Semiorthogonal decompositions of stable $\infty$-categories

Abstract

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 $D^b(\text{Coh}(\mathbb{P}^n))$.