arXiv subjects
Publications and source records attributed to Bernhard Keller.
Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen and Toen-Vaquie.
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the Calabi-Yau property of the cluster category.
In this survey, we first present basic facts on A-infinity algebras and modules including their use in describing triangulated categories. Then we describe the Quillen model approach to A-infinity structures following K. Lefevre's thesis. Finally, starting from an idea of V. Lyubashenko's, we give a conceptual construction of A-infinity functor categories using a suitable closed monoidal category of cocategories. In particular, this yields a natural construction of the bialgebra structure on the bar construction of the Hochschild complex of an associative algebra.
We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R. Marsh and I. Reiten which appeared in their study with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (closely related to work by Caldero-Chapoton-Schiffler) and a question by H. Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many easy examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel, whose triangulated structure goes back to Auslander-Reiten's work on the representation-theoretic approach to rational singularities.
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of the cluster category. This realization provides a natural basis for A. We prove new results and formulate conjectures on `good basis' properties, positivity, denominator theorems and toric degenerations.
We interpret Hochschild cohomology as the Lie algebra of the derived Picard group (in the sense of Rouquier-Zimmermann and Yekutieli) and deduce that it is preserved under derived equivalences.
Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and contravariant homological functors C --> Ab which are not restrictions of representables.
These are expanded notes of four introductory talks on A-infinity algebras, their modules and their derived categories.
We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.
We present a new and explicit method for lifting a tilting complex to a bimodule complex. The key ingredient of our method is the notion of a strong homotopy action in the sense of Stasheff.