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Let A be a finite-dimensional algebra, and M be a d -cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d -cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d -abelian category, known to be equivalent to a d -cluster tilting subcategory of an abelian category A. The completion of M , denoted by Ind (M) , is defined as the universal completion of M with respect to filtered colimits. We explore Ind (M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M , comprising left d -exact functors. Notably, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. For a d -cluster tilting subcategory M of mod A , M → consists of all filtered colimits of objects from M , is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M → is a d -rigid subcategory remains unanswered. However, if it is indeed d -rigid, it qualifies as a d -cluster tilting subcategory. In the case d = 2 , employing cotorsion theory, we establish that M → is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M → is a d -cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]

In this paper, we introduce |$n\mathbb{Z}$| -abelian and |$n\mathbb{Z}$| -exact categories by axiomatizing properties of |$n\mathbb{Z}$| -cluster tilting subcategories. We study these categories and show that every |$n\mathbb{Z}$| -cluster tilting subcategory of an abelian (resp. exact) category has a natural structure of an |$n\mathbb{Z}$| -abelian (resp. |$n\mathbb{Z}$| -exact) category. Also, we show that every small |$n\mathbb{Z}$| -abelian category arises in this way, and discuss the problem for |$n\mathbb{Z}$| -exact categories. [ABSTRACT FROM AUTHOR]

We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types A n or A ˜ n is equal to the corresponding lower bound cluster algebra generated by projective cluster variables. [ABSTRACT FROM AUTHOR]

Let Λ be an artin algebra and M be an n -cluster tilting subcategory of mod Λ. We show that M has an additive generator if and only if the n -almost split sequences form a basis for the relations for the Grothendieck group of M if and only if every effaceable functor M → A b has finite length. As a consequence we show that if mod Λ has an n -cluster tilting subcategory of finite type then the n -almost split sequences form a basis for the relations for the Grothendieck group of Λ. [ABSTRACT FROM AUTHOR]

From the viewpoint of higher homological algebra, we introduce pure semisimple n -abelian categories, which are analogs of pure semisimple abelian categories. Let Λ be an Artin algebra and M be an n -cluster tilting subcategory of Mod-Λ. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let m be an n -cluster tilting subcategory of mod-Λ. We show that Add (m) is an n -cluster tilting subcategory of Mod-Λ if and only if m has an additive generator if and only if Mod (m) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras. [ABSTRACT FROM AUTHOR]

Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and let B be one of the associated tilted algebras. We show that the B-modules, ordered from right to left in the Auslander-Reiten quiver of Λ form a maximal forward hom-orthogonal sequence of Λ-modules whose dimension vectors form the c-vectors of a maximal green sequence for Λ. Thus we give a proof of Igusa-Todorov’s conjecture. [ABSTRACT FROM AUTHOR]

For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e., gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a clustertilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons’s conjecture. [ABSTRACT FROM AUTHOR]