050 Brussels, Belgium, jvercruy@vub.ac.be.
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Abstract

We introduce the category of bicomodules for a comonad on a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [14], and the characterization of coseparable coalgebra coextensions stated in [19].

Keywords

ComonadbicomoduleSeparable functorMaschke functor
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 1 , February 2009 , pp. 123 - 152
Copyright
Copyright © ISOPP 2009

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References

1.Ardizzoni, A., Separable functors and Formal Smoothness, Journal of K-Theory, to appearGoogle Scholar
2.Ardizzoni, A., Menini, C., Stefan, D., Hochschild cohomology and “smoothness” in monoidal categories J. Pure Appl. Algebra 208 (2007), 297–330CrossRefGoogle Scholar
3.Barr, M. and Rinehart, G., Cohomology as the derived functor of derivations, Trans. Amer. Math. Soc. 122 (1966), 416–426CrossRefGoogle Scholar
4.Bautista, R., Colavita, L., and Salmerón, L., On adjoint functors in representation theory, Lecture Notes in Math., vol. 903, Springer-Verlag, 1981, pp. 9–25Google Scholar
5.Brzeziński, T. and Wisbauer, R., Corings and Comodules. Cambridge University Press, LMS 309, (2003)CrossRefGoogle Scholar
6.Caenepeel, S., Militaru, G., Maschke functors, semisimple functors and separable functors of the second kind: Aplications, J. Pure Appl. Algebra 178 (2003) 131–157CrossRefGoogle Scholar
7.Doi, Y., Homological coalgebra, J. Math. Soc. Japan 33 No 1 (1981), 31–50CrossRefGoogle Scholar
8.Iglesias, F. Castaño, Gómez-Torrecillas, J., and Nǎstǎsescu, C., Frobenius functors: Aplications, Commun. Algebra 27 (1999), No 10, 4879–4900CrossRefGoogle Scholar
9.Eilenberg, S. and Moore, J. C., Adjoint functors and triples, Illinois J. Math. 9 (1965), 381–398CrossRefGoogle Scholar
10.Eilenberg, S. and Moore, J. C., Fundations of relative homological algebra, Mem. Amer. Math. Soc. 55 (1965)Google Scholar
11.El Kaoutit, L., Corings over rings with local units, Mathematische Nachrichten, to appearGoogle Scholar
12.Gómez-Torrecillas, J., Separable functors in corings, Int. J. Math. Math. Sc. 30 (2002), No 4, 203–225CrossRefGoogle Scholar
13.Gómez-Torrecillas, J., Comonads and Galois corings, Appl. Categor. Struct. 14 (2006), 579–598CrossRefGoogle Scholar
14.Guzman, F., Cointegrations, Relative Cohomology for Comodules, and Coseparable Corings, J. Algebra 126 (1989) 211–224CrossRefGoogle Scholar
15.Huber, P. J., Homotopy theory in general categories, Math. Ann. 144 (1961), 361–385CrossRefGoogle Scholar
16.Jonah, D. W., Cohomology of coalgebras, Mem. Amer. Math. Soc. 82 (1968)Google Scholar
17.Kleiner, M., Integration and cohomology theory, J. Pure Appl. Algebra, 38 (1985) 71–86CrossRefGoogle Scholar
18.Kleisli, H., Every standard construction is induced by a pair of adjoint functors, Proc. Amer. Math. Soc. 16 (1965), 544–546CrossRefGoogle Scholar
19.Nakajima, A., Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ. 22 (1980), 145–149Google Scholar
20.Nǎstǎsescu, C., Van den Bergh, M., and Van Oystaeyen, F., Separable functors applied to graded rings, J. Algebra 123 (1989), 397–413CrossRefGoogle Scholar
21.Rafael, M. D., Separable functors revisited, Commun. Algebra 18 (1990), No 5, 1445–1459CrossRefGoogle Scholar
22.Sweedler, M. E., The predual theorem to the Jacobson-Bourbaki theorem, Tran. Amer. Math. Soc. 213 (1975), 391–406CrossRefGoogle Scholar
23.Watts, C. E., Intrinsic characterization of some additive functors, Proc. Amer. Math. Soc. 11 (1960), 5–8CrossRefGoogle Scholar