Cluster structures for the $A_{\infty}$ singularity

Jenny August, Man-Wai Cheung, Eleonore Faber, Sira Gratz, Sibylle Schroll

Publikation: Working paper/Preprint Preprint

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Abstract

We study a category $\mathcal{C}_2$ of $\mathbb{Z}$-graded MCM modules over the $A_\infty$ curve singularity and demonstrate it has infinite type $A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type $A$ cluster categories of Holm-Jorgensen, Fisher and Paquette-Yildirim. As a consequence, $\mathcal{C}_2$ has cluster tilting subcategories modelled by certain triangulations of the (completed) $\infty$-gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed $\infty$-gon.
OriginalsprogUdefineret/Ukendt
StatusUdgivet - 30 maj 2022

Emneord

  • math.RT
  • math.AC

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