负簇范畴的Cluster Morita定理
Cluster Morita theorem for negative cluster categories
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中文总结 AI 辅助
本文证明了负簇范畴的Cluster Morita定理：通过dg模型和简单minded系统刻画Hom-有限三角范畴，并在Calabi-Yau假设下给出无环负簇范畴的刻画，同时建立左右Calabi-Yau对称性。
中文摘要 AI 辅助
固定整数 $d\leq-2$。我们刻画了容许一个 $(-d)$-简单 minded 系统的 Hom-有限代数三角范畴，将其刻画为适当的 $(-d)$-自内射非正 dg 代数的稳定范畴 $\underline{\mathrm{CM}}(B)$；等价地，每个这样的范畴都有一个 $d$-稳定的局部有限严格正 dg 模型，其余奇异 dg 商恢复了所选的增强。在 $d$-Calabi--Yau 假设下，我们给出了无环负簇范畴的一个刻画定理，该定理用简单 minded 系统的有限分次扩张代数来表述。在链层面，Hochschild 和约化循环局部化将有限维部分上的右 $(d+1)$-Calabi--Yau 结构等同于余奇异商上的规范化右 $d$-Calabi--Yau 结构。最后，当 Koszul 对偶是适当的时，Brav--Dyckerhoff 求值态射是混合复形的拟同构，从而产生左-右 Calabi--Yau 对称性。
英文摘要
Fix an integer $d\leq-2$. We characterize Hom-finite algebraic triangulated categories admitting a $(-d)$-simple-minded system as stable categories $\underline{\mathrm{CM}}(B)$ of proper $(-d)$-self-injective non-positive dg algebras; equivalently, each admits a $d$-stable locally finite strictly positive dg model whose cosingular dg quotient recovers the chosen enhancement. Under a $d$-Calabi--Yau hypothesis, we give a characterization theorem for acyclic negative cluster categories in terms of the finite graded extension algebra of a simple-minded system. At chain level, Hochschild and reduced cyclic localization identify right $(d+1)$-Calabi--Yau structures on the finite-dimensional part with normalized right $d$-Calabi--Yau structures on the cosingular quotient. Finally, when the Koszul dual is proper, the Brav--Dyckerhoff evaluation morphism is a quasi-isomorphism of mixed complexes, yielding left--right Calabi--Yau symmetry.
发表机构
- Nagoya University（名古屋大学）
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