Higher Auslander-Reiten Translates

• Higher Auslander-Reiten translates are generalizations of classical AR translation, extending almost split sequence theory to monomorphism and epimorphism categories.
• They leverage explicit functorial operations like minimal monomorphisms and cokernels to compute AR orbits and uncover homological periodicities.
• In selfinjective and Nakayama algebra settings, these translates yield precise periodicity formulas and dualities, enriching triangulated and cluster-tilting frameworks.

Higher Auslander-Reiten translates generalize the classical Auslander-Reiten translation, τ, by extending the machinery of almost split sequences and AR duality to structured submodule categories, higher homological algebra, and higher categorical settings. These concepts are central to the paper of homological periodicity, categorical lifts, and connections between module categories and more general exact, Frobenius, or angulated categories. They refine and expand the control provided by classical AR theory, revealing intricate periodicities and functorial behaviors in contexts such as monomorphism categories, higher cluster-tilting theory, and Calabi-Yau triangulated categories.

1. Generalization of the Auslander–Reiten Translation in Monomorphism Categories

The classical AR translate τ in the module category A–mod, for a finite-dimensional algebra A, is extended to the monomorphism category $\mathcal{S}_n(A)$. In this setting, an object $X_{(\varphi_i)} \in \mathcal{S}_n(A)$, represented by a chain of modules and monomorphisms,

$$X_{1} \xrightarrow{\varphi_1} X_2 \xrightarrow{\varphi_2} \cdots \xrightarrow{\varphi_{n-1}} X_n,$$

has its higher Auslander-Reiten translate in $\mathcal{S}_n(A)$ given by the explicit formula: $$\tau_{\mathcal{S}} X_{(\varphi_i)} \cong \operatorname{Mimo}\left( \tau\; \operatorname{Cok} X_{(\varphi_i)} \right)$$ where $\operatorname{Cok} X_{(\varphi_i)}$ is an object in the dual epimorphism category $\mathcal{F}_n(A)$, τ is the classical AR translate in $A$–mod, and the functor $\operatorname{Mimo}$ returns to $\mathcal{S}_n(A)$ via a minimal monomorphism construction that canonically "lifts" back the τ-translated cokernel. This formula generalizes the $n=2$ result of Ringel–Schmidmeier to arbitrary $n$ (Xiong et al., 2011).

2. Existence and Transfer of Almost Split Sequences

Functorial finiteness of $\mathcal{S}_n(A)$ in the morphism category $\operatorname{Mor}_n(A)$ ensures the existence of almost split (i.e., AR) sequences in $\mathcal{S}_n(A)$: $0 \to X' \to Y \to X \to 0$ for every non-projective indecomposable $X \in \mathcal{S}_n(A)$. The AR sequences in $\mathcal{S}_n(A)$ can be constructed directly from those in $\operatorname{Mor}_n(A)$ via the kernel or cokernel functors. This transfer process relies on the closure of $\mathcal{S}_n(A)$ under direct summands and extensions and is essential for extending AR combinatorics to monomorphism categories. Dual results hold for the epimorphism category $\mathcal{F}_n(A)$, showing a robust interplay between submodule and factor module settings (Xiong et al., 2011).

3. Periodicity Phenomena and Higher Powers in Selfinjective Settings

For selfinjective algebras $A$, the situation becomes richer due to additional homological symmetries. The stable category $A$–mod is triangulated (suspension $\Omega^{-1}$), and the classical AR translate τ behaves as a triangle functor, with periodicity arising from the isomorphism τ ≅ $\Omega^2 \mathcal{N}$ (Nakayama functor). The higher power of the AR translate in the monomorphism category is expressed as: $$\tau_{\mathcal{S}}^{\,z} X_{(\varphi_i)} \cong \tau^z \left( \operatorname{Rot}^z X_{(\underline{\varphi_i})} \right)$$ where $\operatorname{Rot}$ is a rotation operation in the morphism category. For $z = s(n+1)$, there is a stable isomorphism: $$\tau_{\mathcal{S}}^{s(n+1)} X_{(\varphi_i)} \cong \tau^{s(n+1)}\; \Omega^{-s(n-1)} X_{(\underline{\varphi_i})}$$

For selfinjective Nakayama algebras $\mathcal{A}(m,t)$, explicit periodicity is established:

• If $n$ odd: $\tau_{\mathcal{S}}^{m(n+1)} X \cong X$,
• If $n$ even: $\tau_{\mathcal{S}}^{2m(n+1)} X \cong X$.

These results reveal deep homological periodicities tied to the parameters of the Nakayama algebra and the length of the monomorphism "branch" $n$ (Xiong et al., 2011).

4. Serre Functors and Their Periodicity

The Serre functor $F_{\mathcal{S}}$ of the stable monomorphism category $\underline{\mathcal{S}_n(A)}$ plays a parallel role to $\tau_{\mathcal{S}}$, and its relation to AR translate is central: $$F_{\mathcal{S}} \cong [1]\,\widetilde{\tau}_{\mathcal{S}}$$ on objects, meaning it coincides with a shifted AR translate. In the stable setting,

$$F_{\mathcal{S}}^{s(n+1)} X \cong \operatorname{Mimo} \left( \tau^{s(n+1)} \Omega^{-2sn} X \right)$$

In selfinjective Nakayama algebras, the periodicity of $F_{\mathcal{S}}$ is computable by explicit greatest common divisors and least common multiples of the algebra parameters. For $(m,t)$ with $t=2$, $N=m/\gcd(m, n-1)$ and $F_{\mathcal{S}}^{N(n+1)} X \cong X$; for $t \geq 3$, $N = m/\gcd(m, t, n+1)$ (Xiong et al., 2011).

5. Applications to Concrete Examples and Combinatorial Modelling

For $A = k[x]/\langle x^2 \rangle$ (the Nakayama algebra $(1,2)$), the AR quiver of $\mathcal{S}_n(A)$ and the periodicity of its AR translate can be described explicitly using the combinatorial language of quivers and their representation types. In higher-dimensions (e.g., $n=3$), examples illustrate both the minimal monomorphism construction and the resulting periodic orbits under the AR translation, with the Mimo procedure becoming computationally concrete.

Moreover, this general framework unifies and both generalizes and recovers the classical AR theory, demonstrating how categorical lifts and higher-derived analogues, such as those for submodule or monomorphism categories, inherit and reflect the homological structure and symmetries of the underlying module categories. These constructions also link to invariants in cluster categories, higher tilting theory, and more generally in higher representation theory (Xiong et al., 2011).

6. Summary Table: Key Formulas in Higher AR Translates

Setting/Operation	Formula	Context
AR translation in $\mathcal{S}_n(A)$	$$\tau_{\mathcal{S}} X_{(\varphi_i)} \cong \operatorname{Mimo}(\tau\,\operatorname{Cok} X_{(\varphi_i)})$$	Generalization of $\mathcal{S}_2$ case; minimal monomorphism correction
Higher powers (selfinjective)	$$\tau_{\mathcal{S}}^{s(n+1)} X_{(\varphi_i)} \cong \tau^{s(n+1)} \Omega^{-s(n-1)} X_{(\underline{\varphi_i})}$$	Periodicity, triangulated structure in selfinjective case
Periodicity in Nakayama algebra	$n$ odd: $\tau_{\mathcal{S}}^{m(n+1)}X \cong X$<br>$n$ even: $\tau_{\mathcal{S}}^{2m(n+1)}X \cong X$	Concrete periodicity results for selfinjective Nakayama algebras
Serre functor periodicity	$$F_{\mathcal{S}}^{s(n+1)} X \cong \operatorname{Mimo}(\tau^{s(n+1)} \Omega^{-2sn} X)$$	Stable monomorphism category

These formulas formalize the lift of classical AR translation, reflect higher homological structures, and can be leveraged in computations of AR orbits, periodicities, and the construction of almost split sequences in categorical settings beyond module categories.

7. Broader Impact and Further Applications

The detailed understanding of higher AR translates in monomorphism categories not only recovers and sharpens the classical Ringel–Schmidmeier theory but also directly connects to contemporary advancements in higher homological algebra, cluster-tilting theory, and the structure of triangulated and (d+2)-angulated categories. The functorial approach—passing between module, monomorphism, and epimorphism categories via minimal monomorphisms, kernels, or cokernels—illustrates that the key homological periodicities and categorical dualities are deeply structural and persist under appropriate categorical lifts (Xiong et al., 2011).