Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties

Caenepeel, S.; De Groot, E.; Vercruysse, J.

doi:10.1090/S0002-9947-06-03857-8

Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties
HTML articles powered by AMS MathViewer

by S. Caenepeel, E. De Groot and J. Vercruysse PDF

Trans. Amer. Math. Soc. 359 (2007), 185-226 Request permission

Abstract:

El Kaoutit and Gómez-Torrecillas introduced comatrix corings, generalizing Sweedler’s canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the coring is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius.

References

Jawad Abuhlail, Morita contexts for corings and equivalences, Hopf algebras in noncommutative geometry and physics, Lecture Notes in Pure and Appl. Math., vol. 239, Dekker, New York, 2005, pp. 1–19. MR 2106920
Jawad Y. Abuhlail, On the linear weak topology and dual pairings over rings, Topology Appl. 149 (2005), no. 1-3, 161–175. MR 2130862, DOI 10.1016/j.topol.2004.09.006
P. N. Ánh and L. Márki, Morita equivalence for rings without identity, Tsukuba J. Math. 11 (1987), no. 1, 1–16. MR 899719, DOI 10.21099/tkbjm/1496160500
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
M. Beattie, S. Dăscălescu, and Ş. Raianu, Galois extensions for co-Frobenius Hopf algebras, J. Algebra 198 (1997), no. 1, 164–183. MR 1482980, DOI 10.1006/jabr.1997.7146
Tomasz Brzeziński, The structure of corings: induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Algebr. Represent. Theory 5 (2002), no. 4, 389–410. MR 1930970, DOI 10.1023/A:1020139620841
Tomasz Brzeziński, Towers of corings, Comm. Algebra 31 (2003), no. 4, 2015–2026. MR 1972904, DOI 10.1081/AGB-120018520
Tomasz Brzeziński, Galois comodules, J. Algebra 290 (2005), no. 2, 503–537. MR 2153266, DOI 10.1016/j.jalgebra.2005.05.009
Tomasz Brzeziński and Piotr M. Hajac, Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999), no. 3, 1347–1367. MR 1669095, DOI 10.1080/00927879908826498
Tomasz Brzeziński and José Gómez-Torrecillas, On comatrix corings and bimodules, $K$-Theory 29 (2003), no. 2, 101–115. MR 2029757, DOI 10.1023/B:KTHE.0000006751.77914.50
Tomasz Brzeziński and Shahn Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492. MR 1604340, DOI 10.1007/s002200050274
Tomasz Brzezinski and Robert Wisbauer, Corings and comodules, London Mathematical Society Lecture Note Series, vol. 309, Cambridge University Press, Cambridge, 2003. MR 2012570, DOI 10.1017/CBO9780511546495
S. Caenepeel, Galois corings from the descent theory point of view, Galois theory, Hopf algebras, and semiabelian categories, Fields Inst. Commun., vol. 43, Amer. Math. Soc., Providence, RI, 2004, pp. 163–186. MR 2075585, DOI 10.1090/fic/043
S. Caenepeel and E. De Groot, Modules over weak entwining structures, New trends in Hopf algebra theory (La Falda, 1999) Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 31–54. MR 1800705, DOI 10.1090/conm/267/04263
Stefaan Caenepeel, Gigel Militaru, and Shenglin Zhu, Frobenius and separable functors for generalized module categories and nonlinear equations, Lecture Notes in Mathematics, vol. 1787, Springer-Verlag, Berlin, 2002. MR 1926102, DOI 10.1007/b83849
S. Caenepeel, J. Vercruysse, and Shuanhong Wang, Morita theory for corings and cleft entwining structures, J. Algebra 276 (2004), no. 1, 210–235. MR 2054395, DOI 10.1016/j.jalgebra.2004.02.002
Stefaan Caenepeel, Joost Vercruysse, and Shuanhong Wang, Rationality properties for Morita contexts associated to corings, Hopf algebras in noncommutative geometry and physics, Lecture Notes in Pure and Appl. Math., vol. 239, Dekker, New York, 2005, pp. 113–136. MR 2106926
Stephen U. Chase and Moss E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, Vol. 97, Springer-Verlag, Berlin-New York, 1969. MR 0260724
Michele Cipolla, Discesa fedelmente piatta dei moduli, Rend. Circ. Mat. Palermo (2) 25 (1976), no. 1-2, 43–46 (Italian, with English summary). MR 498685, DOI 10.1007/BF02849557
Miriam Cohen and Davida Fischman, Semisimple extensions and elements of trace $1$, J. Algebra 149 (1992), no. 2, 419–437. MR 1172438, DOI 10.1016/0021-8693(92)90025-H
M. Cohen, D. Fischman, and S. Montgomery, Hopf Galois extensions, smash products, and Morita equivalence, J. Algebra 133 (1990), no. 2, 351–372. MR 1067411, DOI 10.1016/0021-8693(90)90274-R
Yukio Doi, Generalized smash products and Morita contexts for arbitrary Hopf algebras, Advances in Hopf algebras (Chicago, IL, 1992) Lecture Notes in Pure and Appl. Math., vol. 158, Dekker, New York, 1994, pp. 39–53. MR 1289421
L. El Kaoutit and J. Gómez-Torrecillas, Comatrix corings: Galois corings, descent theory, and a structure theorem for cosemisimple corings, Math. Z. 244 (2003), no. 4, 887–906. MR 2000464, DOI 10.1007/s00209-003-0528-9
Max-Albert Knus and Manuel Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lecture Notes in Mathematics, Vol. 389, Springer-Verlag, Berlin-New York, 1974 (French). MR 0417149
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Bachuki Mesablishvili, Pure morphisms of commutative rings are effective descent morphisms for modules—a new proof, Theory Appl. Categ. 7 (2000), No. 3, 38–42. MR 1742958
Claudia Menini and Gigel Militaru, The affineness criterion for Doi-Koppinen modules, Hopf algebras in noncommutative geometry and physics, Lecture Notes in Pure and Appl. Math., vol. 239, Dekker, New York, 2005, pp. 215–227. MR 2106931
P. Schauenburg and H.J. Schneider, Galois type extensions and Hopf algebras, preprint 2003.
Hans-Jürgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 167–195. Hopf algebras. MR 1098988, DOI 10.1007/BF02764619
Moss Sweedler, The predual theorem to the Jacobson-Bourbaki theorem, Trans. Amer. Math. Soc. 213 (1975), 391–406. MR 387345, DOI 10.1090/S0002-9947-1975-0387345-9
M. Takeuchi, as referred to in MR 2000c 16047, by A. Masuoka.
Robert Wisbauer, Foundations of module and ring theory, Revised and translated from the 1988 German edition, Algebra, Logic and Applications, vol. 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991. A handbook for study and research. MR 1144522
Robert Wisbauer, On the category of comodules over corings, Mathematics & mathematics education (Bethlehem, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 325–336. MR 1911244
Robert Wisbauer, On Galois corings, Hopf algebras in noncommutative geometry and physics, Lecture Notes in Pure and Appl. Math., vol. 239, Dekker, New York, 2005, pp. 309–320. MR 2106938