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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the deformation of algebra morphisms and diagrams
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by M. Gerstenhaber and S. D. Schack PDF
Trans. Amer. Math. Soc. 279 (1983), 1-50 Request permission


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 ${\mathbf {(\# A)!}}$. 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 ${\mathbf {(\# A)!}}$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 ${\mathbf {(\# A)!}}$ is induced by one of ${\mathbf {A}}$; if ${\mathbf {A}}$ also takes values in commutative algebras then the deformation theories of ${\mathbf {(\# A)!}}$ 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
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 279 (1983), 1-50
  • MSC: Primary 16A58; Secondary 14A99, 16A61, 18G10, 18G25, 55N35
  • DOI:
  • MathSciNet review: 704600