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Sweedler's two-cocycles and generalizations of theorems on Amitsur cohomology


Author: Dave Riffelmacher
Journal: Trans. Amer. Math. Soc. 251 (1979), 255-265
MSC: Primary 16A62
DOI: https://doi.org/10.1090/S0002-9947-1979-0531978-7
MathSciNet review: 531978
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Abstract: For any (not necessarily commutative) algebra C over a commutative ring k Sweedler defined a cohomology set, denoted here by $ {\mathcal{H}^2}(C/k)$, which generalizes Amitsur's second cohomology group $ {H^2}(C/k)$. In this paper, if I is a nilpotent ideal of C and $ \bar C\, \equiv \,C/I$ is K-projective, a natural bijection $ {\mathcal{H}^2}(C/k)\tilde \to {\mathcal{H}^2}(\bar C{\text{/}}k)$ is established. Also, when $ k \subset B$ are fields and C is a commutative B-algebra, the sequence $ \{ 1\} \to {H^2}(B{\text{/}}k)\xrightarrow{{{l^{\ast}}}}{H^2}(C/k)\xrightarrow{r}{H^2}(C/B)$ is shown to be exact if the natural map $ C{ \otimes _k}C \to C{ \otimes _B}C$ induces a surjection on units, $ {l^ {\ast} }$ is induced by the inclusion, and r is the ``restriction'' map.


References [Enhancements On Off] (What's this?)

  • [1] S. A. Amitsur, Simple algebras and cohomology groups of arbitrary fields, Trans. Amer. Math. Soc. 97 (1959), 73-112. MR 0101265 (21:78)
  • [2] A. J. Berkson, On Amitsur's complex and restricted Lie algebras, Trans. Amer. Math. Soc. 109 (1963), 430-443. MR 0158916 (28:2138)
  • [3] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 0077480 (17:1040e)
  • [4] G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58-67. MR 0011076 (6:114f)
  • [5] D. Riffelmacher, Multiplication alteration and related rigidity properties of algebras, Pacific J. Math. 71 (1977), 139-157. MR 0485954 (58:5746)
  • [6] A. Rosenberg and D. Zelinsky, Amitsur's complex for inseparable fields, Osaka Math. J. 14 (1962), 219-240. MR 0142604 (26:173)
  • [7] M. Sweedler, Groups of simple algebras, Inst. Hautes Études Sci. Publ. Math. 44 (1975), 79-189. MR 0364332 (51:587)
  • [8] -, Multiplication alteration by two-cocycles, Illinois J. Math. 15 (1971), 302-323. MR 0288150 (44:5348)
  • [9] -, Purely inseparable algebras, J. Algebra 35 (1975), 342-355. MR 0379594 (52:499)
  • [10] S. Yuan, On the theory of p-algebras and the Amitsur cohomology groups for inseparable field extensions, J. Algebra 5 (1967), 280-304. MR 0228548 (37:4128)

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DOI: https://doi.org/10.1090/S0002-9947-1979-0531978-7
Article copyright: © Copyright 1979 American Mathematical Society

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