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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0531978-7
PII: S 0002-9947(1979)0531978-7
Article copyright: © Copyright 1979 American Mathematical Society