A ring with an Auslander dualizing complex is a generalization of an Auslan-der᎐Gorenstein ring. We show that many results which hold for Auslander᎐Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to nongraded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing complexes. ᮊ

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When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). This group acts naturally on the Grothendieck group K 0 (A).We prove that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables us to compute DPic(A) in these cases.Assume A is noetherian. Dualizing complexes over A were defined in [Ye]. These are complexes of bimodules which generalize the commutative definition of [RD]. We prove that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. We use this classification to deduce properties of rigid dualizing complexes, as defined by Van den Bergh in [VdB].Finally we consider finite k-algebras. For the algebra A of upper triangular 2 × 2 matrices over k, we prove that t 3 = s, where t, s ∈ DPic(A) are the classes of A * := Hom k (A, k) and A[1] respectively. In the Appendix by Elena Kreines this result is generalized to upper triangular n × n matrices, and it is shown that the relation t n+1 = s n−1 holds. IntroductionLet A and B be two rings. Recall that according to Morita Theory, any equivalence between the categories of left modules Mod A → Mod B is realized by a B-A-bimodule P , progenerator on both sides, as the functor M → P ⊗ A M .Happel is an equivalence. Here A • denotes the opposite algebra. A complex T with this property is called a tilting complex, and the algebras A and B are said to be derived Morita equivalent.In Section 1 we recall some facts on derived categories of bimodules from [Ye]. Then we reproduce Rickard's results in the formulation needed for this paper. See Remark 1.12 regarding the generalization to an arbitrary commutative base ring k. In Section 2 we prove that if A is either local or commutative then any derived Morita equivalent algebra B is actually Morita equivalent (in the ordinary sense).is a tilting complex then T ∼ = P [n] for some invertible bimodule P and some integer n (in the commutative case Spec A is assumed connected). See Theorems 2.3 and 2.7.When A = B the isomorphism classes of tilting complexes form a group, called the derived Picard group DPic(A). The operation is (T 1 , T 2 ) → T 1 ⊗ L A T 2 , the identity is A and the inverse is T → T ∨ := R Hom A (T, A). Let s ∈ DPic(A) be the class of the complex A[1]. Then the subgroup s is isomorphic to Z. When A is local we show that DPic(A) ∼ = Z × Out k (A), where Out k (A) denotes the group of outer k-algebra automorphisms (see Proposition 3.4). When A is commutative then, where m is the number of connected components of Spec A and Pic A (A) is the usual commutative Picard group (Proposition 3.5). If A is noetherian let K 0 (A) = K 0 (Mod f (A)) be the Grothendieck group. Then there is a representation χ 0 : DPic(A) → Aut(K 0 (A)), with χ 0 (s) = −1.In Section 4 we suppose A is noetherian. Then we have the notion of dualiz- with L an invertible module and n an integer, when A is noncommutative there is no such uniqueness. The question arose how to classify all iso...

Let X be a separated finite type scheme over a noetherian base ring K. There is a complex C · (X) of topological O X -modules, called the complete Hochschild chain complex of X. To any O X -module M -not necessarily quasi-coherent -we assign the complex Hom cont O X C · (X), M of continuous Hochschild cochains with values in M. Our first main result is that when X is smooth over K there is a functorial isomorphismThe second main result is that if X is smooth of relative dimension n and n! is invertible in K, then the standard maps π :When M = O X this is the quasi-isomorphism underlying the Kontsevich Formality Theorem.Combining the two results above we deduce a decomposition of the global Hochschild cohomologywhere T X/K is the relative tangent sheaf. Introduction and Statement of ResultsLet K be a noetherian commutative ring and X a separated K-scheme of finite type. The diagonal morphism ∆ : X → X 2 = X × K X is then a closed embedding. This allows us to identify the category Mod O X of O X -modules with its image inside Mod O X 2 under the functor ∆ * .We shall use derived categories freely in this paper, following the reference [RD].Definition 0.1 (Hochschild Cohomology, First Definition). Given an. This definition of Hochschild cohomology was considered by Kontsevich [Ko] and Swan [Sw] among others. We observe that if K is a field, A is a commutative Kalgebra, A e := A⊗ K A, X := Spec A, M is an A-module and M is the quasi-coherent Date: 8.11.01.