Cluster structures for the A∞ singularity

We study a category C2$\mathcal {C}_2$ of Z$\mathbb {Z}$-graded maximal Cohen-Macaulay (MCM) modules over the A∞$A_\infty$ curve singularity and demonstrate that it has infinite type A$A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type A$A$ cluster categories of Holm–Jørgensen, Fisher and Paquette–Yıldırım. As a consequence, C2$\mathcal {C}_2$ 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.