### Algebra invariants for finite directed graphs with relations.

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Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\le 2$. We construct a triangulated category ${\mathcal{C}}_{A}$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When ${\mathcal{C}}_{A}$ is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also...

We construct derived equivalences between generalized matrix algebras. We record several corollaries. In particular, we show that the $n$-replicated algebras of two derived equivalent, finite-dimensional algebras are also derived equivalent.

We describe the endomorphism rings of maximal rigid objects in the cluster categories of tubes. Moreover, we show that they are gentle and have Gorenstein dimension 1. We analyse their representation theory and prove that they are of finite type. Finally, we study the relationship between the module category and the cluster tube via the Hom-functor.

Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $\mathbf{K}\left(R\text{-}\mathrm{Proj}\right)$ of projective $R$-modules. We produced a set of generators for this category, proved that the category is ${\aleph}_{1}$-compactly generated for any ring $R$, and showed that it need not always be compactly generated, but is for sufficiently nice $R$. We furthermore analyzed the inclusion ${j}_{!}^{}:\mathbf{K}\left(R\text{-}\mathrm{Proj}\right)\to \mathbf{K}\left(R\text{-}\mathrm{Flat}\right)$ and the orthogonal subcategory $\mathcal{S}={\mathbf{K}\left(R\text{-}\mathrm{Proj}\right)}^{\perp}$. And we even showed that the inclusion $\mathcal{S}\to \mathbf{K}\left(R\text{-}\mathrm{Flat}\right)$ has a right adjoint; this forces some natural map to be an equivalence...

We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander-Reiten sequences. We also prove an Auslander-Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when this category...

We study equivalences for category ${\mathcal{O}}_{p}$ of the rational Cherednik algebras ${\mathbf{H}}_{p}$ of type ${G}_{\ell}\left(n\right)={\left({\mu}_{\ell}\right)}^{n}\u22ca{\U0001d516}_{n}$: a highest weight equivalence between ${\mathcal{O}}_{p}$ and ${\mathcal{O}}_{\sigma \left(p\right)}$ for $\sigma \in {\U0001d516}_{\ell}$ and an action of ${\U0001d516}_{\ell}$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between ${\mathcal{O}}_{p}$ and ${\mathcal{O}}_{{p}^{\text{'}}}$ whenever $p$ and ${p}^{\text{'}}$ have integral difference; a highest weight equivalence between ${\mathcal{O}}_{p}$ and a parabolic category $\mathcal{O}$ for the general linear group, under a non-rationality assumption on the parameter $p$. As a consequence, we confirm special cases of conjectures...

Let $A$ be a finite-dimensional $k$-algebra and $K/k$ be a finite separable field extension. We prove that $A$ is derived equivalent to a hereditary algebra if and only if so is $A{\otimes}_{k}K$.

Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $({Q}^{sg},{I}^{sg})$ and $({Q}^{g},{I}^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K{Q}^{sg}/\u27e8{I}^{sg}\u27e9$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K{Q}^{g}/\u27e8{I}^{g}\u27e9$. As a corollary, we find that $gldimK{Q}^{sg}/\u27e8{I}^{sg}\u27e9<\infty $ if and only if $gldimKQ/\u27e8I\u27e9<\infty $ if and only if $gldimK{Q}^{g}/\u27e8{I}^{g}\u27e9<\infty $.