Producing "new" semi-orthogonal decompositions in arithmetic geometry

TL;DR

This paper develops methods to construct new semi-orthogonal decompositions in derived categories of coherent sheaves, generalizing previous results and establishing correspondences between different categories in arithmetic geometry.

Contribution

It introduces a framework for creating new semi-orthogonal decompositions from existing ones and extends these to various derived categories, broadening the scope of prior theorems.

Findings

Established existence of new semi-orthogonal decompositions

Proved correspondence between decompositions of different derived categories

Generalized a theorem of Karmazyn, Kuznetsov, and Shinder

Abstract

This paper is devoted to constructing "new" admissible subcategories and semi-orthogonal decompositions of triangulated categories out of "old" ones. For two triangulated subcategories $T$ and $T'$ of a certain $D$ and a decomposition $(L,R)$ of $T$ we look either for a decomposition $(L',R')$ of $T'$ such that there are no non-zero $D$-morphisms from $L$ into $L'$ and from $R$ into $R'$, or for a decomposition $(L_D,R_D)$ of $D$ such that $L_D\cap T=L$ and $R_D\cap T=R$. We prove some general existence statements (that also extend to semi-orthogonal decompositions with any number of components) and apply them to various derived categories of coherent sheaves over a scheme $X$ that is proper over a noetherian ring $R$. This gives a one-to-one correspondence between semi-orthogonal decompositions of $D_{perf}(X)$ and $D^b_{coh}(X)$; the latter extend to $D^-_{coh}(X)$, $D^+_{coh}({Qcoh}(X))$, $D_{coh}({Qcoh}(X))$, and $D({Qcoh}(X))$ under very mild conditions. In particular, we obtain a vast generalization of a theorem of J. Karmazyn, A. Kuznetsov, and E. Shinder. These applications rely on recent results of Neeman that express $D^b_{coh}(X)$ and $D^-_{coh}(X)$ in terms of $D_{perf}(X)$ along with its new variations corresponding to $D^+_{coh}({Qcoh}(X))$ and $D_{coh}({Qcoh}(X))$. We also discuss an application of this theorem to the construction of certain adjoint functors.