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We show that pretriangulated dg categories enjoy a universal property and deduce that the passage to an orbit quotient commutes with the dg quotient. In particular, for a triangulated category with dg enhancement and an endofunctor, there exists a unique triangulated orbit category. As an application, we prove that for any connective, smooth and proper dg algebra $A$, its perfect derived category is equivalent to the generalized $(\mathbb{X}-1)$-cluster category of $A$. This implies that the orbit $m$-cluster category of $A$ is equivalent to the generalized $m$-cluster category of $A$, which implies a conjecture by Ikeda-Qiu for the case when $A$ is a smooth proper graded gentle algebra.

In a survey paper in 2011, Amiot proposed a conjectural characterisation of the cluster categories which were conceived in the mid 2000s to lift the combinatorics of Fomin--Zelevinsky's cluster algebras to the categorical level. This paper is devoted to a proof of (a variant of) her conjecture. More generally, cluster categories admit higher dimensional and relative variants, the so-called Higgs categories recently introduced by Wu. We also prove higher-dimensional and relative variants of the conjecture., Comment: 42 pages. arXiv admin note: text overlap with arXiv:2307.16222

In 2015, Van den Bergh showed that complete 3-Calabi-Yau algebras over an algebraically closed field of characteristic 0 are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields. The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism. For example, every ice quiver with potential gives rise to a Ginzburg morphism. We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism. In particular, in dimension 3 and over an algebraically closed ground field of characteristic 0, it is given by an ice quiver with potential. Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a non-commutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry., Comment: 39 pages; v2: more accurate historical account in introduction, reference to Joyce-Safronov's work added, many minor changes

Cluster algebras *with coefficients* are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, ... . The approach of Geiss-Leclerc-Schr\"oer often yields Frobenius exact categories which allow to categorify such cluster algebras. In previous work, the third-named author has constructed Higgs categories and relative cluster categories in the relative Jacobi-finite setting (arXiv:2109.03707). Higgs categories generalize the Frobenius categories used by Geiss-Leclerc-Schr\"oer. In this article, we construct the Higgs category and the relative cluster category in the relative Jacobi-infinite setting under suitable hypotheses. These cover for example the case of Jensen-King-Su's Grassmannian cluster category. As in the relative Jacobi-finite case, the Higgs category is no longer exact but still extriangulated in the sense of Nakaoka-Palu. We also construct a cluster character refining Plamondon's. In the appendix, Chris Fraser and the second-named author categorify quasi-cluster morphisms using Frobenius categories. A recent application of this result is due to Matthew Pressland, who uses it to prove a conjecture by Muller-Speyer., Comment: 44 pages, with an appendix by Chris Fraser and Bernhard Keller; v2: Corrections in abstract, references and address; v3: Corrections in abstract and references, Thm 4.14 added

In 2009, Claire Amiot gave a construction of Calabi-Yau structures on Verdier quotients. We sketch how to lift it to the dg setting. We use this construction as an important step in an outline of the proof of her conjecture on the structure of 2-Calabi-Yau triangulated categories with a cluster-tilting object., Comment: v1: 8 pages; v2: minor changes, accepted for publication in Comptes Rendus Math\'ematique; v3: added acknowledgment and reference to expanded version; v4: minor changes, appeared in Comptes Rendus Math\'ematique

We provide an outline of the proof of the Donovan--Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds. The proof relies on results of August, of Hua and the second-named author, Wemyss, and on the Derived Auslander--Iyama Correspondence -- a recent result by the first- and third-named authors., Comment: 25 pages. v4: Corrected several minor typos. v3: New title; final version; to appear in the proceedings of the Abel Symposium 2022: Triangulated categories in representation theory and beyond

We obtain a multiplication formula for cluster characters on (stably) 2-Calabi-Yau (Frobenius or) triangulated categories. This formula generalizes those known for arbitrary pairs of objects and for Auslander-Reiten triangles. As an application, we show that for cluster algebras of acyclic types, specialization of a cluster variable to 1 sends all cluster variables to elements of a cluster algebra of smaller rank. We also obtain application to the reduction of friezes of acyclic type., Comment: 18 pages, v2 updated references

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how -- combined with crucial results of August and Hua-Keller -- our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds., Comment: Appendix by Bernhard Keller. 117 pp., 2 figs. v2: added refs; corrected minor typos and other small changes; new subsec 5.5.4 on the comparison with topological enhancements. v3: corrected minor typos; new subsec 4.6 on an example of a universal Massey product. v4: new title; added further details to the proof of Thm B; several minor edits and typos corrected. v5: corrected typos

In 2016, Leclerc constructed conjectural cluster structures on open Richardson varieties using representations of preprojective algebras. A variant with more explicit seeds was obtained by M\'enard in his thesis. We show that M\'enard's seeds do yield *upper* cluster algebra structures on open Richardson varieties and discuss the problems that remain in order to prove that they are cluster algebra structures., Comment: 11 pages

It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed $t$: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in $t$ and another element not divisible by the cluster variables in $t$. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, $d$-vectors, $F$-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of $d$-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its $F$-polynomial; the $F$-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.

These are notes taken by the second author for a series of three lectures by the first author on absolute and relative Calabi-Yau completions and Calabi-Yau structures given at the workshop of the International Conference on Representations of Algebras which was held online in November 2020. Such structures are relevant for (higher) representation theory as well as for the categorification of cluster algebras with coefficients. After a quick reminder on dg categories and their Hochschild and cyclic homologies, we present examples of absolute and relative Calabi-Yau completions (in the sense of Yeung). In many examples, these are related to higher preprojective algebras in the sense of Iyama-Oppermann. We conclude with the definition of relative (left and right) Calabi-Yau structures after Brav-Dyckerhoff., Comment: 17 pages; v2: Added reminder on cyclic homology, corrected typos, to appear in the proceedings of the ICRA 2020 (online); v3: References corrected and updated

For any finite dimensional algebra $\Lambda$ given by a quiver with relations, we prove that its dg singularity category is quasi-equivalent to the perfect dg derived category of a dg Leavitt path algebra. The result might be viewed as a deformed version of the known description of the dg singularity category of a radical-square-zero algebra in terms of a Leavitt path algebra with trivial differential. The above result is achieved in two steps. We first introduce the singular Yoneda dg category of $\Lambda$, which is quasi-equivalent to the dg singularity category of $\Lambda$. The construction of this new dg category follows from a general operation for dg categories, namely an explicit dg localization inverting a natural transformation from the identity functor to a dg endofunctor. This localization turns out to be quasi-equivalent to a dg quotient category. Secondly, we prove that the endomorphism algebra of the quotient of $\Lambda$ modulo its Jacobson radical in the singular Yoneda dg category is isomorphic to the dg Leavitt path algebra. The appendix is devoted to an alternative proof of the result using Koszul-Moore duality and derived localizations., Comment: Appendix by Bernhard Keller and Yu Wang, v2 some changes in exposition, 51 pages

The $\mathbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\mathbf{g}$-vectors of its $2$-term presilting objects. We prove that the $\mathbf{g}$-vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster $\mathbf{g}$-vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster scattering diagrams only differ by wall-crossing functions on the walls contained in a separating hyperplane. The appendix is devoted to the construction of truncated twist functors and their adjoints., Comment: Appendix by Bernhard Keller. 34 pages

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., Comment: 8 pages. Extended the introduction, including some historical comments. Revised the proof of Theorem 1, fixing an error. Added details about derived (co)limits. Accepted for publication with Comptes Rendus Math\'ematique

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., Comment: 5 pages; v2: references completed, acknowledgments added

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., Comment: 51 pages. This version contains a proof of the transformation formula of DT invariants (weighted by Behrend function), and two new subsections on perverse F-series and anti-cluster character

Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic bases. We survey recent progress on their existence and properties and give a representation-theoretic proof of Greg Muller's theorem stating that full subquivers inherit maximal green sequences. In the appendix, Laurent Demonet describes maximal chains of torsion classes in terms of bricks generalizing a theorem by Igusa., Comment: 15 pages, submitted to the proceedings of the ICRA 18, Prague, comments welcome; v2: misquotation in section 6 corrected; v3: minor changes, final version; v4: reference to Jiarui Fei's work added, post-final version; v4: formulation of Remark 4.3 corrected; v5: misquotation of Hermes-Igusa's 2019 paper corrected; v5: reference to Kim-Yamazaki's paper added

We prove that derived equivalent algebras have isomorphic differential calculi in the sense of Tamarkin--Tsygan., Comment: 7 pages

We study rational curves on smooth complex Calabi--Yau threefolds via noncommutative algebra. By the general theory of derived noncommutative deformations due to Efimov, Lunts and Orlov, the structure sheaf of a rational curve in a smooth CY 3-fold $Y$ is pro-represented by a nonpositively graded dg algebra $\Gamma$. The curve is called nc rigid if $H^0\Gamma$ is finite dimensional. When $C$ is contractible, $H^0\Gamma$ is isomorphic to the contraction algebra defined by Donovan and Wemyss. More generally, one can show that there exists a $\Gamma$ pro-representing the (derived) multi-pointed deformation (defined by Kawamata) of a collection of rational curves $C_1,\ldots,C_t$ so that ${\mathrm{dim}}({\rm{Hom}}_Y({\mathcal{O}}_{C_i},{\mathcal{O}}_{C_j}))=\delta_{ij}$. The collection is called nc rigid if $H^0\Gamma$ is finite dimensional. We prove that $\Gamma$ is a homologically smooth bimodule 3CY algebra. As a consequence, we define a (2CY) cluster category ${\mathcal{C}}_\Gamma$ for such a collection of rational curves in $Y$. It has finite-dimensional morphism spaces iff the collection is nc rigid. When $\bigcup_{i=1}^tC_i$ is (formally) contractible by a morphism $\hat{Y}\to \hat{X}$, ${\mathcal{C}}_\Gamma$ is equivalent to the singularity category of $\hat{X}$ and thus categorifies the contraction algebra of Donovan and Wemyss. The Calabi-Yau structure on $Y$ determines a canonical class $[w]$ (defined up to right equivalence) in the zeroth Hochschild homology of $H^0\Gamma$. Using our previous work on the noncommutative Mather--Yau theorem and singular Hochschild cohomology, we prove that the singularities underlying a 3-dimensional smooth flopping contraction are classified by the derived equivalence class of the pair $(H^0\Gamma, [w])$. We also give a new necessary condition for contractibility of rational curves in terms of $\Gamma$., Comment: This is a final version where several errors have been corrected. To appear in Geometry and Topology

We show that for a noetherian algebra $A$ whose bounded dg derived category is smooth, the singular Hochschild cohomology (=Tate--Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category of $A$. The existence of such an isomorphism is suggested by recent work of Zhengfang Wang., Comment: v4: 7 pages, statement of Lemma 2.4 corrected, v5: error at the beginning of the proof of Theorem 1.1 corrected, v6: commutative ground ring re-established, v7: Proof of Theorem 1.3 expanded and corrected, v8: references corrected, v9: Smoothness hypothesis added following a counterexample by Xiao-Wu Chen, v10: Hyp. on finite gldim of ground ring added following a remark by Xiao-Wu Chen

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., Comment: 5 pages

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., Comment: 3 pages

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., Comment: 35 pages, with appendices by Tobias Barthel, Henning Krause and Bernhard Keller. Version 3: Added a new appendix by Tobias Barthel and Henning Krause that refines for a separated noetherian scheme the results of the main text. Added a discussion of completions of abelian categories and their derived categories. Version 4: Minor revision, correcting some examples

We prove derived invariance of the cap product for associative algebras projective over a commutative ring., Comment: 4 pages

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., Comment: 12 pages, slightly corrected version of a submission to FPSAC 2013

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., Comment: 27 pages

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., Comment: 39 pages; v2: title in metadata corrected

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., Comment: 14 pages. v2: Bibliography expanded. v3: Added results 3.7, 3.8, 4.7, 4.8 and 5.6. v4: Added reference and minor changes