Producing new semi-orthogonal decompositions in arithmetic geometry

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 $D_{\mathrm{perf}}(X)^{\mathrm{op}}\to R-\mathrm{mod}$. Moreover, we discuss an application of this theorem to the construction of certain adjoint functors. Bibliography: 30 titles.