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Odd H-depth and H-separable extensions

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Central European Journal of Mathematics

Abstract

Let C n (A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if C n+1(A,B) is isomorphic to a direct summand of a multiple of C n (A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B. In this paper the H-depth of semisimple subalgebra pairs is derived from the transpose inclusion matrix, and for QF extensions it is derived from the odd depth of the endomorphism ring extension. For general extensions characterizations of H-depth are possible using the H-equivalence generalization of Morita theory.

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References

  1. Boltje R., Danz S., Külshammer B., On the depth of subgroups and group algebra extensions, J. Algebra, 2011, 335, 258–281

    Article  MathSciNet  MATH  Google Scholar 

  2. Boltje R., Külshammer B., On the depth 2 condition for group algebra and Hopf algebra extensions, J. Algebra, 2010, 323(6), 1783–1796

    Article  MathSciNet  MATH  Google Scholar 

  3. Boltje R., Külshammer B., Group algebra extensions of depth one, Algebra Number Theory, 2011, 5(1), 63–73

    Article  MathSciNet  MATH  Google Scholar 

  4. Brzezinski T., Wisbauer R., Corings and Comodules, London Math. Soc. Lecture Note Ser., 309, Cambridge University Press, 2003

  5. Burciu S., On some representations of the Drinfel’d double, J. Algebra, 2006, 296(2), 480–504

    Article  MathSciNet  MATH  Google Scholar 

  6. Burciu S., Depth one extensions of semisimple algebras and Hopf subalgebras, preprint available at http://arxiv.org/abs/1103.0685

  7. Burciu S., Kadison L., Külshammer B., On subgroup depth, Int. Electron. J. Algebra, 2011, 9, 133–166

    MathSciNet  Google Scholar 

  8. Danz S., The depth of some twisted group algebra extensions, Comm. Algebra, 2011, 39(5), 1631–1645

    Article  MathSciNet  MATH  Google Scholar 

  9. Fritzsche T., The depth of subgroups of PSL(2,q), J. Algebra, 2011, 349, 217–233

    Article  Google Scholar 

  10. Hirata K., Some types of separable extensions of rings, Nagoya Math. J., 1968, 33, 107–115

    MathSciNet  MATH  Google Scholar 

  11. Hirata K., Sugano K., On semisimple extensions and separable extensions over non commutative rings, J. Math. Soc. Japan, 1966, 18(4), 360–373

    Article  MathSciNet  MATH  Google Scholar 

  12. Hochschild G., Relative homological algebra, Trans. Amer. Math. Soc., 1956, 82, 246–269

    Article  MathSciNet  MATH  Google Scholar 

  13. Kac G.I., Paljutkin V.G., Finite ring groups, Trudy Moskov. Mat. Obshch., 1966, 15, 224–261 (in Russian)

    MathSciNet  Google Scholar 

  14. Kadison L., New Examples of Frobenius Extensions, Univ. Lecture Ser., 14, American Mathematical Society, Providence, 1999

    MATH  Google Scholar 

  15. Kadison L., Note on Miyashita-Ulbrich action and H-separable extension, Hokkaido Math. J., 2001, 30(3), 689–695

    MathSciNet  MATH  Google Scholar 

  16. Kadison L., Hopf algebroids and H-separable extensions, Proc. Amer. Math. Soc., 2003, 131(10), 2993–3002

    Article  MathSciNet  MATH  Google Scholar 

  17. Kadison L., Depth two and the Galois coring, In: Noncommutative Geometry and Representation Theory in Mathematical Physics, Karlstad, July 5–10, 2004, Contemp. Math., 391, American Mathematical Society, Providence, 2005, 149–156

    Chapter  Google Scholar 

  18. Kadison L., Finite depth and Jacobson-Bourbaki correspondence, J. Pure Appl. Algebra, 2008, 212(7), 1822–1839

    Article  MathSciNet  MATH  Google Scholar 

  19. Kadison L., Infinite index subalgebras of depth two, Proc. Amer. Math. Soc., 2008, 136(5), 1523–1532

    Article  MathSciNet  MATH  Google Scholar 

  20. Kadison L., Subring depth, Frobenius extensions and their towers, unpublished manuscript

  21. Kadison L., Külshammer B., Depth two, normality and a trace ideal condition for Frobenius extensions, Comm. Algebra, 2006, 34(9), 3103–3122

    Article  MathSciNet  MATH  Google Scholar 

  22. Kadison L., Szlachányi K., Bialgebroid actions on depth two extensions and duality, Adv. Math., 2003, 179(1), 75–121

    Article  MathSciNet  MATH  Google Scholar 

  23. Masuoka A., Semisimple Hopf algebras of dimension 6, 8, Israel J. Math., 1995, 92(1–3), 361–373

    Article  MathSciNet  MATH  Google Scholar 

  24. Morita K., The endomorphism ring theorem for Frobenius extensions, Math. Z., 1967, 102, 385–404

    Article  MathSciNet  MATH  Google Scholar 

  25. Müller B., Quasi-Frobenius Erweiterungen, Math. Z., 1964, 85, 345–368

    Article  MathSciNet  MATH  Google Scholar 

Download references

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  1. Departamento de Matematica, Faculdade de Ciências da Universidade do Porto, Rua Campo Alegre 687, 4169-007, Porto, Portugal

    Lars Kadison

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  1. Lars Kadison
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Correspondence to Lars Kadison.

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Kadison, L. Odd H-depth and H-separable extensions. centr.eur.j.math. 10, 958–968 (2012). https://doi.org/10.2478/s11533-012-0013-y

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  • DOI: https://doi.org/10.2478/s11533-012-0013-y

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