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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. [ABSTRACT FROM AUTHOR]

These are slightly expanded notes of lectures given in April 2019 at the Isfahan school and conference on representations of algebras. We recall the formalism of derived categories and functors and survey invariance results for the Hochschild (co)homology of differential graded algebras and categories. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We correct a mistake that occurred in the proof of the main theorem of "Singular Hochschild cohomology via singularity categories" and some inaccuracies in the proof of the reconstruction theorem. [ABSTRACT FROM AUTHOR]

Abstract We prove that derived equivalent algebras have isomorphic differential calculi in the sense of Tamarkin–Tsygan. [ABSTRACT FROM AUTHOR]

Abstract We show that the singular Hochschild cohomology (= Tate–Hochschild cohomology) of an algebra A is isomorphic, as a graded algebra, to the Hochschild cohomology of the differential graded enhancement of the singularity category of A. The existence of such an isomorphism is suggested by recent work by Zhengfang Wang. [ABSTRACT FROM AUTHOR]

Abstract Claire 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. [ABSTRACT FROM AUTHOR]

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 δ-functor Φ taking each finitedimensional representation of S to an object of the derived category of Q. We show that the functor Φ establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Φ. 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. [ABSTRACT FROM AUTHOR]

The amoeba-resistant bacterium Legionella pneumophila causes Legionnaires' disease and employs a type IV secretion system (T4SS) to replicate in the unique, ER-associated Legionella-containing vacuole (LCV). The large fusion GTPase Sey1/atlastin is implicated in ER dynamics, ER-derived lipid droplet (LD) formation, and LCV maturation. Here, we employ cryo-electron tomography, confocal microscopy, proteomics, and isotopologue profiling to analyze LCV-LD interactions in the genetically tractable amoeba Dictyostelium discoideum. Dually fluorescence-labeled D. discoideum producing LCV and LD markers revealed that Sey1 as well as the L. pneumophila T4SS and the Ran GTPase activator LegG1 promote LCV-LD interactions. In vitro reconstitution using purified LCVs and LDs from parental or ?sey1 mutant D. discoideum indicated that Sey1 and GTP promote this process. Sey1 and the L. pneumophila fatty acid transporter FadL were implicated in palmitate catabolism and palmitate-dependent intracellular growth. Taken together, our results reveal that Sey1 and LegG1 mediate LD- and FadL-dependent fatty acid metabolism of intracellular L. pneumophila. [ABSTRACT FROM AUTHOR]

We prove the derived invariance of the cap product for associative algebras projective over a commutative ring. [ABSTRACT FROM AUTHOR]

The |$\textbf{g}$| -vector fan of a finite-dimensional algebra is a fan whose rays are the |$\textbf{g}$| -vectors of its two-term presilting objects. We prove that the |$\textbf{g}$| -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 |$\textbf{g}$| -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. [ABSTRACT FROM AUTHOR]

The original paper did not clearly state that Pierre-Guy Plamondon and Toshiya Yurikusa were the author of the paper, but that Bernhard Keller was the author of the appendix. One digram originally appeared as Graph It should have appeared as Graph In reference 12 "Yildirim" should have appeared as "Yildirim". [Extracted from the article]

For any finite dimensional algebra Λ 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 Λ, which is quasi-equivalent to the dg singularity category of Λ. 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 Λ 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. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field and A a finite-dimensional k-algebra. In this note, we determine complexes which compute the Hochschild homology of the canonical dg enhancement of the bounded derived category of A and of the canonical dg enhancement of the singularity category of A. As an application, we obtain a new approach to the computation of Hochschild homology of Leavitt path algebras. [ABSTRACT FROM AUTHOR]

Let t ∈ N, let K be a field and let VKt denote the class of all algebraic vector bundles over the projective space PKt The cohomology table of a bundle ε ∈ VKt is defined as the family of nonnegative integers hε := (hi(PKt, ε(n)))(i,n)∈N0×Z. A set S ⊆ {0, . . ., t}×Z is said to be a bounding pattern for the cohomology of vector bundles over PKt if for each family (h(i,n))(i,n)∈S of non-negative integers, the set of cohomology tables {hε | ε ∈ VKt : hεi (n) ≤ h(i,n) for all (i, n) ∈ S} is finite. Our main result says that this is the case if and only if S contains a quasi-diagonal of width t, that is, a set of the form {(i, ni)| i = 0, . . ., t} with integers n0 > n1 > · · · > nt. [ABSTRACT FROM AUTHOR]

The University has quite a big campus located not far from the city center, in a quiet and convenient area with many facilities. The School was supported by Centre International de Mathématiques Pures et Appliquées (CIMPA), Agence Universitaire de la Francophonie (AUF), Institute for Research in Fundamental Sciences-Isfahan branch (IPM-Isfahan), Iran National Science Fundation (INSF) and University of Isfahan. Representation theory is a central branch of modern mathematics that studies realizations of abstract non-linear structures using classical linear and combinatorial concrete structures like matrices, linear operators and quivers. [Extracted from the article]