Gluing of n-Cluster Tilting Subcategories for Representation-directed Algebras

Given $n\leq d<\infty $ n ≤ d < ∞ , we investigate the existence of algebras of global dimension d which admit an n-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras A and B, a projective A-module P and an injective B-module I satisfying certain conditions, we show how we can construct a new representation-directed algebra "Image missing" in such a way that the representation theory of Λ is completely described by the representation theories of A and B. Next we introduce n-fractured subcategories which generalize n-cluster tilting subcategories for representation-directed algebras. We then show how one can construct an n-cluster tilting subcategory for Λ by using n-fractured subcategories of A and B. As an application of our construction, we show that if n is odd and d ≥ n then there exists an algebra admitting an n-cluster tilting subcategory and having global dimension d. We show the same result if n is even and d is odd or d ≥ 2n.

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

For an exact category $${{\mathcal {E}}}$$ E with enough projectives and with a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory, we show that the singularity category of $${{\mathcal {E}}}$$ E admits a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $${{\mathcal {E}}}$$ E and $${\underline{{{\mathcal {E}}}}}$$ E ̲ . We also deduce that the Gorenstein projectives of $${{\mathcal {E}}}$$ E admit a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory under some assumptions. Finally, we compute the $$d\mathbb {Z}$$ d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

d-Auslander–Reiten sequences in subcategories

Let Φ be a finite-dimensional algebra over a field k. Kleiner described the Auslander–Reiten sequences in a precovering extension closed subcategory ${\rm {\cal X}}\subseteq {\rm mod }\,\Phi $. If $X\in \mathcal {X}$ is an indecomposable such that ${\rm Ext}_\Phi ^1 (X,{\rm {\cal X}})\ne 0$ and $\zeta X$ is the unique indecomposable direct summand of the $\mathcal {X}$-cover $g:Y\to D\,{\rm Tr}\,X$ such that ${\rm Ext}_\Phi ^1 (X,\zeta X)\ne 0$, then there is an Auslander–Reiten sequence in $\mathcal {X}$ of the form $${\rm \epsilon }:0\to \zeta X\to {X}^{\prime}\to X\to 0.$$Moreover, when ${\rm En}{\rm d}_\Phi (X)$ modulo the morphisms factoring through a projective is a division ring, Kleiner proved that each non-split short exact sequence of the form $$\delta :0\to Y\to {Y}^{\prime}\buildrel \eta \over \longrightarrow X\to 0$$is such that η is right almost split in $\mathcal {X}$, and the pushout of δ along g gives an Auslander–Reiten sequence in ${\rm mod}\,\Phi $ ending at X.In this paper, we give higher-dimensional generalizations of this. Let $d\geq 1$ be an integer. A d-cluster tilting subcategory ${\rm {\cal F}}\subseteq {\rm mod}\,\Phi $ plays the role of a higher ${\rm mod}\,\Phi $. Such an $\mathcal {F}$ is a d-abelian category, where kernels and cokernels are replaced by complexes of d objects and short exact sequences by complexes of d + 2 objects. We give higher versions of the above results for an additive ‘d-extension closed’ subcategory $\mathcal {X}$ of $\mathcal {F}$.

Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories

Let 𝒞 be a triangulated category which has Auslander-Reiten triangles, and ℛ a functorially finite rigid subcategory of 𝒞. It is well known that there exist Auslander-Reiten sequences in mod ℛ. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod ℛ and the Auslander-Reiten functors, triangles in 𝒞, respectively. Furthermore, if 𝒯 is a cluster-tilting subcategory of 𝒞 and mod 𝒯 is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod 𝒯 corresponding to the ones in 𝒞. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Frobenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d=2, and then by Dugas in the general case, using different methods.