On a property of $t$-structures generated by non-classical tilting modules

Abstract

Let $R$ be a ring and $T\in \mathrm{Mod}-R$ be a (non-classical) tilting module of finite projective dimension. Let $\mathcal{T}=({\mathcal{T}}^{\le 0},{\mathcal{T}}^{\ge 0})$ be the $t$-structure on $D(R)$ generated by $T$ and $\mathcal{D}=({\mathcal{D}}^{\le 0},{\mathcal{D}}^{\ge 0})$ be the natural $t$-structure. We show that the pair $(\mathcal{D},\mathcal{T})$ is right filterable in the sense of [FMT14], that is, for any $i\in \mathbb{Z}$ the intersection ${\mathcal{D}}^{\ge i}\cap {\mathcal{T}}^{\ge 0}$ is the co-aisle of a $t$-structure. As a consequence, the heart of $\mathcal{T}$ is derived equivalent to $\mathrm{Mod}-R$.

7 pages