{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T09:33:02Z","timestamp":1775208782483,"version":"3.50.1"},"reference-count":55,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2022,2,23]],"date-time":"2022-02-23T00:00:00Z","timestamp":1645574400000},"content-version":"unspecified","delay-in-days":53,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Compositio Math."],"published-print":{"date-parts":[[2022,1]]},"abstract":"\n Let\n \n \n \n $Q$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be an acyclic quiver and\n \n \n \n $w \\geqslant 1$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be an integer. Let\n \n \n \n $\\mathsf {C}_{-w}({\\mathbf {k}} Q)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be the\n \n \n \n $(-w)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -cluster category of\n \n \n \n ${\\mathbf {k}} Q$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We show that there is a bijection between simple-minded collections in\n \n \n \n $\\mathsf {D}^b({\\mathbf {k}} Q)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n lying in a fundamental domain of\n \n \n \n $\\mathsf {C}_{-w}({\\mathbf {k}} Q)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n \n $w$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -simple-minded systems in\n \n \n \n $\\mathsf {C}_{-w}({\\mathbf {k}} Q)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . This generalises the same result of Iyama\u2013Jin in the case that\n \n \n \n $Q$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is Dynkin. A key step in our proof is the observation that the heart\n \n \n \n $\\mathsf {H}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of a bounded t-structure in a Hom-finite, Krull\u2013Schmidt,\n \n \n \n ${\\mathbf {k}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -linear saturated triangulated category\n \n \n \n $\\mathsf {D}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is functorially finite in\n \n \n \n $\\mathsf {D}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n if and only if\n \n \n \n $\\mathsf {H}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n has enough injectives and enough projectives. We then establish a bijection between\n \n \n \n $w$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -simple-minded systems in\n \n \n \n $\\mathsf {C}_{-w}({\\mathbf {k}} Q)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and positive\n \n \n \n $w$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -noncrossing partitions of the corresponding Weyl group\n \n \n \n $W_Q$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1112\/s0010437x21007648","type":"journal-article","created":{"date-parts":[[2022,2,23]],"date-time":"2022-02-23T07:32:10Z","timestamp":1645601530000},"page":"211-243","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions"],"prefix":"10.1017","volume":"158","author":[{"given":"Raquel","family":"Coelho Sim\u00f5es","sequence":"first","affiliation":[]},{"given":"David","family":"Pauksztello","sequence":"additional","affiliation":[]},{"given":"David","family":"Ploog","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,2,23]]},"reference":[{"key":"S0010437X21007648_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2007.06.002"},{"key":"S0010437X21007648_ref27","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-01-00385-X"},{"key":"S0010437X21007648_ref32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2010.02.003"},{"key":"S0010437X21007648_ref10","doi-asserted-by":"publisher","DOI":"10.1070\/IM1990v034n01ABEH000583"},{"key":"S0010437X21007648_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s00209-011-0906-7"},{"key":"S0010437X21007648_ref28","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(91)90107-J"},{"key":"S0010437X21007648_ref35","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/7213"},{"key":"S0010437X21007648_ref42","doi-asserted-by":"crossref","first-page":"403","DOI":"10.4171\/dm\/451","article-title":"Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras","volume":"19","author":"Koenig","year":"2014","journal-title":"Doc. 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Compositio Mathematica is \u00a9 Foundation Compositio Mathematica.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}