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On the deformation of algebra morphisms and diagrams


Authors: M. Gerstenhaber and S. D. Schack
Journal: Trans. Amer. Math. Soc. 279 (1983), 1-50
MSC: Primary 16A58; Secondary 14A99, 16A61, 18G10, 18G25, 55N35
DOI: https://doi.org/10.1090/S0002-9947-1983-0704600-5
MathSciNet review: 704600
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Abstract: A diagram here is a functor from a poset to the category of associative algebras. Important examples arise from manifolds and sheaves. A diagram $ {\mathbf{A}}$ has functorially associated to it a module theory, a (relative) Yoneda cohomology theory, a Hochschild cohomology theory, a deformation theory, and two associative algebras $ {\mathbf{A}}!$ and $ {\mathbf{(\char93 A)!}}$. We prove the Yoneda and Hochschild cohomologies of $ {\mathbf{A}}$ to be isomorphic. There are functors from $ {\mathbf{A}}$-bimodules to both $ {\mathbf{A}}!$-bimodules and $ {\mathbf{(\char93 A)!}}$bimodules which, in the most important cases (e.g., when the poset is finite), induce isomorphisms of Yoneda cohomologies. When the poset is finite every deformation of $ {\mathbf{(\char93 A)!}}$ is induced by one of $ {\mathbf{A}}$; if $ {\mathbf{A}}$ also takes values in commutative algebras then the deformation theories of $ {\mathbf{(\char93 A)!}}$ and $ {\mathbf{A}}$ are isomorphic. We conclude the paper with an example of a noncommutative projective variety. This is obtained by deforming a diagram representing projective $ 2$-space to a diagram of noncommutative algebras.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0704600-5
Keywords: Associative algebra, diagram, Hochschild cohomology, Yoneda cohomology, deformation, simplicial cohomology
Article copyright: © Copyright 1983 American Mathematical Society

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