On the deformation of algebra morphisms and diagrams

Gerstenhaber, M.; Schack, S. D.

doi:10.1090/S0002-9947-1983-0704600-5

On the deformation of algebra morphisms and diagrams
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by M. Gerstenhaber and S. D. Schack

Trans. Amer. Math. Soc. 279 (1983), 1-50

DOI: https://doi.org/10.1090/S0002-9947-1983-0704600-5

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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 ${\mathbf {(\# A)!}}$. 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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