Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞-algebras,

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Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.

The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D in terms of the nerve of a certain category of (C, D)-bimodules. We also prove that the homotopy category Ho(dg −Cat) possesses internal Hom’s relative to the (derived) tensor product of dg-categories. We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories C and D as the dg-category of (C, D)-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the classifying space of dgcategories (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dgcategories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent

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The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated projective modules over an algebra it specializes to the cyclic homology of the algebra. However, we show that McCarthy's theory cannot be both compatible with localizations and invariant under functors inducing equivalences in the derived category. This is our motivation for introducing a new theory for which all three properties hold: extension, invariance and localization. Thanks to these properties, the new theory can be computed explicitly for a number of categories of modules and sheaves.

Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday. Abstract. For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal Cohen-Macaulay approximations, a construction of Tate cohomology, and an extension of the classical Grothendieck duality. In addition, the relevance of the stable derived category in modular representation theory is indicated.

We establish equivalences of the following three triangulated categories: Dquantum(g) ← → D G coherent (Ñ) ← → Dperverse(Gr). Here, Dquantum(g) is the derived category of the principal block of finite dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g; the category DG coherent (Ñ) is defined in terms of coherent sheaves on the cotangent bundle on the (finite dimensional) flag manifold for G ( = semisimple group with Lie algebra g), and the category Dperverse(Gr) is the derived category of perverse sheaves on the Grassmannian Gr associated with the loop group LG ∨ , where G ∨ is the Langlands dual group, smooth along the Schubert stratification. The equivalence between Dquantum(g) and DG coherent (Ñ) is an ‘enhancement ’ of the known expression (due to Ginzburg-Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between Dperverse(Gr) and DG coherent (Ñ) can be viewed as a ‘categorification ’ of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag

These notes are based on a series of five lectures given during the

Using one of Wodzicki’s examples of H-unital algebras [14] we exhibit a ring whose derived category contains a smashing subcategory which is not generated by small objects. This disproves the generalization to arbitrary triangulated categories of a conjecture due to Ravenel [8, 1.33] and, originally, Bousfield [2, 3.4]. 1. Statement of the conjecture We refer to [7] for a nicely written analysis of the following setup: Let S be a triangulated category [13] admitting arbitrary (set-indexed) coproducts. An object X ∈ S is small if the functor Hom (X,?) commutes with arbitrary coproducts. We denote the full subcategory on the small objects of S by Sb. We suppose that Sb is equivalent to a small category. A full subcategory of S is localizing if it is a triangulated subcategory in the sense of Verdier which is closed under forming coproducts with respect to S. WeKeller suppose that S is generated by S b, i.e. coincides with its smallest