×

Co-Frobenius Hopf algebras: Integrals, Doi-Koppinen modules and injective objects. (English) Zbl 0942.16040

Let \(H\) be a Hopf algebra over a field \(k\) and consider a right \(H\) comodule algebra \(A\) and a left \(H\) module coalgebra \(C\). The paper is devoted to the study of the category \(_A\mathbb{M}^C\) of Doi-Koppinen modules [see Y. Doi, J. Algebra 153, No. 2, 373-385 (1992; Zbl 0782.16025) and M. Koppinen, J. Pure Appl. Algebra 104, No. 1, 61-80 (1995; Zbl 0838.16035)] under the assumption that \(C\) enjoys some extra conditions, like being semiperfect, quasi co-Frobenius or co-Frobenius [see B. I. Lin, J. Algebra 49, 357-373 (1977; Zbl 0369.16010) and J. Gómez Torrecillas and C. Năstăsescu, J. Algebra 174, No. 3, 909-923 (1995; Zbl 0833.16038)]. In these cases, the rational part \({C^*}^{\text{rat}}\) of \(C^*=\text{Hom}_k(C,k)\) does not depend on the side, which is used to prove that the largest rational module \(M^{\text{rat}}\) of any left \(C^*\)-module \(M\) can be computed as \(M^{\text{rat}}{C^*}^{\text{rat}}M\). This gives the main tool developed in the paper: An exact functor \(t\colon{_{A\#C^*}\mathbb{M}}\to{_A\mathbb{M}^C}\) from the category of left modules over the smash product \(A\#C^*\) to the category of Doi-Koppinen modules is supplied. An explicit description of the right adjoint to \(t\) is given in Section 3, which allows to extend to the Doi-Koppinen setting a result proved for comodule categories by C. Năstăsescu and the reviewer [Theorem 2.3 in loc. cit.]. Two applications of these techniques are given in Section 4. It is proved that the forgetful functor \({_A\mathbb{M}^C}\to{_A\mathbb{M}}\) preserves injective objects of finite support (a Doi-Koppinen module \(M\) is said to have finite support if its coefficient subcoalgebra in \(C\) is finite-dimensional). Finally, an easy coalgebraic proof of the uniqueness of integrals is given [see J. B. Sullivan, J. Algebra 19, 426-440 (1971; Zbl 0239.16006) for the original proof].

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16L60 Quasi-Frobenius rings
16D90 Module categories in associative algebras
16D50 Injective modules, self-injective associative rings
16S40 Smash products of general Hopf actions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albu, T.; Năstăsescu, C., Infinite group-graded rings, rings of endomorphisms, and localization, J. Pure Appl. Algebra, 59, 125-150 (1989) · Zbl 0676.16001
[2] M. Beattie, S. Dăscălescu, and, L. Grunenfelder, Constructing Pointed Hopf Algebras by Ore Extensions, preprint.; M. Beattie, S. Dăscălescu, and, L. Grunenfelder, Constructing Pointed Hopf Algebras by Ore Extensions, preprint. · Zbl 0948.16026
[3] Beattie, M.; Dăscălescu, S.; Grunenfelder, L.; Năstăsescu, C., Finiteness conditions, co-Frobenius Hopf algebras, and quantum groups, J. Algebra, 200, 312-333 (1998) · Zbl 0902.16028
[4] Beattie, M.; Dăscălescu, S.; Raianu, Ş.; Van Oystaeyen, F., The categories of Yetter-Drinfel’d modules, Doi-Hopf modules, and two-sided two-cosided Hopf modules, Appl. Categorical Structures, 6, 223-237 (1998) · Zbl 0983.16029
[5] Bulacu, D., Injective modules graded by \(G\)-sets, Comm. Algebra, 27, 3537-3544 (1999) · Zbl 0929.16040
[6] Caenepeel, S.; Raianu, S., Induction functors for the Doi-Koppinen unified Hopf modules, (Facchini, A.; Menini, C., Abelian Group and Modules (1995), Kluwer Academic: Kluwer Academic Dordrecht) · Zbl 0843.16035
[7] Caenepeel, S.; Militaru, G.; Zhu, S., Crossed modules and Doi-Hopf modules, Israel J. Math., 100, 221-247 (1997) · Zbl 0888.16018
[8] Dăscălescu, S.; Năstăsescu, C.; del Rio, A.; van Oystaeyen, F., Gradings of finite support. Applications to injective objects, J. Pure Appl. Algebra, 107, 193-206 (1996) · Zbl 0859.16036
[9] Doi, Y., Homological coalgebra, J. Math. Soc. Jpn., 33, 31-50 (1981) · Zbl 0459.16007
[10] Doi, Y., Unifying Hopf modules, J. Algebra, 153, 373-385 (1992) · Zbl 0782.16025
[11] Gabriel, P., Des categories abeliennes, Bull. Soc. Math. France, 90, 323-448 (1962) · Zbl 0201.35602
[12] Torrecillas, J. Gomez; Năstăsescu, C., Quasi-co-Frobenius coalgebras, J. Algebra, 174, 909-923 (1995) · Zbl 0833.16038
[13] Koppinen, M., Variations on the smash product with applications to group-graded rings, J. Pure Appl. Algebra, 104, 61-80 (1995) · Zbl 0838.16035
[14] Lin, B. J., Semiperfect coalgebras, J. Algebra, 49, 357-373 (1977) · Zbl 0369.16010
[15] C. Menini, B. Torrecillas, and, R. Wisbauer, Strongly Rational Comodules and Semiperfect Hopf Algebras over QF Rings, J. Pure Appl. Algebra, to appear.; C. Menini, B. Torrecillas, and, R. Wisbauer, Strongly Rational Comodules and Semiperfect Hopf Algebras over QF Rings, J. Pure Appl. Algebra, to appear. · Zbl 0976.16029
[16] Menini, C.; Zuccoli, M., Equivalence theorems and Hopf-Galois extensions, J. Algebra, 194, 245-274 (1997) · Zbl 0884.16024
[17] Montgomery, S., Hopf Algebras and Their Actions on Rings. Hopf Algebras and Their Actions on Rings, CMS Conference Proceedings, 82 (1993), American Mathematical Society: American Mathematical Society Providence · Zbl 0804.16041
[18] Năstăsescu, C., Smash products and applications to finiteness conditions, Rev. Roumaine Math. Pures Appl., 34, 825-837 (1989) · Zbl 0689.16001
[19] Năstăsescu, C.; Raianu, Ş.; Van Oystaeyen, F., Modules graded by \(G\)-sets, Math. Z., 203, 605-627 (1990) · Zbl 0721.16025
[20] Ştefan, D., The uniqueness of integrals: a homological approach, Comm. Algebra, 23, 1657-1662 (1995) · Zbl 0824.16029
[21] Sweedler, M. E., Hopf Algebras (1969), Benjamin: Benjamin New York
[22] Takeuchi, M., A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math., 7, 251-270 (1972) · Zbl 0238.16011
[23] R. Wisbauer, Introduction to Coalgebras and Comodules, preprint.; R. Wisbauer, Introduction to Coalgebras and Comodules, preprint. · Zbl 1233.16027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.