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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 Buchweitzs 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.
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Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(mathcal A ,mathcal B,mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $mathcal A$ and $mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $mathcal B$ under certain conditions.
Let $mathscr C$ be a Krull-Schmidt $(n+2)$-angulated category and $mathscr A$ be an $n$-extension closed subcategory of $mathscr C$. Then $mathscr A$ has the structure of an $n$-exangulated category in the sense of Herschend-Liu-Nakaoka. This construction gives $n$-exangulated categories which are not $n$-exact categories in the sense of Jasso nor $(n+2)$-angulated categories in the sense of Geiss-Keller-Oppermann in general. As an application, our result can lead to a recent main result of Klapproth.
Let $mathscr{F}$ be an $(n+2)$-angulated Krull-Schmidt category and $mathscr{A} subset mathscr{F}$ an $n$-extension closed, additive and full subcategory with $operatorname{Hom}_{mathscr{F}}(Sigma_n mathscr{A}, mathscr{A}) = 0$. Then $mathscr{A}$ naturally carries the structure of an $n$-exact category in the sense of Jasso, arising from short $(n+2)$-angles in $mathscr{F}$ with objects in $mathscr{A}$ and there is a binatural and bilinear isomorphism $operatorname{YExt}^{n}_{(mathscr{A},mathscr{E}_{mathscr{A}})}(A_{n+1},A_0) cong operatorname{Hom}_{mathscr{F}}(A_{n+1}, Sigma_n A_{0})$ for $A_0, A_{n+1} in mathscr{A}$. For $n = 1$ this has been shown by Dyer and we generalize this result to the case $n > 1$. On the journey to this result, we also develop a technique for harvesting information from the higher octahedral axiom (N4*) as defined by Bergh and Thaule. Additionally, we show that the axiom (F3) for pre-$(n+2)$-angulated categories, introduced by Geiss, Keller and Oppermann and stating that a commutative square can be extended to a morphism of $(n+2)$-angles, implies a stronger version of itself.
The notion of right semi-equivalence in a right $(n+2)$-angulated category is defined in this article. Let $mathscr C$ be an $n$-exangulated category and $mathscr X$ is a strongly covariantly finite subcategory of $mathscr C$. We prove that the standard right $(n+2)$-angulated category $mathscr C/mathscr X$ is right semi-equivalence under a natural assumption. As an application, we show that a right $(n+2)$-angulated category has an $n$-exangulated structure if and only if the suspension functor is right semi-equivalence. Besides, we also prove that an $n$-exangulated category $mathscr C$ has the structure of a right $(n+2)$-angulated category with right semi-equivalence if and only if for any object $Xinmathscr C$, the morphism $Xto 0$ is a trivial inflation.
Let $n$ be an integer greater or equal than $3$. We give a simultaneous generalization of $(n-2)$-exact categories and $n$-angulated categories, and we call it one-sided $n$-suspended categories. One-sided $n$-angulated categories are also examples of one-sided $n$-suspended categories. We provide a general framework for passing from one-sided $n$-suspended categories to one-sided $n$-angulated categories. Besides, we give a method to construct $n$-angulated quotient categories from Frobenius $n$-prile categories. These results generalize their works by Jasso for $n$-exact categories, Lin for $(n+2)$-angulated categories and Li for one-sided suspended categories.
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