This article is cited in
2 papers
Producing new semi-orthogonal decompositions in arithmetic geometry
M. V. Bondarkoab a St. Petersburg University, St. Petersburg, Russia
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
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
$\mathcal{T}$ and
$\mathcal{T}'$ of some category
$\mathcal{D}$ and a semi-orthogonal decomposition
$(\mathcal{A},\mathcal{B})$ of
$\mathcal{T}$ we look either for a decomposition
$(\mathcal{A}',\mathcal{B}')$ of
$\mathcal{T}'$ such that there are no nonzero
$\mathcal{D}$-morphisms from
$\mathcal{A}$ into
$\mathcal{A}'$ and from
$\mathcal{B}$ into
$\mathcal{B}'$, or for a decomposition $(\mathcal{A}_{\mathcal{D}},\mathcal{B}_{\mathcal{D}})$ of
$\mathcal{D}$ such that $\mathcal{A}_{\mathcal{D}}\cap \mathcal{T}=\mathcal{A}$ and $\mathcal{B}_{\mathcal{D}}\cap \mathcal{T}=\mathcal{B}$. We prove some general existence statements (that also extend to semi-orthogonal decompositions of arbitrary length) and apply them to various derived categories of coherent sheaves over a scheme
$X$ that is proper over the spectrum of a Noetherian ring
$R$. This produces a one-to-one correspondence between semi-orthogonal decompositions of
$D_{\mathrm{perf}}(X)$ and
$D^{\mathrm{b}}(\operatorname{coh}(X))$; the latter extend to
$D^-(\operatorname{coh}(X))$,
$D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))$,
$D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ and
$D(\operatorname{Qcoh}(X))$ under very mild assumptions. In particular, we obtain a broad generalization of a theorem of Karmazyn, Kuznetsov and Shinder.
These applications rely on some recent results of Neeman that express
$D^{\mathrm{b}}(\operatorname{coh}(X))$ and
$D^-(\operatorname{coh}(X))$ in terms of
$D_{\mathrm{perf}}(X)$. We also prove a rather similar new theorem that relates
$D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ and
$D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ (these} are certain modifications of the bounded below and the unbounded derived category of coherent sheaves on
$X$ to homological functors
$D_{\mathrm{perf}}(X)^{\mathrm{op}}\to R\text{-}\mathrm{mod}$. Moreover, we discuss an application of this theorem to the construction of certain adjoint functors.
Bibliography: 30 titles.
Keywords:
triangulated category, adjoint functor, semi-orthogonal decomposition, admissible subcategory, quasi-coherent sheaves, perfect complexes.
MSC: Primary
18G80,
14F08,
18A40; Secondary
18E10,
14A15,
14G40 Received: 16.03.2022 and 08.02.2024
DOI:
10.4213/sm9752
Fulltext:
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