Abstract not found

Abstract. For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Morita-like equivalence between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial description meets the expectations and delivers a series of new results, which allows us to establish a well-rounded theory for Gorenstein dimensions. For any pair of adjoint functors, C

This is the first in a series of papers meant to introduce a notion of regularity on abelian varieties and more general irregular varieties. This notion, called Mukai regularity, is based on Mukai’s concept of Fourier transform, and in a very particular form (called Theta regularity) it parallels and strengthens the usual Castelnuovo-Mumford

Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homologies of Connes' convolution algebra of the groupoid, and derive some spectral sequences which serve as a tool for the computation of these homologies.

In this review we study BPS D-branes on Calabi–Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Π-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter

Abstract. We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition. These ideas are central to constructing realization functors on the homotopy theory of schemes [DI1, Is]. We give a few other applications for this new description of the homotopy theory of simplicial presheaves. Contents

We define a particular class of topological field theories associated to open strings and prove the resulting D-branes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0-brane on any Calabi–Yau threefold must become unstable along some path in the Kähler moduli space. As a byproduct of this analysis we see how the derived category can be invariant under a birational transformation The idea that a D-brane is simply some subspace of the target space where open strings are allowed to end is clearly too simple. Even at zero string coupling, we are faced with carefully analyzing the non-linear σ-model of maps from the string worldsheet into the target space. It is well-known that the non-linear σ-model modifies the usual rules of classical geometry

We consider the compactification of the E8×E8 heterotic string on a K3 surface with “the spin connection embedded in the gauge group ” and the dual picture in the type IIA string (or F-theory) on a Calabi–Yau threefold X. It turns out that the same X arises also as dual to a heterotic compactification on 24 point-like instantons. X is necessarily singular, and we see that this singularity allows the Ramond-Ramond moduli on X to split into distinct components, one containing the (dual of the heterotic) tangent bundle, while another component contains the point-like instantons. As a practical application we derive the result that a heterotic string compactified on the tangent bundle of a K3 with ADE singularities acquires nonperturbatively enhanced gauge symmetry in just the same fashion as a type IIA string on a singular K3 surface. On a more philosophical level we discuss how it appears to be natural to say that the heterotic string is compactified using an object in the derived category of coherent sheaves. This is necessary to properly extend the notion of T-duality to the heterotic

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

We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the Feigin-Tsygan-Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the non-Hausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...