On the phantom stable categories of $n$-Frobenius categories

Let $n$ be a non-negative integer. {Motivated by the universal property of the stable category of Frobenius categories, the authors in \cite{bfss} extended the stabilization of Frobenius categories to $n$-Frobenius categories, and called it the phantom stable categories. Precisely, assume that $\C$ is an $n$-Frobenius category.} The phantom stable category of $\C$ is a pair $(\C_{\p}, T)$, with $\C_{\p}$ an additive category having the same objects as $\C$ and $T$ an additive covariant functor from $\C$ to $\C_{\p}$, vanishing over $n$-$\Ext$-phantom morphisms and $T(f)$ is an isomorphism, for any $n$-$\Ext$-invertible morphism $f$, {and $T$ has the universal property with respect to these conditions. The existence of the phantom stable category $(\C_{\p}, T)$ and its several interesting properties have appeared in \cite{bfss}. This paper is devoted to further study of phantom stable categories. In particular, it is shown that } the syzygy functor $\syz$, using $n$-projective objects, from $\C$ to $\C_{\p}$ is not only an additive functor, but also it induces an auto-equivalence functor $\Syz$ on $\C_{\p}$. These results would be the first evidence to show that phantom stable categories are triangulated, with the shift functor $\Syz$. At the end of the paper we give a 1-Frobenius subcategory of the category of coherent sheaves over the projective line.