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We introduce cohomology and deformation theories for a bialgebra A (over a commutative unital ring k) such that the second cohomology group is the space of infinitesimal deformations. Our theory gives a natural identification between the underlying k-modules of the original and the deformed bialgebra. Certain explicit deformation formulas are given for the construction of quantum groups-i.e., Hopf algebras that are neither commutative nor cocommutative (whether or not they arise from quantum Yang-Baxter operators). These formulas yield, in particular, all GLq(n) and SLq(n) as deformations of GL(n) and SL(n). Using a Hodge decomposition of the underlying cochain complex, we compute our cohomology for GL(n). With this, we show that every deformation of GL(n) is equivalent to one in which the comultiplication is unchanged, not merely on elements of degree one but on all elements (settling in the strongest way a decade-old conjecture) and in which the quantum determinant, as an element of the underlying k-module, is identical with the usual one. Section 1. Bialgebra Cohomology and the LaplacianLet A be a k-bialgebra with multiplication ,u and comultiplication A. We write A®" for the q-fold tensor product A k. ..

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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 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.