closed under extensions, direct sums and direct summands. For an object $T$ of $\CC$ we denote by $\add(T)$ the *additive envelope* of $T$, that is, the full subcategory whose objects are finite direct sums of direct summands of $T$. We say that

-   $T$ is $\CC$-[*maximal rigid*]{} if $
    \Ext_\L^1(T \oplus X,X) = 0
    $ with $X \in \CC$ implies $X \in \add(T)$;

-   $T$ is a $\CC$-[*cluster-tilting module*]{} if $
    \Ext_\L^1(T,X) = 0
    $ with $X \in \CC$ implies $X \in \add(T)$.

Since $\Ext_\L^1(T \oplus X,X) \cong \Ext_\L^1(T,X) \oplus \Ext_\L^1(X,X)$, we see that the second property is a priori stronger than the first.

\[main9\] For a rigid $\L$-module $T$ in $\CC_w$ the following are equivalent:

-   $T$ has $r$ pairwise non-isomorphic indecomposable direct summands;

-   $T$ is $\CC_w$-maximal rigid;

-   $T$ is a $\CC_w$-cluster-tilting module.

Let $T = T_1\oplus \cdots \oplus T_r$ be a $\CC_w$-cluster-tilting module, with each summand $T_i$ indecomposable. Consider the endomorphism algebra $A_T:={\operatorname{End}}_\L(T)^{\rm op}$. This is a basic algebra, with indecomposable projective modules $$P_{T_i} := {\operatorname{Hom}}_\L(T,T_i),\qquad (1\le i\le r).$$ The simple $A_T$-modules are the heads $S_{T_i}$ of the projectives $P_{T_i}$. One defines a quiver $\G_T$ with vertex set $\{1,\ldots,r\}$, and $d_{ij}$ arrows from $i$ to $j$, 