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In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof, of Serre duality on any reduced pure n -dimensional paracompact complex space X . At the core of the paper is the introduction of certain fine sheaves B X n , q of currents on X of bidegree ( n , q ) , such that the Dolbeault complex ( B X n , • , ∂ ¯ ) becomes, in a certain sense, a dualizing complex. In particular, if X is Cohen–Macaulay then ( B X n , • , ∂ ¯ ) is an explicit fine resolution of the Grothendieck dualizing sheaf. [ABSTRACT FROM AUTHOR]

We study the role of the Serre functor in the theory of derived equivalences. Let A be an abelian category and let ( U , V ) be a t -structure on the bounded derived category D b A with heart H . We investigate when the natural embedding H → D b A can be extended to a triangle equivalence D b H → D b A . Our focus of study is the case where A is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the t -structure is bounded and the aisle U of the t -structure is closed under the Serre functor. [ABSTRACT FROM AUTHOR]

Abstract: An Auslander–Reiten formula for complexes of modules is presented. This formula contains as a special case the classical Auslander–Reiten formula. The Auslander–Reiten translate of a complex is described explicitly, and various applications are discussed. [Copyright &y& Elsevier]