Admissible subcategories and metric techniques
Kabeer Manali Rahul
TL;DR
The paper develops a framework of metric techniques in compactly generated triangulated categories to lift semiorthogonal decompositions from compact objects to larger ambient subcategories via compressible metrics and closure-of-compacts constructions. Central to the approach are the notions of extended good metrics, generating sequences, and (weak) co-approximability, which yield a systematic way to obtain SODs on $oldsymbol{T}^{-}_c$, $oldsymbol{T}^{+}_c$, and $oldsymbol{T}^{b}_c$ under Hypotheses (I) and (II). The results apply to algebro-geometric contexts—schemes, algebras, and stacks—providing explicit decompositions under relatively mild hypotheses and computing the relevant compact-closure subcategories in concrete categories like $ extbf{D}^{p}_{ ext{Qcoh},Z}(X)$, $ extbf{D}^{-}_{ ext{coh},Z}(X)$, and their noncommutative analogues. By comparing to recent works (Sun–Zhang, Bondarko, Kuznetsov–Shinder) and embracing co-approximability, the framework offers a versatile, representability-light route to constructing and analyzing SODs in broad geometric settings.
Abstract
In this work, we provide a way of constructing new semiorthogonal decompositions using metric techniques (à la Neeman). Given a semiorthogonal decomposition on a category with a special kind of metric, which we call a compressible metric, we can construct new semiorthogonal decomposition on a category constructed from the given one using the aforementioned metric. In the algebro-geometric setting, this gives us a way of producing new semiorthogonal decompositions on various small triangulated categories associated to a scheme, if we are given one. In the general setting, the work is related to that of Sun-Zhang, while its applications to algebraic geometry are related to the work of Bondarko and Kuznetsov-Shinder.