高阶稳定dg范畴与簇Morita理论的Auslander对应
Auslander correspondence for higher stable dg categories and cluster Morita theory
浏览论文内容
中文总结 AI 辅助
该研究建立高阶稳定dg范畴的Auslander对应，引入d-簇dg范畴并证明其包含原d-稳定dg范畴作为d-簇倾斜子范畴，发展簇Morita理论并证明Amiot猜想的Morita变体与Calabi-Yau对应。
中文摘要 AI 辅助
d-稳定dg范畴的概念将稳定dg范畴的d-簇倾斜子范畴公理化。我们建立了d-稳定dg范畴的Auslander对应：通过凝聚性、弱整体维数及有限表现模上的对偶性，刻画加法连通dg范畴的d-稳定性，这为d-稳定性提供了同调刻画，并揭示其为(d+1)-Calabi-Yau对偶的扭曲形式。对于局部有限连通dg代数，在Koszul对偶下该解释尤为清晰，其中d-稳定性对应于Koszul对偶上的平移自内射条件。遵循Amiot、Guo和Keller的构造，对d-稳定dg范畴M，我们引入其d-簇dg范畴$\boldsymbol{\frak C}_{d,{\rm dg}}(M):=\text{per}_{\rm dg}M/^\boldsymbol{\frak L}\boldsymbol{\frak D}^b_{\rm fp, dg}(M)$。利用我们的Auslander对应，证明$\boldsymbol{\frak C}_{d,{\rm dg}}(M)$包含M作为d-簇倾斜子范畴，即每个d-稳定dg范畴都可实现为稳定dg范畴的d-簇倾斜子范畴。随后我们发展簇Morita理论：配备d-簇倾斜子范畴M的预三角dg范畴与$\boldsymbol{\frak C}_{d,{\rm dg}}(M)$拟等价，故簇倾斜子范畴的连通dg结构决定其环境dg范畴的拟等价类。作为簇Morita理论的应用，我们证明Amiot猜想的Morita理论变体，具体而言，建立Calabi-Yau对应：域上局部有限d-稳定dg范畴M的$\boldsymbol{\frak D}^b_{\rm fp, dg}(M)$上的右(d+1)-Calabi-Yau结构，与$\boldsymbol{\frak C}_{d,{\rm dg}}(M)$上的右d-Calabi-Yau结构一一对应。
英文摘要
The notion of $d$-stable dg categories axiomatizes $d$-cluster tilting subcategories of stable dg categories. We establish an Auslander correspondence for $d$-stable dg categories: we characterize the $d$-stability of an additive connective dg category in terms of coherence, weak global dimension, and a duality on finitely presented modules. This gives a homological characterization of $d$-stability and reveals it as a twisted form of $(d+1)$-Calabi--Yau duality. For locally finite connective dg algebras, this interpretation becomes particularly transparent under Koszul duality, where $d$-stability corresponds to a shifted self-injectivity condition on the Koszul dual. Following the constructions of Amiot, Guo and Keller, for a $d$-stable dg category $M$, we introduce its $d$-cluster dg category $\mathcal C_{d,{\rm dg}}(M):=\operatorname{per}_{\rm dg}M/^\mathbb{L}\mathcal D^b_{\rm fp, dg}(M)$. Using our Auslander correspondence, we show that $\mathcal C_{d,{\rm dg}}(M)$ contains $M$ as a $d$-cluster tilting subcategory. In particular, every $d$-stable dg category can be realized as a $d$-cluster tilting subcategory of a stable dg category. We then develop cluster Morita theory: a pretriangulated dg category equipped with a $d$-cluster tilting subcategory $M$ is quasi-equivalent to $\mathcal C_{d,{\rm dg}}(M)$. Thus the connective dg structure of a cluster tilting subcategory determines its ambient dg category up to quasi-equivalence. As an application of cluster Morita theory, we prove a Morita-theoretic variant of Amiot's conjecture. More precisely, we establish a Calabi--Yau correspondence: for a locally finite $d$-stable dg category $M$ over a field, right $(d+1)$-Calabi--Yau structures on $\mathcal D^b_{\rm fp, dg}(M)$ are in bijection with right $d$-Calabi--Yau structures on $\mathcal C_{d,{\rm dg}}(M)$.
发表机构
- Graduate School of Mathematical Sciences, The University of Tokyo（东京大学大学院数理科学研究科）
机构由 AI 辅助整理，请以论文原文为准。
