On the phantom stable categories of $n$-Frobenius categories
Abdolnaser Bahlekeh, Fahimeh Sadat Fotouhi, Shokrollah Salarian, Atousa Sartipzadeh
TL;DR
This work extends the stabilization of Frobenius categories to $n$-Frobenius categories by studying phantom stable categories $(\mathscr{C}_{\mathcal{P}}, T)$, which satisfy a universal property with respect to $n$-$\operatorname{Ext}$-phantom and invertible morphisms. It proves that the syzygy construction using $n$-projectives induces an additive endofunctor on $\mathscr{C}_{\mathcal{P}}$ that is an auto-equivalence, supporting a triangulated-structure paradigm with shift given by the auto-equivalence (to be developed in a separate paper). The paper also provides a concrete description of the $\operatorname{Ext}$-relative morphism sets in the phantom stable category, and demonstrates the preservation of phantom/invertible morphisms under syzygy, ensuring the universal property yields a well-defined shift. As an application, it constructs a $1$-Frobenius subcategory inside $\operatorname{coh}(\mathbf{P^1})$ and places the theory within a broader landscape of algebraic triangulated categories arising from higher Frobenius contexts.
Abstract
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.