arXiv subjects
Publications and source records attributed to Bernhard Keller.
Given a tilting object of the derived category of an abelian category of finite global dimension, we give (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring.
Given a quiver $Q$, a formal potential is called analytic if its coefficients are bounded by the terms of a geometric series. As shown by Toda, the potentials appearing in the deformation theory of complexes of coherent sheaves on complex projective Calabi-Yau threefolds are analytic. Our paper consists of two parts. In the first part, we establish the foundations of the differential calculus of quivers with analytic potentials and prove two fundamental results: the inverse function theorem and Moser's trick. As an application, we prove finite determinacy of analytic potentials with finite-dimensional Jacobi algebra, answering a question of Ben Davison. We also prove a Mather-Yau type theorem for analytic potentials with finite-dimensional Jacobi algebra, extending previous work by the first author with Zhou. In the second part, we study Donaldson-Thomas theory of quivers with analytic potentials. First, we construct a canonical perverse sheaf of vanishing cycles on the moduli stack of finite dimensional modules over the Jacobi algebra of an arbitrary iterated mutation via a separation lemma for analytic potentials. Finally, we provide a transformation formula for DT invariants (weighted by the Behrend function) under iterated mutations as a counterpart of Nagao's result on topological DT invariants. Our result leads to a perverse analogue of the $F$-series in cluster algebras. It also yields a Behrend--weighted Caldero--Chapoton formula for what we call anti-cluster algebras.
Applying recent results by Lowen-Van den Bergh we show that Hochschild cohomology is preserved under Koszul-Moore duality as a Gerstenhaber algebra. More precisely, the corresponding Hochschild complexes are linked by a quasi-isomorphism of B-infinity-algebras.
This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a proper foundation for completing a triangulated category.
We define $n$-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller's parametrization of pre-triangulations extends to pre-$n$-angulations. We obtain a large class of examples of $n$-angulated categories by considering $(n-2)$-cluster tilting subcategories of triangulated categories which are stable under the $(n-2)$nd power of the suspension functor. As an application, we show how $n$-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension. Finally, we sketch a link to algebraic geometry and string theory.
We prove that derived equivalent algebras have isomorphic differential calculi in the sense of Tamarkin--Tsygan.
We correct an error and some inaccuracies that occurred in "Deformed Calabi-Yau completions''. The most important point is that, as pointed out by W. K. Yeung, to show that the deformed Calabi-Yau completion has the Calabi-Yau property, one needs to assume that the deformation parameter comes from negative cyclic homology. Notice that this does hold in the case of Ginzburg dg algebras.
C. Amiot has classified the connected triangulated k-categories with finitely many isoclasses of indecomposables satisfying suitable hypotheses. We remark that her proof shows that these triangulated categories are determined by their underlying k-linear categories. We observe that if the connectedness assumption is dropped, the triangulated categories are still determined by their underlying k-categories together with the action of the suspension functor on the set of isoclasses of indecomposables.
We prove derived invariance of the cap product for associative algebras projective over a commutative ring.
A quiver is an oriented graph. Quiver mutation is an elementary operation on quivers. It appeared in physics in Seiberg duality in the nineties and in mathematics in the definition of cluster algebras by Fomin-Zelevinsky in 2002. We show, for large classes of quivers Q, using quiver mutation and quantum dilogarithms, one can construct the combinatorial DT-invariant, a formal power series intrinsically associated with Q. When defined, it coincides with the "total" Donaldson-Thomas invariant of Q (with a generic potential) provided by algebraic geometry (work of Joyce, Kontsevich-Soibelman, Szendroi and many others). We illustrate combinatorial DT-invariants on many examples and point out their links to quantum cluster algebras and to (infinite) generalized associahedra.
Inspired by recent work of Cerulli-Feigin-Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.
Inspired by recent work of Hernandez-Leclerc and Leclerc-Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc-Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category S. We determine the quiver of S and the number of minimal relations between any two given vertices. We construct a delta-functor Phi taking each finite-dimensional representation of S to an object of the derived category of Q. We show that the functor Phi establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Phi. If the underlying graph of Q is an ADE Dynkin diagram, the image is the whole derived category; otherwise, it is the category of "line bundles over the non commutative curve given by Q". We show that the degeneration order between strata corresponds to Jensen-Su-Zimmermann's degeneration order on objects of the derived category. Moreover, if Q is an ADE Dynkin quiver, the singular category S is weakly Gorenstein of dimension 1 and its derived category of singularities is equivalent to the derived category of Q.
Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.
We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky's cluster algebras and their (additive) categorification via triangulated categories.
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coefficients, c-vectors and g-vectors. We show how c-vectors appear in the study of quantum cluster algebras and their links to the quantum dilogarithm. We then present the framework of additive categorification of cluster algebras based on the notion of quiver with potential and on the derived category of the associated Ginzburg algebra. We show how the combinatorics introduced previously lift to the categorical level and how this leads to proofs, for cluster algebras associated with quivers, of some of Fomin-Zelevinsky's fundamental conjectures.
These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin-Zelevinsky's theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock-Goncharov, Kontsevich-Soibelman and Nagao.
Using techniques due to Dwyer-Greenlees-Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category. This allows us, in certain situations, to deduce the existence of a canonical t-structure on the perfect derived category of a dg algebra. From this, we can obtain a bijection between hearts of t-structures and sets of so-called simple-minded objects for some dg algebras (including Ginzburg algebras associated to quivers with potentials). In three appendices, we elucidate the relation between Milnor colimits and homotopy colimits and clarify the construction of t-structures from sets of compact objects in triangulated categories as well as the construction of a canonical weight structure on the unbonded derived category of a non positive dg category.
Integral cluster categories of acyclic quivers have recently been used in the representation-theoretic approach to quantum cluster algebras. We show that over a principal ideal domain, such categories behave much better than one would expect: They can be described as orbit categories, their indecomposable rigid objects do not depend on the ground ring and the mutation operation is transitive.