Hom-configurations in triangulated categories generated by spherical objects

Coelho, Simoes, Raquel

doi:10.1016/j.jpaa.2014.10.017

Cited by 13 publications

(3 citation statements)

References 14 publications

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“…It was proved in [ 4 ] that the algebras and are derived equivalent if and only if there exists a tilting complex such that is isomorphic to as a -algebra. In the same paper it is explained how to construct an equivalence from to sending to X using the tilting complex X and an algebra isomorphism .…”

Section: Application To Derived Picard Groupsmentioning

confidence: 99%

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Groups generated by two twists along spherical sequences

Volkov¹

2022

Math. Proc. Camb. Phil. Soc.

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We describe all groups that can be generated by two twists along spherical sequences in an enhanced triangulated category. It will be shown that with one exception such a group is isomorphic to an abelian group generated by not more than two elements, the free group on two generators or the braid group of one of the types $A_2$ , $B_2$ and $G_2$ factorised by a central subgroup. The last mentioned subgroup can be nontrivial only if some specific linear relation between length and sphericity holds. The mentioned exception can occur when one has two spherical sequences of length 3 and sphericity 2. In this case the group generated by the corresponding two spherical twists can be isomorphic to the nontrivial central extension of the symmetric group on three elements by the infinite cyclic group. Also we will apply this result to give a presentation of the derived Picard group of selfinjective algebras of the type $D_4$ with torsion 3 by generators and relations.

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Section: Application To Derived Picard Groupsmentioning

confidence: 99%

“…On the other hand, m -spherical objects with deserve attention too. They were considered, for example, in [ 4, 5, 6, 10 ].…”

Section: Introductionmentioning

confidence: 99%

Groups generated by two twists along spherical sequences

Volkov¹

2022

Math. Proc. Camb. Phil. Soc.

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“…Simple‐minded systems (SMSs) in stable module categories were introduced in [30] and studied for negative CY triangulated categories in [15, 17]. Recently, there is increasing interest in negative CY triangulated categories (see, for example, [7, 12–16, 19]), including the stable categories of Cohen–Macaulay (CM) dg modules [21].…”

Section: Introductionmentioning

confidence: 99%

Reductions of triangulated categories and simple‐minded collections

Jin

2023

Journal of London Math Soc

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Silting and Calabi–Yau reductions are important processes in representation theory to construct new triangulated categories from given ones, which are similar to Verdier quotient. In this paper, first we introduce a new reduction process of triangulated category, which is analogous to the silting (Calabi–Yau) reduction. For a triangulated category T$\mathcal {T}$ with a pre‐simple‐minded collection (pre‐SMC) R$\mathcal {R}$, we construct a new triangulated category U$\mathcal {U}$ such that the SMCs in U$\mathcal {U}$ bijectively correspond to those in T$\mathcal {T}$ containing R$\mathcal {R}$. Second, we give an analogue of Buchweitz's theorem for the singularity category scriptTprefixsg$\mathcal {T}_{\operatorname{sg}\nolimits }$ of a SMC quadruple false(scriptT,Tp,double-struckS,scriptSfalse)$(\mathcal {T},\mathcal {T}^{\mathrm{p}},\mathbb {S}, \mathcal {S})$: the category scriptTprefixsg$\mathcal {T}_{\operatorname{sg}\nolimits }$ can be realized as the stable category of an extriangulated subcategory F$\mathcal {F}$ of T$\mathcal {T}$. Finally, we show the simple‐minded system (SMS) reduction due to Coelho Simões and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi–Yau reduction is the shadow of silting reduction due to Iyama and Yang.

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Abelian subcategories of triangulated categories induced by simple minded systems

Jørgensen¹

2022

Math. Z.

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Hom-configurations in triangulated categories generated by spherical objects

Cited by 13 publications

References 14 publications

Groups generated by two twists along spherical sequences

Groups generated by two twists along spherical sequences

Reductions of triangulated categories and simple‐minded collections

Abelian subcategories of triangulated categories induced by simple minded systems

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