Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: type A

TL;DR

This paper discusses the invalidation of a combinatorial method for computing Calabi-Yau dimensions in certain triangulated categories, highlighting the need for revised approaches in classifying stable module categories.

Contribution

It identifies a flaw in a previously claimed combinatorial formula for Calabi-Yau dimensions, emphasizing the challenge in classifying categories via these dimensions.

Findings

Theorem 2.2 in arXiv:math/0610728 is false due to a counterexample.

Current methods cannot reliably compute Calabi-Yau dimensions for certain categories.

The invalidation impacts the classification of higher cluster categories.

Abstract

The preprints arXiv:math/0610728 and arXiv:math/0612451 are withdrawn due to a problem with Theorem 2.2 in arXiv:math/0610728. The theorem claims that for certain triangulated categories with finitely many indecomposable objects, the Calabi-Yau dimension can be computed combinatorially, by finding the smallest d for which the Serre functor and the d'th power of the suspension functor have the same action on the Auslander-Reiten quiver. This is false, and we are grateful to Alex Dugas for pointing out a counterexample; see Section 5 of his paper arXiv:math/0808.1311 for more details. Unfortunately, we are not presently able to come up with a corrected version of the theorem, and this means that we cannot compute the Calabi-Yau dimensions of concrete stable module categories. Since these dimensions are necessary for identifying the categories with higher cluster categories, we presently have no means to achieve such identifications.