On the Deformation of Algebra Morphisms and Diagrams

A diagram here is a functor from a poset to the category of associative algebras. Important examples arise from manifolds and sheaves. A diagram <!-- MATH ${\mathbf{A}}$ --> has functorially associated to it a module theory, a (relative) Yoneda cohomology theory, a Hochschild cohomology theory, a deformation theory, and two associative algebras <!-- MATH ${\mathbf{A}}!$ --> and <!-- MATH ${\mathbf{(\# A)!}}$ --> . We prove the Yoneda and Hochschild cohomologies of <!-- MATH ${\mathbf{A}}$ --> to be isomorphic. There are functors from <!-- MATH ${\mathbf{A}}$ --> -bimodules to both <!-- MATH ${\mathbf{A}}!$ --> -bimodules and <!-- MATH ${\mathbf{(\# A)!}}$ --> 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 <!-- MATH ${\mathbf{(\# A)!}}$ --> is induced by one of <!-- MATH ${\mathbf{A}}$ --> ; if <!-- MATH ${\mathbf{A}}$ --> also takes values in commutative algebras then the deformation theories of <!-- MATH ${\mathbf{(\# A)!}}$ --> and <!-- MATH ${\mathbf{A}}$ --> are isomorphic. We conclude the paper with an example of a noncommutative projective variety. This is obtained by deforming a diagram representing projective -space to a diagram of noncommutative algebras.