Cluster structures for the A-infinity singularity

Abstract: This talk is about a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules over the ring $\mathbb{C}[x,y]/(x^k).$ This yields an infinite rank analog of the Grassmannian cluster categories introduced by Jensen, King, and Su. In the special case, $k=2,$ $\text{Spec}(\mathbb{C}[x,y]/(x^2))$ is a type $A$-infinity singularity and the ungraded version of the category of maximal Cohen-Macaulay modules over $\mathbb{C}[x,y]/(x^2))$ has been studied by Buchweitz, Greuel, and Schreyer in the 1980s. We demonstrate that his category has infinite type $A$ cluster combinatorics. In particular, we show that it has cluster-tilting subcategories modeled by certain triangulations of the (completed) infinity-gon and we can also interpret certain mutations of the category in this model. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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