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

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.

Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories. Contents

Abstract. We prove a sheaf-theoretic derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendieck’s local duality theorem): for a quasi-compact separated scheme X and a “proregular ” subscheme Z—for example, any separated noetherian scheme and any closed subscheme—there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, left-derived completion can be identified with local homology, i.e., the homology of RHom • (RΓ Z OX, −). Sheafified generalizations of a number of duality theorems scattered about the literature result: the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality

Abstract. We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relating products and distinguished triangles. The axioms hold in the examples and shed light on generalized homology and cohomology theories. Contents 1. Generalized trace maps 2 2. Triangulated categories 6 3. Weak pushouts and weak pullbacks 9 4. The compatibility axioms 11

We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of Thomason-Trobaugh’s [51] and Yao’s [58] localization theorems in K-theory (cf. also [55]).

Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)-module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kG-module M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.

Abstract. We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne’s method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about “Brown Representability, ” is indicated as well. A section on applications and examples illustrates how our results synthesize a number of different duality-related topics (local duality, formal duality, residue theorems, dualizing complexes,...). A flat-base-change theorem for pseudo-proper maps leads in particular to sheafified versions of duality for bounded-below complexes with quasi-coherent homology. Thanks to Greenlees-May duality, the results take a specially nice form for proper maps and bounded-below complexes with coherent homology. Contents 1. Preliminaries and main theorems. 4

Abstract. Using hypercohomology, we can extend cyclic homology from algebras to all schemes over a ring k. By ‘extend ’ we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme. The purpose of this paper is to show that there is a cyclic homology theory HC ∗ of schemes over a commutative ring k, extending the usual cyclic homology HC ∗ of k-algebras. By a cyclic homology theory for schemes over k we mean a family of graded k-modules HCn(X) associated to every scheme X over k which satisfy: (0.1) they are natural and contravariant in X; (0.2) for each affine scheme X = Spec A, there are natural isomorphisms HCn(X) ∼ = HCn(A) for all n; (0.3) if X = U ∪ V, there is a Mayer-Vietoris sequence · · · HCn(X) → HCn(U) ⊕ HCn(V) → HCn(U ∩ V) → HCn−1(X) · · ·. We discuss uniqueness of a cyclic homology theory briefly in Remark 0.5 below. We have chosen homological indexing because of axiom (0.2), and because cohomological indexing (HC n = HC−n) would concentrate the nonzero groups in negative degrees.

Abstract. We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus T as an ind-object in the category of holomorphic vector bundles on T. We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative