2021
DOI: 10.1017/s0013091521000109
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Report on the finiteness of silting objects

Abstract: We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$ -tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$ -tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$ -tilting-finite algebra is representatio… Show more

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Cited by 16 publications
(34 citation statements)
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“…Unfortunately, the cardinality of 2silt Γ 2 is 28 by [AHMW,Theorem 2]. Since Γ 2 admits an anti-automorphism fixing the vertex 2, a similar argument as the proof of Theorem 2.10 explains that T − (P 2 ) Λ ≃ T + (P 2 ) Γ 2 anti ≃ T − (P 2 ) Γ 2…”
Section: Selfinjective Nakayama Algebrasmentioning
confidence: 96%
“…Unfortunately, the cardinality of 2silt Γ 2 is 28 by [AHMW,Theorem 2]. Since Γ 2 admits an anti-automorphism fixing the vertex 2, a similar argument as the proof of Theorem 2.10 explains that T − (P 2 ) Λ ≃ T + (P 2 ) Γ 2 anti ≃ T − (P 2 ) Γ 2…”
Section: Selfinjective Nakayama Algebrasmentioning
confidence: 96%
“…As a strategy to determine that an algebra is τ -tilting finite or not, we use the Happel-Vossieck's list [HV] since algebras containing a tame concealed algebra as a factor are τ -tilting infinite [AHMW,Proposition 4.5].…”
Section: Tensor Product Of Algebrasmentioning
confidence: 99%
“…(5 2 ), (5,4), (4 2 , 1), (3 3 ), (3 2 , 2), (3, 2 3 ), (2 5 ), (2 4 , 1). Otherwise, A(λ) is of wild representation type.…”
Section: Staircase Algebrasmentioning
confidence: 99%
“…Therefore, what kind of algebras satisfies the condition: τ -tilting finite implies representation-finite becomes an interesting question. There are some answers, for example, cycle-finite algebras [44], gentle algebras [48], tilted and cluster-tilted algebras [64] and commutative ladders [5].…”
Section: Introductionmentioning
confidence: 99%