{"title":"Remarks on \n \n τ\n $\\tau$\n -tilted versions of the second Brauer–Thrall conjecture","authors":"Calvin Pfeifer","doi":"10.1112/blms.70048","DOIUrl":null,"url":null,"abstract":"<p>In this short note, we state a stable and a <span></span><math>\n <semantics>\n <mi>τ</mi>\n <annotation>$\\tau$</annotation>\n </semantics></math>-reduced version of the second Brauer–Thrall conjecture. The former is a slight strengthening of a brick version of the second Brauer–Thrall conjecture raised by Mousavand and Schroll–Treffinger–Valdivieso. The latter is stated in terms of Geiß–Leclerc–Schröer's generically <span></span><math>\n <semantics>\n <mi>τ</mi>\n <annotation>$\\tau$</annotation>\n </semantics></math>-reduced components and provides a geometric interpretation of a question of Demonet. It follows that the stable second Brauer–Thrall conjecture implies our <span></span><math>\n <semantics>\n <mi>τ</mi>\n <annotation>$\\tau$</annotation>\n </semantics></math>-reduced second Brauer–Thrall conjecture. Finally, we prove the reversed implication for the class of <span></span><math>\n <semantics>\n <mi>E</mi>\n <annotation>$E$</annotation>\n </semantics></math>-tame algebras recently introduced by Asai–Iyama.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 5","pages":"1568-1583"},"PeriodicalIF":0.9000,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70048","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.70048","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
In this short note, we state a stable and a -reduced version of the second Brauer–Thrall conjecture. The former is a slight strengthening of a brick version of the second Brauer–Thrall conjecture raised by Mousavand and Schroll–Treffinger–Valdivieso. The latter is stated in terms of Geiß–Leclerc–Schröer's generically -reduced components and provides a geometric interpretation of a question of Demonet. It follows that the stable second Brauer–Thrall conjecture implies our -reduced second Brauer–Thrall conjecture. Finally, we prove the reversed implication for the class of -tame algebras recently introduced by Asai–Iyama.