Caenepeel, S.; De Groot, E.; Vercruysse, J. Galois theory for comatrix corings: descent theory, Morita theory, Frobenius and separability properties. (English) Zbl 1115.16017 Trans. Am. Math. Soc. 359, No. 1, 185-226 (2007). L. El Kaoutit and J. Gómez-Torrecillas [Math. Z. 244, No. 4, 887-906 (2003; Zbl 1037.16026)] studied a general version of descent theory, where the descent data are comodules over a comatrix coring. In the paper under review, the authors continue this investigation, by proving a generalization of an unpublished result of Joyal and Tierney. Some inspiration is taken from a proof of B. Mesablishvili [Theory Appl. Categ. 7, 38-42 (2000; Zbl 0937.13002)]. Generalizations of the Galois coring structure theorem are given. A Morita context is associated to a comodule over a coring, and this construction is used to investigate Galois type properties of the coring. More properties related to the Morita context are given in the case where the coring is Frobenius or co-Frobenius. Reviewer: Sorin Dascalescu (Bucureşti) Cited in 1 ReviewCited in 19 Documents MSC: 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16D90 Module categories in associative algebras Keywords:Galois corings; comatrix corings; descent theory; Morita contexts; comodules Citations:Zbl 1037.16026; Zbl 0937.13002 PDFBibTeX XMLCite \textit{S. Caenepeel} et al., Trans. Am. Math. Soc. 359, No. 1, 185--226 (2007; Zbl 1115.16017) Full Text: DOI arXiv References: [1] J. Abuhlail, Morita contexts for corings and equivalences, in “Hopf algebras in non-commutative geometry and physics”, S. Caenepeel and F. Van Oystaeyen, eds., Lect. Notes Pure Appl. Math. 239, Dekker, New York, 2004. [2] J. Abuhlail, On the weak linear topology and dual pairings over rings, Topology Appl. 149 (2005), 161-175. · Zbl 1086.16024 [3] P.N. \'Anh and L. M\'arki, Morita equivalence for rings without identity, Tsukuba J. Math. 11 (1987), 1-16. · Zbl 0627.16031 [4] H. Bass, “Algebraic K-theory”, Benjamin, New York, 1968. · Zbl 0174.30302 [5] M. Beattie, S. D\v asc\v alescu and \c S. Raianu, Galois extensions for co-Frobenius Hopf algebras, J. Algebra 198 (1997), 164-183. · Zbl 0901.16017 [6] T. Brzezi\'nski, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Representat. Theory 5 (2002), 389-410. · Zbl 1025.16017 [7] T. Brzezi\'nski, Towers of corings, Comm. Algebra 31 (2003), 2015-2026. · Zbl 1031.16021 [8] T. Brzezi\'nski, Galois comodules, J. Algebra 290 (2005), 503-537. · Zbl 1078.16039 [9] T. Brzezi\'nski and P. Hajac, Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999), 1347-1367. · Zbl 0923.16031 [10] T. Brzezi\'nski and J. G\'omez-Torrecillas, On comatrix corings and bimodules, K-Theory 29 (2003), 101-115. · Zbl 1041.16025 [11] T. Brzezi\'nski and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467-492. · Zbl 0899.55016 [12] T. Brzezi\'nski and R. Wisbauer, “Corings and comodules”, London Math. Soc. Lect. Note Ser. 309, Cambridge University Press, Cambridge, 2003. · Zbl 1035.16030 [13] S. Caenepeel, Galois corings from the descent theory point of view, Fields Inst. Comm. 43 (2004), 163-186. · Zbl 1082.16043 [14] S. Caenepeel and E. De Groot, Modules over weak entwining structures, Contemp. Math. 267 (2000), 31-54. · Zbl 0978.16033 [15] S. Caenepeel, G. Militaru and Zhu Shenglin “Frobenius and separable functors for generalized module categories and nonlinear equations”, Lect. Notes in Math. 1787, Springer-Verlag, Berlin, 2002. · Zbl 1008.16036 [16] S. Caenepeel, J. Vercruysse and Shuanhong Wang, Morita Theory for corings and cleft entwining structures, J. Algebra 276 (2004), 210-235. · Zbl 1064.16037 [17] S. Caenepeel, J. Vercruysse and Shuanhong Wang, Rationality properties for Morita contexts associated to corings, in “Hopf algebras in non-commutative geometry and physics”, Caenepeel S. and Van Oystaeyen, F. (eds.), Lect. Notes Pure Appl. Math. 239, Dekker, New York, 2004. · Zbl 1080.16039 [18] S. Chase and M.E. Sweedler, “Hopf algebras and Galois theory”, Lect. Notes in Math. 97, Springer-Verlag, Berlin, 1969. · Zbl 0197.01403 [19] M. Cipolla, Discesa fedelmente piatta dei moduli, Rendiconti del Circolo Matematico di Palermo, Serie II 25 (1976), 43-46. · Zbl 0391.16022 [20] M. Cohen and D. Fischman, Semisimple extensions and elements of trace 1, J. Algebra 149 (1992), 419-437. · Zbl 0788.16029 [21] M. Cohen, D. Fischman and S. Montgomery, Hopf Galois extensions, smash products, and Morita equivalence, J. Algebra 133 (1990), 351-372. · Zbl 0706.16023 [22] Y. Doi, Generalized smash products and Morita contexts for arbitrary Hopf algebras, in “Advances in Hopf algebras”, Bergen, J. and Montgomery, S. (eds.), Lect. Notes Pure Appl. Math. 158, Dekker, New York, 1994. · Zbl 0831.16023 [23] L. El Kaoutit and J. G\'omez Torrecillas, Comatrix corings: Galois corings, descent theory, and a structure theorem for cosemisimple corings, Math. Z. 244 (2003), 887-906. · Zbl 1037.16026 [24] M. Knus and M. Ojanguren, “Th\'eorie de la Descente et Alg\`ebres d”Azumaya”, Lect. Notes in Math. 389, Springer-Verlag, Berlin, 1974. · Zbl 0284.13002 [25] S. Mac Lane, “Categories for the working mathematician”, second edition, Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, 1997. · Zbl 0232.18001 [26] B. Mesablishvili, Pure morphisms of commutative rings are effective descent morphisms for modules - a new proof, Theory Appl. Categories 7 (2000), 38-42. · Zbl 0937.13002 [27] C. Menini and G. Militaru, An affineness criterion for Doi-Hopf modules, in “Hopf algebras in non-commutative geometry and physics”, Caenepeel S. and Van Oystaeyen, F. (eds.), Lect. Notes Pure Appl. Math. 239, Dekker, New York, 2004. [28] P. Schauenburg and H.J. Schneider, Galois type extensions and Hopf algebras, preprint 2003. [29] H.J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 70 (1990), 167-195. · Zbl 0731.16027 [30] M.E. Sweedler, The predual theorem to the Jacobson-Bourbaki Theorem, Trans. Amer. Math. Soc. 213 (1975), 391-406. · Zbl 0317.16007 [31] M. Takeuchi, as referred to in MR 2000c 16047, by A. Masuoka. [32] R. Wisbauer, “Foundations of module and ring theory”, Gordon and Breach, Philadelphia, 1991. · Zbl 0746.16001 [33] R. Wisbauer, On the category of comodules over corings, in “Mathematics and mathematics education (Bethlehem, 2000)”, World Sci. Publishing, River Edge, NJ, 2002, 325-336. · Zbl 1021.16027 [34] R. Wisbauer, On Galois corings, in “Hopf algebras in non-commutative geometry and physics”, Caenepeel S. and Van Oystaeyen, F. (eds.), Lect. Notes Pure Appl. Math. 239, Dekker, New York, 2004. 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