Completing hearts of triangulated categories via weight-exact localizations

TL;DR

This paper explores how weight-exact localizations of triangulated categories determine the relationships between hearts of associated t-structures, generalizing non-commutative ring localizations and extending previous work on compactly generated subcategories.

Contribution

It provides a comprehensive description of functors between hearts in weight-exact localizations, broadening the understanding of localizations beyond compactly generated cases.

Findings

Functors between hearts are determined by the heart of the weight structure and the inverted morphisms.

Generalizes earlier descriptions of non-commutative localizations of rings.

Connects weight structures with classical localizations like Bousfield localization.

Abstract

We study a weight-exact localization pi of a well generated triangulated category C along with the embedding of the hearts of adjacent t-structures coming from the functor right adjoint to pi. We prove that the functors relating the corresponding four hearts are completely determined by the heart Hw of the weight structure on C along with the set of Hw-morphisms that we invert via pi; it also suffices to know the corresponding embedding of the hearts of t-structures. Our results generalize the description of non-commutative localizations of rings in terms of weight-exact localizations given in an earlier paper of the first author. That paper was essentially devoted to weight-exact localizations by compactly generated subcategories, whereas in the current text we focus on "more complicated" localizations. We recall that two types of localizations of the sort we are interested in were studied by several authors. They took C=D(R-mod); the heart of the first t-structure was equivalent to R-mod, and the second heart was equivalent to the exact abelian category $U_{contra}\subset R-mod$ of U-contramodules (corresponding to a set of Proj R-mod-morphisms U related either to a homological ring epimorphism $u:R\to UU$ or to an ideal of I of R). The functor $R-mod\to U_{contra}$ induced by pi is a certain completion one. Consequently, the hearts of the corresponding weight structures are equivalent to Proj R-mod and to the subcategory of projective objects of U_{contra}, respectively. Moreover, the connecting functors between these categories are isomorphic to ones coming from any weight structure class-generated by a single compact object whose endomorphism ring is R^{op}; in particular, one can take R=Z and C=SH and re-prove some important statements due to Bousfield.