Even-dimensional extension of the Auslander–Reiten angle local-finiteness criterion

Determine whether Zhou’s local-finiteness criterion for an $(n+2)$-angulated category remains valid when $n$ is even: specifically, establish whether the generation of $ \operatorname{Ker}\pi$ by the classes of all Auslander–Reiten $(n+2)$-angles in $K_0(C,0)$ implies that $C$ is locally finite without imposing Condition \ref{cco1}.

Background

For an $(n+2)$-angulated category $C$, Zhou’s theorem proves that if $n$ is odd and $\operatorname{Ker}\pi\subseteq K_0(C,0)$ is generated by the classes of all Auslander–Reiten $(n+2)$-angles, then $C$ is locally finite. The paper explains that Zhou’s proof relies on an alternating sum of Hom-space dimensions whose two relevant end contributions have the same sign when $n$ is odd.

When $n$ is even, the two end contributions have opposite signs and may cancel, so the detection argument used in the odd case does not extend directly. The paper establishes the desired implication under the additional non-cancellation Condition \ref{cco1} and shows that this condition holds for a broad class of examples, but it does not settle whether the implication holds for all even $n$ without that additional hypothesis.