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 $\text{C}$ is an $n$-Frobenius category.} The phantom stable category of $\text{C}$ is a pair $({\text{C}}_{\text{p}},T)$, with ${\text{C}}_{\text{p}}$ an additive category having the same objects as $\text{C}$ and $T$ an additive covariant functor from $\text{C}$ to ${\text{C}}_{\text{p}}$, vanishing over $n$-$\text{Ext}$-phantom morphisms and $T(f)$ is an isomorphism, for any $n$-$\text{Ext}$-invertible morphism $f$, {and $T$ has the universal property with respect to these conditions. The existence of the phantom stable category $({\text{C}}_{\text{p}},T)$ 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 $\text{syz}$, using $n$-projective objects, from $\text{C}$ to ${\text{C}}_{\text{p}}$ is not only an additive functor, but also it induces an auto-equivalence functor $\text{Syz}$ on ${\text{C}}_{\text{p}}$. These results would be the first evidence to show that phantom stable categories are triangulated, with the shift functor $\text{Syz}$. At the end of the paper we give a 1-Frobenius subcategory of the category of coherent sheaves over the projective line.