On the deformation of algebra morphisms and diagrams

Abstract

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 symmetriĉ -modules, i.e. the left and right actions coincide. Tensor products will always be taken over k unless otherwise indicated.

The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set

“…), from the set of deformations of A to the set of deformations of A!. In [GSl,§21] we analyze this map. We show there that it is an injection if, for example, A factors through the category of commutative algebras and ui': Ext*_A( -,-) -► ExtA!.Ai( -!, -!)…”

*-Quantizations of Fourier–Mukai Transforms

“…Of course, this is a version of the well-known statement that deformations of unital algebras are unital (see for instance [GS83,Section 20] It suffices to check that the unit exists locally on M since a unit element in an associative algebra is unique. Also, it suffices to prove existence of a right unit.…”

Diextensions et panextensions de catégories

“…Ceci a été généralisé par Gerstenhaber et Schack [5] qui ont décrit une cohomologie des diagrammes d'algèbres.…”

“…La théorie de Baues-Wirsching que l'on retrouve ici est reliée par Pavešić dans [12] à l'extension de la cohomologie de Hochschild décrite par Gerstenhaber-Schack dans [5].…”

“…Malgré le contexte bivariant qui va apparaître, on ne parle donc pas de biextensions. Signalons ici la notion d'appariement de fibrations considérée par [12] qui relie les constructions de [2] sur la cohomologie des catégories et celles de [5] qui décrit une cohomologie des diagrammes d'algèbres généralisant la cohomologie de Hochschild [7] dont nous reparlerons.…”