This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim X ≥ 2, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail. 1.

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

These notes are based on lectures given at the Workshop on Structured ring spectra and

Abstract. We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.

Abstract. A spectral sequence is constructed whose nonzero E1-terms are the Witt groups of the residue fields of a regular scheme X, arranged in Gersten-Witt complexes, and whose limit is the four global Witt groups of X. There are several immediate consequences concerning purity for Witt groups of low-dimensional schemes. The Witt groups of punctured spectra of regular local rings are also computed. Let X be a regular integral separated noetherian scheme in which 2 is everywhere invertible. (We will maintain these hypotheses throughout the introduction.) It is now known to the experts that the Witt groups of the residue fields of X form a nonexact cochain complex

Abstract. We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation of the center of the resulting deformed ring. This connects with earlier calculations by Vafa and Witten of chiral numbers and deformations of a similar orbifold. 1.

Abstract. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on a manifold X. Let TwFuk(T ∗ X) be the triangulated A∞-category of twisted complexes in the Fukaya category of the cotangent bundle T ∗ X. We prove that Sh(X) embeds as an A∞-subcategory of TwFuk(T ∗ X). Taking cohomology gives an embedding of the corresponding derived categories.

Abstract. We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K-theory. We also deduce versions of Thomason’s blow-up formula and the projective bundle formula for THH and TC. 1.

It known from the work of Feigin-Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite group G acting on X the same procedure applied to G-equivariant sheaves gives the orbifold cohomology of X/G. As an application, in some cases we are able to obtain simple proofs of an additive isomorphism between the orbifold cohomology of X/G and the usual cohomology of its crepant resolution (the equality of Euler and Hodge numbers was obtained earlier by various authors). We also state some conjectures on the product structures, as well as the singular case; and a connection with a recent work by Kawamata. 1

We construct an A∞-category D(C|B) from a given A∞-category C and its full subcategory B. The construction resembles a particular case of Drinfeld’s quotient of differential graded categories [Dri02]. We use D(C|B) to construct an A∞-functor of K-injective resolutions of a complex. The conventional derived category is obtained as the 0-th cohomology of the quotient of differential graded category of complexes over acyclic complexes. We continue to study the 2-category of A∞-categories introduced in [Lyu02]. Our main subject is a quotient-like A∞-category D(C|B) obtained from a given A∞-category C and its full subcategory B. Originally it has been defined by Drinfeld for differential graded categories [Dri02]. Bondal and Kapranov proposed to produce triangulated categories out of differential graded categories [BK90]. Drinfeld’s construction deals with their quotients, in particular, it produces derived categories. The usefulness of A∞-approach is explained by our construction of an A∞-functor, which assigns to a complex its K-injective resolution, when the ground ring is a field. Plan of the article with comments. In the first section we describe conventions and notations used in the article. In particular, we recall some conventions and useful formulas from [Lyu02].