The $N$-Stable Category

TL;DR

This paper extends Buchweitz's equivalence theorem to $N$-complexes, establishing an $N$-stable category framework, and explores its properties over Frobenius and self-injective categories, including Serre functors and Calabi-Yau characteristics.

Contribution

It introduces the $N$-stable category via the monomorphism category and proves Buchweitz's theorem for $N$-complexes over Frobenius categories, extending classical results.

Findings

Established the $N$-stable category as an analogue of the stable category.

Proved Buchweitz's theorem for $N$-complexes in Frobenius categories.

Computed the Serre functor and analyzed fractional Calabi-Yau properties.

Abstract

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to $N$-complexes, one must find an appropriate candidate for the $N$-analogue of the stable category. We identify this "$N$-stable category" via the monomorphism category and prove Buchweitz's theorem for $N$-complexes over a Frobenius exact abelian category. We also compute the Serre functor on the $N$-stable category over a self-injective algebra and study the resultant fractional Calabi-Yau properties.