稳定∞-范畴的半正交分解
Semiorthogonal decompositions of stable $\infty$-categories
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中文总结 AI 辅助
本文扩展半正交分解理论，构造Waldhausen图表与凝聚复形的∞-范畴等价，证明半正交分解的重构定理并将其应用于Beilinson特殊集合以重构Dᵇ(Coh(ℙⁿ))。
中文摘要 AI 辅助
我们定义了长度为n的稳定∞-范畴的半正交分解，扩展了arXiv:2106.02873中提出的半正交分解理论。在此之前，我们明确构造了将Waldhausen图表与稳定∞-范畴中的凝聚复形联系起来的∞-范畴等价，这让我们能从两种不同视角看待半正交分解，各有其独特优缺点。在温和条件下，我们证明了半正交分解的重构定理，将稳定∞-范畴重构为构成其分解的子范畴所形成图表的（余） lax极限。我们将该重构应用于Beilinson特殊集合的情形，得到了Dᵇ(Coh(ℙⁿ))的重构。
英文摘要
We define semiorthogonal decompositions of stable $\infty$-categories of length $n$, extending the theory of semiorthogonal decompositions presented in arXiv:2106.02873. Prior to this, we explicitly construct an equivalence of $\infty$-categories relating Waldhausen diagrams to coherent complexes in stable $\infty$-categories. This allows us to view semiorthogonal decompositions from two different perspectives, each of which has its unique advantages and disadvantages. Under mild conditions, we prove a reconstruction theorem for semiorthogonal decompositions, recovering a stable $\infty$-category as the (op)lax limit of a diagram formed by the subcategories constituting its decomposition. We apply this reconstruction in the case of Beilinson's exceptional collection, to obtain a reconstruction of $D^b(\text{Coh}(\mathbb{P}^n))$.
