Alg\`ebres amass\'ees et applications (2009)
Sergey Fomin and Andrei Zelevinsky have invented cluster algebras at the beginning of this decade with the aim of creating an algebraic framework for the study of canonical bases in quantum groups...
ON CLUSTER ALGEBRAS WITH COEFFICIENTS AND 2-CALABI-YAU CATEGORIES (2009)
Abstract. Building on work by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories....
THE HALL ALGEBRA OF A SPHERICAL OBJECT (2009)
Bernhard Keller, Dong Yang, Guodong Zhou
Abstract. We determine the Hall algebra, in the sense of Toën, of the algebraic triangulated category generated by a spherical object. 1.
A BRIEF INTRODUCTION TO A-INFINITY ALGEBRAS (2009)
Abstract. These are notes of a 90-minute talk given at the workshop on
Cluster-tilted algebras are Gorenstein and stably (2009)
Abstract. We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show...
Bernhard Keller, Dieter Vossieck
Abstract (1)- We investigate the relations between the hearts of two t-structures on one triangulated category. Under suitable compatibility conditions, we obtain a common generalization of the...
DERIVED INVARIANCE OF HIGHER STRUCTURES ON THE HOCHSCHILD COMPLEX (2009)
Abstract. We show that derived equivalences preserve the homotopy type of the (cohomological) Hochschild complex as a B∞-algebra. More generally, we prove that, as an object of the homotopy...
EQUALIZERS IN THE CATEGORY OF COCOMPLETE COCATEGORIES (2009)
Bernhard Keller, Oleksandr Manzyuk
Abstract. We prove existence of equalizers in certain categories of cocomplete cocategories. This allows us to complete the proof of the fact that A∞-functor categories arise
1 Manus. Math. 67 (1990), 379–417. CHAIN COMPLEXES AND STABLE CATEGORIES (2009)
complexes concentrated in positive degrees. We thereby obtain a new proof for the key result of J. Rickard’s ’Morita theory for Derived categories ‘ [17] and a sharpening of a theorem of Happel...
On triangulated orbit categories (2009)
Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday
From triangulated categories to cluster algebras (2009)
Philippe Caldero, Bernhard Keller
Abstract. 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...
The author is grateful to Guodong Zhou [3] and to Bill Crawley-Boevey and (2009)
Andrew Hubery [1] for pointing out that in the definition of a right Serre functor
cohomology and derived Picard groups (2009)
Dedicated to Idun Reiten on the occasion of her sixtieth birthday Abstract. We interpret Hochschild cohomology as the Lie algebra of the derived Picard group and deduce that it is preserved under...
CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES (2009)
Abstract. This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is...
Categorification of acyclic cluster algebras: an introduction, to appear (2009)
Bernhard Keller, Denis Diderot, To Murray Gerstenhaber, Jim Stasheff
Summary. This is a concise introduction to Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster...
Manuscript Region of Origin: Manuscript BOUNDED DERIVED CATEGORIES AND REPETITIVE ALGEBRAS (2009)
Dieter Happel, Bernhard Keller, Idun Reiten
Let Λ be a finite dimensional algebra over a field k. It was proved in [H1] that there is a full and faithful embedding of the bounded derived category D b (Λ) into the stable category mod ̂ Λ of...
KOSZUL DUALITY AND CODERIVED CATEGORIES (AFTER K. LEFÈVRE) (2009)
Abstract. This is a brief report on a part of Chapter 2 of K. Lefèvre’s thesis [5]. We sketch a framework for Koszul duality [1] where the Koszul dual algebra is replaced by a coalgebra. This...
Cluster tilting for one-dimensional hypersurface singularities (2009)
Igor Burban, Osamu Iyama, Bernhard Keller, Idun Reiten
Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of...
On the combinatorics of rigid objects in 2-Calabi-Yau categories (2009)
Abstract. Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In...
Bernhard Keller, Van Den Bergh
Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in...
From triangulated categories to cluster algebras (2009)
Philippe Caldero, Bernhard Keller
similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called exceptional Hall algebra, of the cluster category. This realization...
DERIVED CATEGORIES AND TILTING (2009)
Abstract. We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel’s theorem which states that each...
Deformed Calabi-Yau Completions (2009)
We define and investigate deformed n-Calabi-Yau completions of homologically smooth differential graded (=dg) categories. Important examples are: deformed preprojective algebras of connected non...
THE HALL ALGEBRA OF A SPHERICAL OBJECT (2009)
Bernhard Keller, Dong Yang, Guodong Zhou
Abstract. We determine the Hall algebra, in the sense of Toën, of the triangulated category generated by a spherical object of dimension greater or equal to 3. 1.
ON CLUSTER ALGEBRAS WITH COEFFICIENTS AND 2-CALABI-YAU CATEGORIES (2009)
Abstract. Building on work by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott we investigate the link between cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These...
CLUSTER ALGEBRAS AND QUIVER REPRESENTATIONS (2009)
Cluster algebras were invented by Fomin-Zelevinsky in 2000, cf. [23], with the aim to find a combinatorial approach to the objects constructed by Lusztig in two series of articles: the first one on...
Derived equivalences from mutations of quivers with potential (2009)
We show that Derksen-Weyman-Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi-Yau triangulated categories. Our approach is related to that of Iyama-Reiten and...
On the (non)vanishing of some "derived" categories of curved dg algebras (2009)
Keller, Bernhard, Lowen, Wendy, Nicolas, Pedro
Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes,...
On Hochschild Cohomology and Morita Deformations (2009)
Keller, Bernhard, Lowen, Wendy
In this paper we show that, in general, first-order Morita deformations are too limited to capture the second Hochschild cohomology of a differential graded category. For differential graded...
The Hall algebra of a spherical object (2009)
Keller, Bernhard, Yang, Dong, Zhou, Guodong
We determine the Hall algebra, in the sense of B. Toën, of the algebraic triangulated category generated by a spherical object.
The Hall algebra of a spherical object (2008)
Keller, Bernhard, Yang, Dong, Zhou, Guodong
We determine the Hall algebra, in the sense of Toen, of the algebraic triangulated category generated by a spherical object.
Cluster algebras, quiver representations and triangulated categories (2008)
This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on...
Categorification of acyclic cluster algebras: an introduction (2008)
This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras...
On the Combinatorics of Rigid Objects in 2-Calabi-Yau Categories (2008)
Given a triangulated 2-Calabi–Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this...
CORRECTION TO ‘ON TRIANGULATED ORBIT CATEGORIES’ (2008)
The description of the triangulated hull of the orbit category given in section 7 of [2] is probably not correct in general. One obtains a correct description by replacing the quotient D b (B) /...
A-INFINITY ALGEBRAS IN REPRESENTATION THEORY (2007)
Abstract. We give a brief introduction to A1-algebras and show three contexts in which they appear in representation theory: the study of Yoneda algebras and Koszulity, the description of categories...
is an equivalence. We say that the functor (2007)
Abstract. Using classical ideas we show how to construct a universal exact functor from an exact category to an abelian category.
J. Daniel Christensen, Bernhard Keller, Amnon Neeman
dedicated to H. Lenzing on the occasion of his sixtieth birthday
J. Daniel Christensen, Bernhard Keller, Amnon Neeman
dedicated to H. Lenzing on the occasion of his sixtieth birthday
I thank R.-O. Buchweitz for his letter [1], where he points out the following error: Lemma 3.3 of [2] is not true in the form it is stated in the article. The error occurs in the rst line of the...
BIMODULE COMPLEXES VIA STRONG HOMOTOPY ACTIONS (2007)
To Professor Klaus W. Roggenkamp on the occasion of his sixtieth birthday Abstract. We present a new and explicit method for lifting a tilting complex to a bimodule complex. The key ingredient of our...
I thank R.-O. Buchweitz for his letter [1], where he points out and corrects the following mistakes in [2]: In 1.11 of [2], it should be HH (mod k) = k (rather than k[u]). Paragraph 2.5 [2] ends with...
(=DG category). As a first application, we deduce a 'triangulated analogue ` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories...
On cluster algebras with coefficients and 2-Calabi-Yau categories (2007)
Fu, Changjian, Keller, Bernhard
Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include...
On the combinatorics of rigid objects in 2-Calabi-Yau categories (2007)
Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this...
Cluster tilting for one-dimensional hypersurface singularities (2007)
Burban, Igor, Iyama, Osamu, Keller, Bernhard, Reiten, Idun
In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster...
Bounded derived categories and repetitive algebras (2007)
Happel, Dieter, Keller, Bernhard, Reiten, Idun
By a theorem due to the first author, the bounded derived category of a finite-dimensional algebra over a field embeds fully faithfully into the stable category over its repetitive algebra. This...
Equalizers in the category of cocomplete cocategories (2006)
Keller, Bernhard, Manzyuk, Oleksandr
We prove existence of equalizers in certain categories of cocomplete cocategories. This allows us to complete the proof of the fact that A-infinity functor categories arise as internal Hom-objects in...
Acyclic Calabi-Yau categories (2006)
Keller, Bernhard, Reiten, Idun
We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles....
On differential graded categories (2006)
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...
On differential graded categories (2006)
Abstract. 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...
Cluster-tilted algebras are Gorenstein and stably Calabi-Yau (2005)
Keller, Bernhard, Reiten, Idun
We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the...
A-infinity algebras, modules and functor categories (2005)
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...
From triangulated categories to cluster algebras II (2005)
Caldero, Philippe, Keller, Bernhard
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...
From triangulated categories to cluster algebras (2005)
Caldero, Philippe, Keller, Bernhard
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...
On triangulated orbit categories (2005)
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....
Hochschild cohomology and derived Picard groups (2003)
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.
Introduction to {$A$}-infinity algebras and modules (2001)
These are expanded notes of four introductory talks on $A_\infty$-algebras, their modules and their derived categories.
Failure of Brown representability in derived categories, Topology 40 (2001)
J. Daniel Christensen, Bernhard Keller, Amnon Neeman
dedicated to H. Lenzing on the occasion of his sixtieth birthday Abstract. Let T be a triangulated category with coproducts, T c ⊂ T the full subcategory of compact objects in T. If T is the...
Zuckermodifizierte DNA-Analoga : von Bicyclo(3.2.1)DNA zu Tricyclo(5.3.1.0 1.5)DNA / (2001)
Diss. Naturwiss. Bern.
Failure of Brown representability in derived categories (2000)
Christensen, J. Daniel, Keller, Bernhard, Neeman, Amnon
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...
Bimodule complexes via strong homotopy actions (1999)
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.
Introduction to A-infinity algebras and modules (1999)
These are expanded notes of four introductory talks on A-infinity algebras, their modules and their derived categories.
On the cyclic homology of exact categories (1999)
Abstract. 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...
Würzburg, Universiẗat, Diss., 1999.
On the cyclic homology of ringed spaces and schemes (1998)
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...
On the Cyclic Homology of Ringed Spaces and Schemes (1998)
. In their recent proof [2] of Schapira-Schneider's conjecture [19], Bressler-Nest-Tsygan construct a (generalized) Chern character from the K- theory of perfect complexes to the negative cyclic...
and localization for cyclic homology of DG algebras (1998)
Abstract. 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...
Bernhard Keller, Communicated Peter Schneider
Abstract. 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...
On The Cyclic Homology Of Exact Categories (1996)
. The cyclic homology of an exact category was defined by R. McCarthy [17] using the methods of F. Waldhausen [26]. McCarthy's theory enjoys a number of desirable properties, the most basic...
A remark on the generalized smashing conjecture (1994)
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...
Diss. med. Basel (kein Austausch).
Würzburg, Univ., Fachbereich Medizin, Diss., 1976.