[*Frobenius category*]{}.

For such a Frobenius category $\CC$ we can define its [*stable category*]{} ${\stCC}$. The objects of ${\stCC}$ are the same as the objects of $\CC$, and the morphism spaces ${\operatorname{Hom}}_{\stCC}(X,Y)$ are the morphism spaces in $\CC$ modulo morphisms factoring through $\CC$-projective-injective objects. The category ${\stCC}$ is a triangulated category in a natural way [@H2], where the shift is given by the relative inverse syzygy functor $
\Omega^{-1}\colon {\stCC} \to {\stCC}.
$ (Recall that the syzygy functor $\Omega$ maps an object to the kernel of its projective cover. This induces an auto-equivalence of the stable category.) For all $X$ and $Y$ in $\CC$ there is a functorial isomorphism $$\label{functorial1}
\Ext_\CC^1(X,Y) \cong {\operatorname{Hom}}_{\stCC}(X,\Omega^{-1}(Y)).$$ The category $\CC$ is called *stably $2$-Calabi-Yau* if the stable category $\stCC$ is a [*$2$-Calabi-Yau category*]{}, that is, if for all $X,Y \in \CC$ there is a functorial isomorphism $$\label{functorial2}
\Ext_\CC^1(X,Y) \cong {\rm D}\Ext_\CC^1(Y,X),$$ where ${\rm D}$ denotes the duality for vector spaces. We can now state some important properties of the categories $\CC_w$.

\[main1\] The categories $\CC_w$ are Frobenius categories. They are stably $2$-Calabi-Yau. The indecomposable $\CC_w$-projective-injective modules, are the indecomposable direct summands of $I_w$.

Cluster-tilting modules {#sect4.5}
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Let $\CC$ be a subcategory of $\nil(\L)$ 