To my parents Abstract. This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In [3] we introduced

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Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents

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1.1 A∞-algebras as spaces........................ 2

Abstract. We construct a new Quillen model, based on the notions of Drinfeld’s DG quotient, [1], and localization pair, for the Morita homotopy theory of DG categories. This new Quillen model carries a natural closed symmetric monoidal structure and allow us to re-interpret Toën’s construction of the internal Hom-functor for the homotopy category of DG categories as a total right derived internal Hom-functor.

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Abstract. We present a simple description of moduli spaces of torsion-free D-modules (D-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero-Moser quiver varieties. Namely, we show that the moduli of D-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T ∗ X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T ∗ X [n] in the rank one case). The proof is based on the description of the derived category of D-modules on X by a noncommutative version of the Beilinson transform on P 1. 1.

Abstract. We show that if an algebra is n-representation-finite then its (n + 1)-preprojective algebra is self-injective. In this situation, we show that the stable module category is (n + 1)-Calabi-Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra. Finally we show that if the (n + 1)-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n ∈ {1, 2}) the result above still holds for

Abstract. We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the