Related papers: Tilting objects in singularity categories and leve…
In this paper, we introduce a class of twisted matrix algebras of $M_2(E)$ and twisted direct products of $E\times E$ for an algebra $E$. Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, $z\in A_2$ be a regular central element…
For a left artinian ring A, we study the lattice torsA of torsion pairs in the category of finitely generated A-modules by considering an isomorphic lattice formed by certain closed sets in a topological space associated to A, the Ziegler…
Classical affine Lie algebras appear e.g. as symmetries of infinite dimensional integrable systems and are related to certain differential equations. They are central extensions of current algebras associated to finite-dimensional Lie…
We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$ of this category associated with each…
The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When…
We observe that there exists an associative finite dimensional $\mathbb{C}$-algebra $A$ of finite global dimension, such that the bounded derived category $D^b(A)$ of finite dimensional $A$-modules admits an admissible subcategory…
An important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Moreover, cotilting classes are…
We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced…
Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau, and realize its cluster category as a…
We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic $d$-Calabi-Yau triangulated category with a $d$-cluster tilting object $T$ such that its shifted sum…
We say that an algebra $\Lambda$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $\Lambda$-modules. We show…
We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra $\mathcal{H}(\mathcal{E})$ of…
We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(\Gamma)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $\Gamma = P^+ \times J$, $J$ is an…
Let $S$ be an $\mathbb N$-graded Koszul Artin-Schelter regular algebra and let $\sigma$ be a graded algebra automorphism of $S$. We study the stable category of graded maximal Cohen-Macaulay modules over the trivial extension algebra…
We show that tilting modules for quantum groups over local Noetherian domains exist and that the indecomposable tilting modules are parametrized by their highest weight. For this we introduce a model category ${\mathcal X}={\mathcal…
We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be…
We prove that a finite dimensional algebra is $\tau$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $\tau$-tilting finite algebra $A$ there is a bijection between isomorphism classes of…
We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more…
For any finite sequence of elements $s_1, \ldots , s_d$ in a commutative noetherian ring $R$, we show that for $n \gg 0$, the natural map from the Koszul complex $K(s_1^n, \ldots , s_d^n)$ to the Koszul complex $K(s_1, \ldots , s_d)$…
We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated…