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), 150 Request permission
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 ${\mathbf {(\# A)!}}$. 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.References

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Additional Information
 © Copyright 1983 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 279 (1983), 150
 MSC: Primary 16A58; Secondary 14A99, 16A61, 18G10, 18G25, 55N35
 DOI: https://doi.org/10.1090/S00029947198307046005
 MathSciNet review: 704600