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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Cohomological dimension and global dimension of algebras


Author: Joseph A. Wehlen
Journal: Proc. Amer. Math. Soc. 32 (1972), 75-80
MSC: Primary 16A60
DOI: https://doi.org/10.1090/S0002-9939-1972-0291226-1
MathSciNet review: 0291226
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Abstract: Let R be a regular local ring and A an algebra over R which is an R-progenerator. Defining the cohomological dimension of A as $ R$-$ \dim A = {\text{l}} \cdot {\text{hd}_{{A^e}}}(A)$, one obtains the Hochschild cohomological dimension of A as an R-algebra. We show the following under the additional hypothesis that R-dim A is finite: (1) $ R$-$ \dim A = n$ iff $ A/N$ is R-separable and $ {\text{l}} \cdot {\text{hd}_A}(A/N) = n + {\text{gl}}\;\dim R$; (2) $ {\text{gl}}\;\dim A = R{\text{-}}\dim A + {\text{gl}}\,\dim R$; (3) A is R-separable iff $ {\text{gl}}\;\dim {A^e} = {\text{gl}}\;\dim R$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0291226-1
Keywords: Hochschild cohomological dimension, regular local ring, separable algebra, global dimension, progenerator
Article copyright: © Copyright 1972 American Mathematical Society