The $t$-structures generated by objects

Abstract:

Let $\mathscr {T}$ be a well generated triangulated category, and let $S\subset \mathscr {T}$ be a set of objects. We prove that there is a $t$-structure on $\mathscr {T}$ with $\mathscr {T}^{\leq 0}=\overline {\langle S \rangle }^{(-\infty ,0]}$.

This article is an improvement on the main result of Alonso, Jeremías and Souto [Trans. Amer. Soc. 355 (2003), pp. 2523–2543], in which the theorem was proved under the assumption that $\mathscr {T}$ has a nice enough model. It should be mentioned that the result in Alonso, Jeremías and Souto has been influential—it turns out to be interesting to study all of these $t$-structures.