Normality of algebras over commutative rings and the Teichmüller class. I.

The absolute case
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Abstract

Let S be a commutative ring and Q a group that acts on S by ring automorphisms. Given an S-algebra endowed with an outer action of Q, we study the associated Teichmüller class in the appropriate third group cohomology group. We extend the classical results to this general setting. Somewhat more specifically, let R denote the subring of S that is fixed under Q. A Q-normalS-algebra consists of a central S-algebra A and a homomorphism \(\sigma :Q\rightarrow \mathop {\mathrm{Out}}\nolimits (A)\) into the group \(\mathop {\mathrm{Out}}\nolimits (A)\) of outer automorphisms of A that lifts the action of Q on S. With respect to the abelian group \(\mathrm {U}(S)\) of invertible elements of S, endowed with the Q-module structure coming from the Q-action on S, we associate to a Q-normal S-algebra \((A, \sigma )\) a crossed 2-fold extension \(\mathrm {e}_{(A, \sigma )}\) starting at \(\mathrm {U}(S)\) and ending at Q, the Teichmüller complex of \((A, \sigma )\), and this complex, in turn, represents a class, the Teichmüller class of \((A, \sigma )\), in the third group cohomology group \(\mathrm {H}^3(Q,\mathrm {U}(S))\) of Q with coefficients in \(\mathrm {U}(S)\). We extend some of the classical results to this general setting. Among others, we relate the Teichmüller cocycle map with the generalized Deuring embedding problem and develop a seven term exact sequence involving suitable generalized Brauer groups and the generalized Teichmüller cocycle map. We also relate the generalized Teichmüller cocycle map with a suitably defined abelian group Open image in new window of classes of representations of Q in the Q-graded Brauer category Open image in new window of S with respect to the given action of Q on S.

Keywords

Teichmüller cocycle Crossed module Crossed pair Normal algebra Crossed product Deuring embedding problem Group cohomology Galois theory of commutative rings Azumaya algebra Brauer group Galois cohomology Non-commutative Galois theory Non-abelian cohomology 

Mathematics Subject Classification

12G05 13B05 16H05 16K50 16S35 20J06 
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Notes

Acknowledgements

I am indebted to the referee for a number of valuable comments.

References

  1. 1.
    Auslander, B.: The Brauer group of a ringed space. J. Algebra 4, 220–273 (1966)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Auslander, M., Buchsbaum, D.A.: On ramification theory in noetherian rings. Am. J. Math. 81, 749–765 (1959)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Auslander, M., Goldman, O.: The Brauer group of a commutative ring. Trans. Am. Math. Soc. 97, 367–409 (1960)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Auslander, M., Goldman, O.: Maximal orders. Trans. Am. Math. Soc. 97, 1–24 (1960)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bass, H.: Algebraic \(K\)-theory. W. A. Benjamin Inc., New York (1968)MATHGoogle Scholar
  6. 6.
    Bourbaki, N.: Éléments de mathématique. Fascicule XXVII. Algèbre commutative. Chapitre 1: Modules plats. Chapitre 2: Localisation. Actualités Scientifiques et Industrielles, No. 1290, Herman, Paris (1961)Google Scholar
  7. 7.
    Brown, R., Higgins, P.J., Sivera, R.: Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids.With contributions by Christopher D.Wensley and Sergei V. Soloviev. EMS Tracts in Mathematics, vol. 15. European Mathematical Society (EMS), Zürich (2011). doi: 10.4171/083
  8. 8.
    Cegarra, A.M., Garzón, A.R., Grandjean, A.R.: Graded extensions of categories. Category theory and its applications (Montreal, QC, 1997). J. Pure Appl. Algebra 154(1–3), 117–141 (2000). doi: 10.1016/S0022-4049(99)00059-6
  9. 9.
    Cegarra, A.M., Garzón, A.R., Ortega, J.A.: Graded extensions of monoidal categories. J. Algebra 241(2), 620–657 (2001). doi: 10.1006/jabr.2001.8769 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chase, S.U., Sweedler, M.E.: Hopf Algebras and Galois Theory. Lecture Notes in Mathematics, vol. 97. Springer, Berlin (1969)CrossRefMATHGoogle Scholar
  11. 11.
    Chase, S.U., Harrison, D.K., Rosenberg, A.: Galois Theory and Galois Cohomology of Commutative Rings. Mem. Amer. Math. Soc. 52, 15–33 (1965)Google Scholar
  12. 12.
    Childs, L.N.: On normal Azumaya algebras and the Teichmüller cocycle map. J. Algebra 23, 1–17 (1972)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Deuring, M.: Einbettung von Algebren in Algebren mit kleinerem Zentrum. J. Reine Angew. Math. 175, 124–128 (1936). doi: 10.1515/crll.1936.175.124 MathSciNetMATHGoogle Scholar
  14. 14.
    Eilenberg, S., MacLane, S.: Cohomology and Galois theory. I. Normality of algebras and Teichmüller’s cocycle. Trans. Am. Math. Soc. 64, 1–20 (1948)MATHGoogle Scholar
  15. 15.
    Fröhlich, A., Wall, C.: Generalisations of the Brauer group. I (1971) (Preprint)Google Scholar
  16. 16.
    Fröhlich, A., Wall, C.T.C.: Equivariant Brauergroups in algebraic number theory. In: Colloque de Théorie des Nombres (Univ. de Bordeaux, Bordeaux, 1969). Soc. Math. France, Paris, pp. 91–96. Bull. Soc. Math. France, Mém. No. 25 (1971)Google Scholar
  17. 17.
    Fröhlich, A., Wall, C.T.C.: Graded monoidal categories. Compos. Math. 28, 229–285 (1974)MathSciNetMATHGoogle Scholar
  18. 18.
    Fröhlich, A., Wall, C.T.C.: Equivariant Brauer groups. In: Quadratic Forms and Their Applications (Dublin, 1999). Contemp. Math., vol. 272. Amer. Math. Soc., Providence, RI, pp 57–71 (2000). doi: 10.1090/conm/272/04397
  19. 19.
    Hochschild, G., Serre, J.P.: Cohomology of group extensions. Trans. Am. Math. Soc. 74, 110–134 (1953)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Holt, D.F.: An interpretation of the cohomology groups \(H^{n}(G,\, M)\). J. Algebra 60(2), 307–318 (1979). doi: 10.1016/0021-8693(79)90084-X MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Huebschmann, J.: Crossed \(n\)-fold extensions of groups and cohomology. Comment. Math. Helv. 55(2), 302–313 (1980)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Huebschmann, J.: Automorphisms of group extensions and differentials in the Lyndon–Hochschild–Serre spectral sequence. J. Algebra 72(2), 296–334 (1981). doi: 10.1016/0021-8693(81)90296-9 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Huebschmann, J.: Group extensions, crossed pairs and an eight term exact sequence. J. Reine Angew. Math. 321, 150–172 (1981). doi: 10.1515/crll.1981.321.150 MathSciNetMATHGoogle Scholar
  24. 24.
    Huebschmann, J.: Normality of algebras over commutative rings and the Teichmüller class. II. Crossed pairs and the relative theory. J. Homotopy Relat. Struct. (2017). doi: 10.1007/s40062-017-0174-2
  25. 25.
    Huebschmann, J.: Normality of algebras over commutative rings and the Teichmüller class. III. Examples. J. Homotopy Relat. Struct. (2017). doi: 10.1007/s40062-017-0175-1
  26. 26.
    Kanzaki, T.: On commutor rings and Galois theory of separable algebras. Osaka J. Math. 1, 103–115 (1964) (correction, ibid 1:253 (1964))Google Scholar
  27. 27.
    Knus, M.A.: A Teichmüller cocycle for finite extensions ETH-preprint (1975)Google Scholar
  28. 28.
    MacLane, S.: A nonassociative method for associative algebras. Bull. Am. Math. Soc. 54, 897–902 (1948)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    MacLane, S.: Symmetry of algebras over a number field. Bull. Am. Math. Soc. 54, 328–333 (1948)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    MacLane, S.: Homology, 1st edn. die Grundlehren der mathematischen Wissenschaften, Band 114. Springer, Berlin (1967)Google Scholar
  31. 31.
    MacLane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5. Springer, New York (1971)MATHGoogle Scholar
  32. 32.
    MacLane, S.: Historical note. J. Algebra 60(2), 319–320 (1979) (appendix to [20])Google Scholar
  33. 33.
    Montgomery, S.: Hopf Algebras and Their Actions on Rings. CBMS Regional Conference Series in Mathematics, vol. 82. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1993)Google Scholar
  34. 34.
    Nakayama, T.: On a \(3\)-cohomology class in class field theory and the relationship of algebra- and idèle-classes. Ann. Math. (2) 57, 1–14 (1953)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Neukirch, J.: Class Field Theory. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-35437-3 (the Bonn lectures, edited and with a foreword by Alexander Schmidt, Translated from the 1967 German original by F. Lemmermeyer and W. Snyder, Language editor: A. Rosenschon)
  36. 36.
    Oberst, U.: Affine Quotientenschemata nach affinen, algebraischen Gruppen und induzierte Darstellungen. J. Algebra 44(2), 503–538 (1977)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Pareigis, B.: Über normale, zentrale, separable Algebren und Amitsur-Kohomologie. Math. Ann. 154, 330–340 (1964)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Rosenberg, A., Zelinsky, D.: Automorphisms of separable algebras. Pac. J. Math. 11, 1109–1117 (1961)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Schneider, H.J.: Principal homogeneous spaces for arbitrary Hopf algebras. Isr. J. Math. 72(1–2), 167–195 (1990). doi: 10.1007/BF02764619 MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Tate, J.T.: Global class field theory. In: Cassels, J.W.S., Fröheich, A. (eds.) Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 162–203. Thompson, Washington, DC (1967)Google Scholar
  41. 41.
    Taylor, R.L.: Compound group extensions. I. Continuations of normal homomorphisms. Trans. Am. Math. Soc. 75, 106–135 (1953)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Teichmüller, O.: Über die sogenannte nichtkommutative Galoissche Theorie und die Relation \(\xi _{\lambda,\mu,\nu }\xi _{\lambda,\mu \nu,\pi }\xi ^{\lambda }_{\mu,\nu,\pi }=\xi _{\lambda,\mu,\nu \pi }\xi _{\lambda,\mu,\nu,\pi }\). Deutsche Math 5, 138–149 (1940)MathSciNetGoogle Scholar
  43. 43.
    Ulbrich, K.H.: On the Teichmüller cocycle map. J. Algebra 124(2), 461–471 (1989). doi: 10.1016/0021-8693(89)90143-9 MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Ulbrich, K.H.: On the Teichmüller cocycle map and a theorem of Eilenberg-Mac Lane. Bull. Sci. Math. 118(3), 307–324 (1994)MathSciNetMATHGoogle Scholar
  45. 45.
    Villamayor, O.E., Zelinsky, D.: Brauer groups and Amitsur cohomology for general commutative ring extensions. J. Pure Appl. Algebra 10(1), 19–55 (1977/78)Google Scholar
  46. 46.
    Whitehead, J.H.C.: Combinatorial homotopy. II. Bull. Am. Math. Soc. 55, 453–496 (1949)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Zelinsky, D.: Long exact sequences and the Brauer group. In: Zelinski, D. (ed.) Brauer groups (Proc. Conf., Northwestern Univ., Evanston, Ill., 1975). Lecture Notes in Math., vol. 549, pp. 63–70. Springer, Berlin (1976)Google Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2017

Authors and Affiliations

  1. 1.CNRS-UMR 8524, Labex CEMPI (ANR-11-LABX-0007-01), USTL, UFR de MathématiquesVilleneuve d’Ascq CedexFrance

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