  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$~|~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~
0<i<n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is
of finite type if and only if ${\rm
Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M},
{\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several
equivalent conditions for Iyama's question which shows the relation of Iyama's
question with different subjects in representation theory such as purity and
covering theory.
