Auslander-Buchweitz context and co-t-structures

TL;DR

This paper explores the relationship between Auslander-Buchweitz contexts and co-t-structures in triangulated categories, establishing correspondences and characterizations, especially in the presence of silting classes and in derived categories of hereditary categories.

Contribution

It demonstrates that Auslander-Buchweitz contexts coincide with co-t-structures, establishes bijections with certain subcategories, and characterizes bounded co-t-structures using relative homological algebra.

Findings

Auslander-Buchweitz context equals co-t-structure in certain categories

Bijection between co-t-structures and cosuspended, precovering subcategories

Silting classes induce bounded co-t-structures

Abstract

We show that the relative Auslander-Buchweitz context on a triangulated category $\T$ coincides with the notion of co-$t$-structure on certain triangulated subcategory of $\T$ (see Theorem \ref{M2}). In the Krull-Schmidt case, we stablish a bijective correspondence between co-$t$-structures and cosuspended, precovering subcategories (see Theorem \ref{correspond}). We also give a characterization of bounded co-$t$-structures in terms of relative homological algebra. The relationship between silting classes and co-$t$-structures is also studied. We prove that a silting class $\omega$ induces a bounded non-degenerated co-$t$-structure on the smallest thick triangulated subcategory of $\T$ containing $\omega.$ We also give a description of the bounded co-$t$-structures on $\T$ (see Theorem \ref{Msc}). Finally, as an application to the particular case of the bounded derived category $\D(\HH),$ where $\HH$ is an abelian hereditary category which is Hom-finite, Ext-finite and has a tilting object (see \cite{HR}), we give a bijective correspondence between finite silting generator sets $\omega=\add\,(\omega)$ and bounded co-$t$-structures (see Theorem \ref{teoH}).