Related papers: Tilting preserves finite global dimension

In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We introduce the global dimension and extension dimension of extriangulated categories, and give some upper bounds of global…

For abelian length categories the borderline between finite and infinite representation type is discussed. Characterisations of finite representation type are extended to length categories of infinite height, and the minimal length…

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. We investigate the structure properties of the endomorphism algebras of semi-tilting $A$-modules, and prove that the endomorphism algebras arising from the…

We give an upper bound on the dimension of the bounded derived category of an abelian category. We show that if $\CX$ is a sufficiently nice subcategory of an abelian category, then derived dimension of $\CA$ is at most $\CX$-dim$\CA$,…

For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings.

Given a finite dimensional algebra $\Lambda$, we show that a frequently satisfied finiteness condition for the category ${\cal P}^{\infty}(\Lambda\rm{-mod})$ of all finitely generated (left) $\Lambda$-modules of finite projective dimension,…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field $k$. We survey some results on algebras of finite global dimension and address some open problems.

We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander-Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite…

In a compactly generated triangulated category, we introduce a class of tilting objects satisfying certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent…

Let C be a connected noetherian hereditary abelian Ext-finite category with Serre functor over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from…

We consider filtrations of objects in an abelian category $\catA$ induced by a tilting object $T$ of homological dimension at most two. We define three disjoint subcategories with no maps between them in one direction, such that each object…

The strong global dimension of a ring is the supremum of the length of perfect complexes that are indecomposable in the derived category. In this note we characterize the noetherian commutative rings that have finite strong global…

We characterize the modules of infinite projective dimension over the endomorphism algebras of Opperman-Thomas cluster tilting objects $X$ in $(n+2)$-angulated categories $(\mathcal C,\Sigma^n,\Theta)$. For an indecomposable object $M$ of…

Let $A$ be an Auslander algebra of global dimension equal to 2. We provide a necessary and sufficient condition for $A$ to be a tilted algebra. In particular, $A$ is tilted if and only if pd$(\tau_{A}\Omega_{A}DA)\leq1$.

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, $\mathcal {D}^b(A)$ be the bounded derived category of $A$-mod and $A^{(m)}$ be the $m$-replicated algebra of $A$. In this paper, we investigate the structure…

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 give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables. We use this to show that the global dimension of a finite…

Mean dimension may decrease after taking the natural extension. In this paper we show that mean dimension is preserved by natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the…

In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be…

We prove the existence of tilting objects on some global quotient stacks. As a consequence we provide further evidence for a conjecture on the Rouquier dimension of derived categories formulated by Orlov.