On the deformation of algebra morphisms and diagrams

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 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 A! and (#A)!.We prove the Yoneda and Hochschild cohomologies of A to be isomorphic.There are functors from A-bimodules to both A!-bimodules and (#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 (#A)! is induced by one of A; if A also takes values in commutative algebras then the deformation theories of (#A)! and 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.0. Introduction.There is a striking similarity between the formal aspects of the deformation theories of complex manifolds and associative algebras.In this work we link the two with a deformation theory for diagrams and prove a Cohomology Comparison Theorem (CCT) which partially explains the analogy.The CCT enables one to show-among other things-that the deformation theory of a diagram associated to a compact manifold is isomorphic to that of a certain associative algebra.The assignment diagram ~» algebra is functorial while manifold ~* diagram is not.(The CCT has much wider applications; for example, we sketch here ( §7), and will discuss in detail in a later paper, its application to simplicial cohomology.)Here are the basic definitions:We fix a commutative unital ring k and consider the category ALG of associative unital ^-algebras.All algebras and bimodules over them are required to be symmetric ^-modules, i.e. the left and right actions coincide.Tensor products will always be taken over k unless otherwise indicated.