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To each presheaf (over a poset) of associative algebras A we associate an algebra A!. We define a full exact embedding of the category of (presheaf) A-bimodules in that of A!-bimodules. We show that this embedding preserves neither enough (relative) injectives nor enough (relative) projectives, but nonetheless preserves (relative) Yoneda cohomology. The cohomology isomorphism links the deformations of manifolds, algebraic presheaves, and algebras. It also implies that the cohomology of any triangulable space is isomorphic to the Hochschild cohomology of an associative algebra. (The latter isomorphism preserves all known cohomology operations.) We conclude the paper by exhibiting for each associative algebra and triangulable space a "product" which is again an associative algebra. 0. Introduction. Presheaves are rather familiar objects and arise in many situations, some of which are discussed below. Our primary purpose in this paper is to prove a theorem which reduces questions about the cohomology of presheaves of rings to questions about the (classical) cohomology of rings. Specifically, fix a commutative ring k and a partially ordered set (poset) JY. Let A be a presheaf of associative k-algebras over >J hereinafter: a diagram. Then A has naturally associated with it an abelian category of bimodules and a (relative) Yoneda Cohomology bifunctor Ext*A(-,-). (All the foregoing formalism is introduced in ?1.) We define for each such diagram A a single k-algebra A! and a full exact embedding, M M!, of the category of A-bimodules in the category of A!-bimodules (?2). The embedding induces a natural transformation of Yoneda cohomology bifunctors, w*: ExtA (-, -) !A Ext l (-!,-!) (?2). Our central result is the Special Cohomology Comparison Theorem (CCT), namely: for an arbitrary diagram A over an arbitrary poset w*: ExtA (-,-) * ExtA , H _(-!-!) is an isomorphism. The proof occupies ??3, 4. The scheme of the proof will be given shortly. First we describe some of the history and utility of the CCT. In [GS1] we proved a "primitive" version of the CCT, applicable to arbitrary diagrams defined over arbitrary finite posets and to a narrow class of diagrams defined over a limited class of infinite posets. The proofs given there for the finite Received by the editors March 6, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A58, 16A61, 16A62, 18G10, 18G25; Secondary 18E25, 32G05, 55N25, 55U10.

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