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Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of $\Lambda$-mod with $n\ge 2$. From the viewpoint of higher homological algebra, a question that naturally arose in [17] is when $\mathcal{M}$ induces an n-cluster tilting subcategory of $\Lambda$-Mod. In this paper, we answer this question and explore its connection to Iyama's question on the finiteness of n-cluster tilting subcategories of $\Lambda$-mod. In fact, our theorem reformulates Iyama's question in terms of the vanishing of Ext; and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that Add$(\mathcal{M})$ is an n-cluster tilting subcategory of $\Lambda$-Mod if and only if Add$(\mathcal{M})$ is a maximal n-rigid subcategory of $\Lambda$-Mod if and only if $\lbrace X\in \Lambda$-Mod$~|~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0

Let $\Lambda$ be a finite dimensional algebra. In this paper we show that there is a natural bijection between cosilting modules in Mod$\Lambda$ and semibricks in Mod$\Lambda$ satisfying some condition. Also this bijection restricts to a bijection between all semibricks in mod$\Lambda$ and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence [7] between support $\tau^-$-tilting modules and right finite semibricks.

Let $A$ be a finite-dimensional algebra, and $\mathfrak{M}$ be a $d$-cluster tilting subcategory of mod$A$. From the viewpoint of higher homological algebra, a natural question to ask is when $\mathfrak{M}$ induces a $d$-cluster tilting subcategory in Mod$A$. In this paper, we investigate this question in a more general form. Let $\mathcal{M}$ be a small $d$-abelian category of an abelian category $\mathcal{A}$. The completion of $\mathcal{M}$, denoted by Ind$(\mathcal{M})$, is defined as the universal completion of $\mathcal{M}$ with respect to filtered colimits. We explore Ind$(\mathcal{M})$ and demonstrate its equivalence to the full subcategory $\mathcal{L}_d(\mathcal{M})$ of Mod$\mathcal{M}$, comprising left $d$-exact functors. Notably, while Ind$(\mathcal{M})$ as a subcategory of $\frac{Mod\mathcal{M}}{Eff(\mathcal{M})}$, satisfies all properties of a $d$-cluster tilting subcategory except $d$-rigidity, it falls short of being a $d$-cluster tilting category. For a $d$-cluster tilting subcategory $\mathfrak{M}$ of mod$A$, $\overrightarrow{\mathfrak{M}}$, consists of all filtered colimits of objects from $\mathfrak{M}$, is a generating-cogenerating, functorially finite subcategory of Mod$A$. The question of whether $\mathfrak{M}$ 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 $\overrightarrow{\mathfrak{M}}$ is a $2$-cluster tilting subcategory if and only if $\mathfrak{M}$ is of finite type. Thus, the question regarding whether $\overrightarrow{\mathfrak{M}}$ is a $d$-cluster tilting subcategory of Mod$ A$ appears to be equivalent to the Iyama's qestion about the finiteness of $\mathfrak{M}$.

Let $\Lambda$ be an artin algebra and $\mathcal{C}$ be a functorially finite subcategory of mod$\Lambda$ which contains $\Lambda$ or $D\Lambda$. We use the concept of the infinite radical of $\mathcal{C}$ and show that $\mathcal{C}$ has an additive generator if and only if rad$^\infty_{\mathcal{C}}$ vanishes. In this case we describe the morphisms in powers of the radical of $\mathcal{C}$ in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that $\mathcal{C}$ is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in $\mathcal{C}$ is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left $\mathcal{C}$-approximations and right $\mathcal{C}$-approximations of simple $\Lambda$-modules, we give other criteria to describe whether $\mathcal{C}$ is of finite representation type. In addition, we give a nilpotency index of the radical of $\mathcal{C}$ which is independent from the maximal length of indecomposable $\Lambda$-modules in $\mathcal{C}$., Comment: We changed the title and some results of the paper. We also added some new results

In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. We also characterize contravariantly finite resolving subcategories of the module category $\Lambda${\rm -mod} that contain the Jacobson radical of $\Lambda$ of finite type, by their functor categories. We study the pure semisimplicity conjecture for a locally finitely presented category ${\underrightarrow{\lim}}\mathscr{X}$ when $\mathscr{X}$ is a covariantly finite subcategory of $\Lambda$-mod and every simple object in Mod$(\mathscr{X}^{\rm op})$ is finitely presented and give a characterization of covariantly finite subcategories of finite representation type in terms of decomposition properties of their closure under filtered colimits. As a consequence we study finiteness and purity of $n$-cluster tilting subcategories and the subcategory of the Gorenstein projective modules of the module categories. These results extend and unify some known results.

In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories. We study this 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 arise in this way, and discuss the problem for $n\mathbb{Z}$-exact categories.