Related papers: A note on abelian quotient categories

We prove an analogon of the the fundamental homomorphism theorem for certain classes of exact and essentially surjective functors of Abelian categories $\mathscr{Q}:\mathcal{A} \to \mathcal{B}$. It states that $\mathscr{Q}$ is up to…

Let $\CC$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set $\mathcal G^T(\CC)$ of generic values of the cluster character associated to…

We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$…

For an abelian category and a distinguished object with a graded endomorphism ring a necessary and sufficient criterion is given so that the category is equivalent to the abelian quotient of the category of finitely presented graded modules…

Let $R$ be a commutative ring. A full additive subcategory $\C$ of $R$-modules is triangulated if whenever two terms of a short exact sequence belong to $\C$, then so does the third term. In this note we give a classification of…

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting…

First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term…

We study the maximal rigid subcategories in $2-$CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously maximal rigid; we prove that the converse is true if the $2-$CY triangulated categories…

We study certain toric Gorenstein varieties with isolated singularities which are the quotient spaces of generic unimodular representations by the one-dimensional torus, or by the product of the one-dimensional torus with a finite abelian…

We explicitly describe the Albanese morphism of a hyperelliptic variety, i.e., the quotient $X$ of an abelian variety $A$ by a finite group $G$ acting freely and not only by translations, by giving a description of the Albanese variety and…

We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection…

We show that if a (not necessarily algebraic) triangulated category T contains an admissible hereditary abelian subcategory H, then we can lift the inclusion of H into T to a fully faithful triangle functor from the whole of the bounded…

Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext…

We introduce a new cluster character with coefficients for a cluster category $\mathcal{C}$ and rather than using a Frobenius $2$-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster…

We introduce the notion of mutation of $n$-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules…

We give an example of a cluster-tilted algebra A with quiver Q, such that the associated cluster algebra has a denominator vector which is not the dimension vector of any indecomposable A-module. This answers a question posed by T.…

We study the notion of positive and negative complexity of pairs of objects in cluster categories. The first main result shows that the maximal complexity occurring is either one, two or infinite, depending on the representation type of the…

We define the notion of right $n$-angulated category, which generalizes the notion of right triangulated category. Let $\mathcal{C}$ be an additive category or $n$-angulated category and $\mathcal{X}$ a covariantly finite subcategory, we…

We show that for $2\le d\le 4$, every finite geometric simplicial complex $\Delta$ in $\mathbb{R}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $\Delta, T$ and $C$ all have the…

We study a category $\mathcal{C}_2$ of $\mathbb{Z}$-graded MCM modules over the $A_\infty$ curve singularity and demonstrate it has infinite type $A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable…