On t-structures adjacent and orthogonal to weight structures

TL;DR

This paper explores the relationship between t-structures and weight structures in triangulated categories, establishing conditions for their adjacency and orthogonality, with applications to derived categories of sheaves.

Contribution

It characterizes when t-structures adjacent to weight structures exist, especially under Brown representability, and investigates orthogonal t-structures and their hearts.

Findings

Existence of adjacent t-structures characterized by smashing weight structures.

Construction methods for t-structures from weight structures.

Applications to derived categories of sheaves on schemes.

Abstract

We study $t$-structures (on triangulated categories) that are closely related to weight structures. A $t$-structure couple $t=(C_{t\le 0},C_{t\ge 0})$ is said to be adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$. 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 $Hw^{op}\to Ab$ 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 $C'_{t\le 0}$ (resp. $C'_{t\ge 0}$) is characterized by the vanishing of morphisms from $C_{w\ge 1}$ (resp. $C_{w\le -1}$). 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.