Cluster structures for the A∞$A_\infty$ singularity

Author:

August Jenny1,Cheung Man Wai2,Faber Eleonore3,Gratz Sira1,Schroll Sibylle45

Affiliation:

1. Department of Mathematics Aarhus University Aarhus C Denmark

2. School of Mathematics, Kavli IPMU (WPI) UTIAS, The University of Tokyo Chiba Japan

3. School of Mathematics University of Leeds Leeds UK

4. Department of Mathematics Universität zu Köln Köln Germany

5. Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway

Abstract

AbstractWe study a category of ‐graded maximal Cohen‐Macaulay (MCM) modules over the curve singularity and demonstrate that it has infinite type cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type cluster categories of Holm–Jørgensen, Fisher and Paquette–Yıldırım. As a consequence, has cluster tilting subcategories modelled by certain triangulations of the (completed) ‐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 ‐gon.

Funder

London Mathematical Society

National Science Foundation

Association for Women in Mathematics

Alfred P. Sloan Foundation

Villum Fonden

Danmarks Grundforskningsfond

Ministry of Education, Culture, Sports, Science and Technology

Engineering and Physical Sciences Research Council

University Of Leeds

Publisher

Wiley

Subject

General Mathematics

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