Semiorthogonal decompositions of stable ∞-categories
Abstract
We define semiorthogonal decompositions of stable ∞-categories of length n, extending the theory of semiorthogonal decompositions presented in arXiv:2106.02873. Prior to this, we explicitly construct an equivalence of ∞-categories relating Waldhausen diagrams to coherent complexes in stable ∞-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 ∞-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 Db(Coh(ℙn)).
Related
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BibTeX
@misc{albi2026semiorthogonal,
title = {{Semiorthogonal decompositions of stable $\infty$-categories}},
author = {Rio Haeussler Albi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.31020},
eprint = {2608.31020},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31020},
}
RIS
TY - UNPB
TI - Semiorthogonal decompositions of stable ∞-categories
AU - Rio Haeussler Albi
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2608.31020
UR - https://arxiv.org/abs/2608.31020
AB - We define semiorthogonal decompositions of stable ∞-categories of length n, extending the theory of semiorthogonal decompositions presented in arXiv:2106.02873. Prior to this, we explicitly construct an equivalence of ∞-categories relating Waldhausen diagrams to coherent complexes in stable ∞-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 ∞-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(Coh(ℙ^n)).
ER -