Cluster structures for the A∞ singularity

Jenny August; Man Wai Cheung; Eleonore Faber; Sira Gratz; Sibylle Schroll

doi:10.1112/jlms.12735

Cluster structures for the A_∞ singularity

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

^*Corresponding author for this work

Department of Mathematics

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Abstract

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

Original language	English
Journal	Journal of the London Mathematical Society
Volume	107
Issue	6
Pages (from-to)	2121-2149
ISSN	0024-6107
DOIs	https://doi.org/10.1112/jlms.12735
Publication status	Published - Jun 2023

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Journal of London Math Soc - 2023 - August - Cluster structures for the A A infty singularityFinal published version, 563 KBLicence: CC BY

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title = "Cluster structures for the A∞ singularity",

abstract = "We study a category (Formula presented.) of (Formula presented.) -graded maximal Cohen-Macaulay (MCM) modules over the (Formula presented.) curve singularity and demonstrate that it has infinite type (Formula presented.) cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type (Formula presented.) cluster categories of Holm–J{\o}rgensen, Fisher and Paquette–Yıldırım. As a consequence, (Formula presented.) has cluster tilting subcategories modelled by certain triangulations of the (completed) (Formula presented.) -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 (Formula presented.) -gon.",

author = "Jenny August and Cheung, {Man Wai} and Eleonore Faber and Sira Gratz and Sibylle Schroll",

note = "Publisher Copyright: {\textcopyright} 2023 The Authors. Journal of the London Mathematical Society is copyright {\textcopyright} London Mathematical Society.",

year = "2023",

month = jun,

doi = "10.1112/jlms.12735",

language = "English",

volume = "107",

pages = "2121--2149",

journal = "Journal of the London Mathematical Society",

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TY - JOUR

T1 - Cluster structures for the A∞ singularity

AU - August, Jenny

AU - Cheung, Man Wai

AU - Faber, Eleonore

AU - Gratz, Sira

AU - Schroll, Sibylle

PY - 2023/6

Y1 - 2023/6

N2 - We study a category (Formula presented.) of (Formula presented.) -graded maximal Cohen-Macaulay (MCM) modules over the (Formula presented.) curve singularity and demonstrate that it has infinite type (Formula presented.) cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type (Formula presented.) cluster categories of Holm–Jørgensen, Fisher and Paquette–Yıldırım. As a consequence, (Formula presented.) has cluster tilting subcategories modelled by certain triangulations of the (completed) (Formula presented.) -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 (Formula presented.) -gon.

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

U2 - 10.1112/jlms.12735

DO - 10.1112/jlms.12735

M3 - Journal article

AN - SCOPUS:85152916642

SN - 0024-6107

VL - 107

SP - 2121

EP - 2149

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 6

ER -

Cluster structures for the A∞ singularity

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Cluster structures for the A_∞ singularity