Odd H-depth and H-separable extensions

Lars Kadison

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 958-968
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Lars Kadison. "Odd H-depth and H-separable extensions." Open Mathematics 10.3 (2012): 958-968. <http://eudml.org/doc/269169>.

@article{LarsKadison2012,
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.},
author = {Lars Kadison},
journal = {Open Mathematics},
keywords = {Subring depth; H-separable; Frobenius extension; Complex semisimple algebra; ring extensions; depth; separability; Frobenius extensions; bimodules; separable extensions; endomorphism ring extensions; QF-extensions},
language = {eng},
number = {3},
pages = {958-968},
title = {Odd H-depth and H-separable extensions},
url = {http://eudml.org/doc/269169},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Lars Kadison
TI - Odd H-depth and H-separable extensions
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 958
EP - 968
AB - 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.
LA - eng
KW - Subring depth; H-separable; Frobenius extension; Complex semisimple algebra; ring extensions; depth; separability; Frobenius extensions; bimodules; separable extensions; endomorphism ring extensions; QF-extensions
UR - http://eudml.org/doc/269169
ER -

References

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  1. [1] Boltje R., Danz S., Külshammer B., On the depth of subgroups and group algebra extensions, J. Algebra, 2011, 335, 258–281 http://dx.doi.org/10.1016/j.jalgebra.2011.03.019 Zbl1250.20001
  2. [2] Boltje R., Külshammer B., On the depth 2 condition for group algebra and Hopf algebra extensions, J. Algebra, 2010, 323(6), 1783–1796 http://dx.doi.org/10.1016/j.jalgebra.2009.11.043 Zbl1200.16035
  3. [3] Boltje R., Külshammer B., Group algebra extensions of depth one, Algebra Number Theory, 2011, 5(1), 63–73 http://dx.doi.org/10.2140/ant.2011.5.63 Zbl1236.20001
  4. [4] Brzezinski T., Wisbauer R., Corings and Comodules, London Math. Soc. Lecture Note Ser., 309, Cambridge University Press, 2003 
  5. [5] Burciu S., On some representations of the Drinfel’d double, J. Algebra, 2006, 296(2), 480–504 http://dx.doi.org/10.1016/j.jalgebra.2005.09.004 Zbl1094.16025
  6. [6] Burciu S., Depth one extensions of semisimple algebras and Hopf subalgebras, preprint available at http://arxiv.org/abs/1103.0685 Zbl1296.16028
  7. [7] Burciu S., Kadison L., Külshammer B., On subgroup depth, Int. Electron. J. Algebra, 2011, 9, 133–166 Zbl1266.20001
  8. [8] Danz S., The depth of some twisted group algebra extensions, Comm. Algebra, 2011, 39(5), 1631–1645 http://dx.doi.org/10.1080/00927871003738980 Zbl1244.16016
  9. [9] Fritzsche T., The depth of subgroups of PSL(2,q), J. Algebra, 2011, 349, 217–233 http://dx.doi.org/10.1016/j.jalgebra.2011.10.017 
  10. [10] Hirata K., Some types of separable extensions of rings, Nagoya Math. J., 1968, 33, 107–115 Zbl0179.05804
  11. [11] Hirata K., Sugano K., On semisimple extensions and separable extensions over non commutative rings, J. Math. Soc. Japan, 1966, 18(4), 360–373 http://dx.doi.org/10.2969/jmsj/01840360 Zbl0178.36802
  12. [12] Hochschild G., Relative homological algebra, Trans. Amer. Math. Soc., 1956, 82, 246–269 http://dx.doi.org/10.1090/S0002-9947-1956-0080654-0 Zbl0070.26903
  13. [13] Kac G.I., Paljutkin V.G., Finite ring groups, Trudy Moskov. Mat. Obshch., 1966, 15, 224–261 (in Russian) 
  14. [14] Kadison L., New Examples of Frobenius Extensions, Univ. Lecture Ser., 14, American Mathematical Society, Providence, 1999 Zbl0929.16036
  15. [15] Kadison L., Note on Miyashita-Ulbrich action and H-separable extension, Hokkaido Math. J., 2001, 30(3), 689–695 Zbl1005.16036
  16. [16] Kadison L., Hopf algebroids and H-separable extensions, Proc. Amer. Math. Soc., 2003, 131(10), 2993–3002 http://dx.doi.org/10.1090/S0002-9939-02-06876-4 Zbl1033.16017
  17. [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 http://dx.doi.org/10.1090/conm/391/07325 
  18. [18] Kadison L., Finite depth and Jacobson-Bourbaki correspondence, J. Pure Appl. Algebra, 2008, 212(7), 1822–1839 http://dx.doi.org/10.1016/j.jpaa.2007.11.007 
  19. [19] Kadison L., Infinite index subalgebras of depth two, Proc. Amer. Math. Soc., 2008, 136(5), 1523–1532 http://dx.doi.org/10.1090/S0002-9939-08-09077-1 Zbl1189.16027
  20. [20] Kadison L., Subring depth, Frobenius extensions and their towers, unpublished manuscript Zbl1280.16026
  21. [21] Kadison L., Külshammer B., Depth two, normality and a trace ideal condition for Frobenius extensions, Comm. Algebra, 2006, 34(9), 3103–3122 http://dx.doi.org/10.1080/00927870600650291 Zbl1115.16020
  22. [22] Kadison L., Szlachányi K., Bialgebroid actions on depth two extensions and duality, Adv. Math., 2003, 179(1), 75–121 http://dx.doi.org/10.1016/S0001-8708(02)00028-2 Zbl1049.16022
  23. [23] Masuoka A., Semisimple Hopf algebras of dimension 6, 8, Israel J. Math., 1995, 92(1–3), 361–373 http://dx.doi.org/10.1007/BF02762089 Zbl0839.16036
  24. [24] Morita K., The endomorphism ring theorem for Frobenius extensions, Math. Z., 1967, 102, 385–404 http://dx.doi.org/10.1007/BF01111076 Zbl0162.33803
  25. [25] Müller B., Quasi-Frobenius Erweiterungen, Math. Z., 1964, 85, 345–368 http://dx.doi.org/10.1007/BF01110680 Zbl0203.04403

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