Producing new semi-orthogonal decompositions in arithmetic geometry

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 some category D and a semi-orthogonal decomposition (A, A , B ) of T we look either for a decomposition (A ', A ' , B ' ) of T ' ' such that there are no nonzero D-morphisms from A into A ' ' and from B into B ' , or for a decomposition (AD, A D , B D ) of D such that AD D f1 T = A and BD D f1 T = 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 perf (X) and Db(coh(X)); b (coh(X)) ; the latter extend to D-(coh(X)), - (coh(X)), D+coh(Qcoh(X)), + coh (Qcoh(X)), Dcoh(Qcoh(X)) coh (Qcoh(X)) and D(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 Db(coh(X)) b (coh(X)) and D-(coh(X)) - (coh(X)) in terms of Dperf(X). perf (X). We also prove a rather similar new theorem that relates D+coh(Qcoh(X)) + coh (Qcoh(X)) and Dcoh(Qcoh(X)) coh (Qcoh(X)) (these are certain modifications of the bounded below and the unbounded derived category of coherent sheaves on X ) to homological functors Dperf(X)op -> R-mod. perf (X) op -> R-mod. Moreover, we discuss an application of this theorem to the construction of certain adjoint functors. Bibliography: 30 titles.