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

The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set


Authors: Murray Gerstenhaber and Samuel D. Schack
Journal: Trans. Amer. Math. Soc. 310 (1988), 135-165
MSC: Primary 16A58; Secondary 16A61, 18E25, 55N25, 55U10
MathSciNet review: 965749
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Abstract: To each presheaf (over a poset) of associative algebras $ \mathbb{A}$ we associate an algebra $ \mathbb{A}!$. We define a full exact embedding of the category of (presheaf) $ \mathbb{A}$-bimodules in that of $ \mathbb{A}!$-bimodules. We show that this embedding preserves neither enough (relative) injectives nor enough (relative) projectives, but nonetheless preserves (relative) Yoneda cohomology. The cohomology isomorphism links the deformations of manifolds, algebraic presheaves, and algebras. It also implies that the cohomology of any triangulable space is isomorphic to the Hochschild cohomology of an associative algebra. (The latter isomorphism preserves all known cohomology operations.) We conclude the paper by exhibiting for each associative algebra and triangulable space a "product" which is again an associative algebra.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0965749-X
PII: S 0002-9947(1988)0965749-X
Keywords: Associative algebra, bimodule, Yoneda cohomology, Hochschild cohomology, simplicial cohomology, deformation, triangulable space
Article copyright: © Copyright 1988 American Mathematical Society