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Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions

Research output: Contribution to Journal/Magazine › Journal article › peer-review

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Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions. / Coelho Guardado Simoes, Raquel; Pauksztello, David; Ploog, David et al.
In: Compositio Mathematica, Vol. 158, No. 1, 31.01.2022, p. 211-243.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Coelho Guardado Simoes, R, Pauksztello, D, Ploog, D & Zvonareva, A 2022, 'Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions', Compositio Mathematica, vol. 158, no. 1, pp. 211-243. https://doi.org/10.1112/S0010437X21007648

APA

Coelho Guardado Simoes, R., Pauksztello, D., Ploog, D., & Zvonareva, A. (2022). Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions. Compositio Mathematica, 158(1), 211-243. https://doi.org/10.1112/S0010437X21007648

Vancouver

Coelho Guardado Simoes R, Pauksztello D, Ploog D, Zvonareva A. Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions. Compositio Mathematica. 2022 Jan 31;158(1):211-243. doi: 10.1112/S0010437X21007648

Author

Coelho Guardado Simoes, Raquel ; Pauksztello, David ; Ploog, David et al. / Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions. In: Compositio Mathematica. 2022 ; Vol. 158, No. 1. pp. 211-243.

Bibtex

@article{d980533c52c542bd832d2b73fa2cc38f,
title = "Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions",
abstract = "Let Q be an acyclic quiver and w⩾1 be an integer. Let C−w(kQ) be the (−w)-cluster category of kQ. We show that there is a bijection between simple-minded collections in Db(kQ) lying in a fundamental domain of C−w(kQ) and w-simple-minded systems in C−w(kQ). This generalises the same result of Iyama–Jin in the case that Q is Dynkin. A key step in our proof is the observation that the heart H of a bounded t-structure in a Hom-finite, Krull–Schmidt, k-linear saturated triangulated category D is functorially finite in D if and only if H has enough injectives and enough projectives. We then establish a bijection between w-simple-minded systems in C−w(kQ) and positive w-noncrossing partitions of the corresponding Weyl group WQ.",
keywords = "simple-minded system, simple-minded collection, Calabi–Yau triangulated category, noncrossing partition, Riedtmann configuration",
author = "{Coelho Guardado Simoes}, Raquel and David Pauksztello and David Ploog and Alexandra Zvonareva",
note = "http://journals.cambridge.org/action/displayJournal?jid=UHY The final, definitive version of this article has been published in the Journal, Compositio Mathematica, 158 (1), pp 211-243 2022, {\textcopyright} 2022 Cambridge University Press.",
year = "2022",
month = jan,
day = "31",
doi = "10.1112/S0010437X21007648",
language = "English",
volume = "158",
pages = "211--243",
journal = "Compositio Mathematica",
issn = "0010-437X",
publisher = "Cambridge University Press",
number = "1",

}

RIS

TY - JOUR

T1 - Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions

AU - Coelho Guardado Simoes, Raquel

AU - Pauksztello, David

AU - Ploog, David

AU - Zvonareva, Alexandra

N1 - http://journals.cambridge.org/action/displayJournal?jid=UHY The final, definitive version of this article has been published in the Journal, Compositio Mathematica, 158 (1), pp 211-243 2022, © 2022 Cambridge University Press.

PY - 2022/1/31

Y1 - 2022/1/31

N2 - Let Q be an acyclic quiver and w⩾1 be an integer. Let C−w(kQ) be the (−w)-cluster category of kQ. We show that there is a bijection between simple-minded collections in Db(kQ) lying in a fundamental domain of C−w(kQ) and w-simple-minded systems in C−w(kQ). This generalises the same result of Iyama–Jin in the case that Q is Dynkin. A key step in our proof is the observation that the heart H of a bounded t-structure in a Hom-finite, Krull–Schmidt, k-linear saturated triangulated category D is functorially finite in D if and only if H has enough injectives and enough projectives. We then establish a bijection between w-simple-minded systems in C−w(kQ) and positive w-noncrossing partitions of the corresponding Weyl group WQ.

AB - Let Q be an acyclic quiver and w⩾1 be an integer. Let C−w(kQ) be the (−w)-cluster category of kQ. We show that there is a bijection between simple-minded collections in Db(kQ) lying in a fundamental domain of C−w(kQ) and w-simple-minded systems in C−w(kQ). This generalises the same result of Iyama–Jin in the case that Q is Dynkin. A key step in our proof is the observation that the heart H of a bounded t-structure in a Hom-finite, Krull–Schmidt, k-linear saturated triangulated category D is functorially finite in D if and only if H has enough injectives and enough projectives. We then establish a bijection between w-simple-minded systems in C−w(kQ) and positive w-noncrossing partitions of the corresponding Weyl group WQ.

KW - simple-minded system

KW - simple-minded collection

KW - Calabi–Yau triangulated category

KW - noncrossing partition

KW - Riedtmann configuration

U2 - 10.1112/S0010437X21007648

DO - 10.1112/S0010437X21007648

M3 - Journal article

VL - 158

SP - 211

EP - 243

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 1

ER -