On t-structures adjacent and orthogonal to weight structures

We study t-structures (on triangulated categories) that are closely related to weight structures. A t-structure couple t=(Ct≤ 0,Ct≥ 0) is said to be adjacent to a weight structure w=(Cw≤ 0, Cw≥ 0) if Ct≥ 0=Cw≥ 0. For a category C that satisfies the Brown representability property we prove that t that is adjacent to w exists if and only if w is smashing (that is, "respects C-coproducts"). The heart $Ht$ of this t is the category of those functors Hwᵒᵖ→ Ab that respect products (here $Hw$ is the heart of w); the result has important applications. We prove several more statements on constructing t-structures starting from weight structures; we look for a strictly orthogonal t-structure t on some $C'$ (where $C,C'$ are triangulated subcategories of a common D) such that C't≤ 0 (resp. C't≥ 0) is characterized by the vanishing of morphisms from Cw≥ 1 (resp. Cw≤ -1). Some of these results generalize properties of semi-orthogonal decompositions proved in the previous paper, and can be applied to various derived categories of (quasi)coherent sheaves on a scheme X that is projective over an affine noetherian one. We also study hearts of orthogonal t-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal t-structures.