A correspondence and distance of t-structures

Abstract

For two t-structures $D_{1}=(D_{1}^{\leqslant 0},D_{1}^{\geqslant 1})$ and $D_{2}=(D_{2}^{\leqslant 0},D_{2}^{\geqslant 1})$ with $D_{1}^{\leqslant 0} \subseteq D_{2}^{\leqslant 0}$ on a triangulated category $\mathcal{D}$, we give a correspondence between t-structure $D_{i}=(D_{i}^{\leqslant 0},D_{i}^{\geqslant 1})$ which satisfies $D_{1}^{\leqslant 0} \subseteq D_{i}^{\leqslant 0} \subseteq D_{2}^{\leqslant 0}$ and a pair of full subcategories of $D_{1}^{\geqslant 1}\bigcap D_{2}^{\leqslant 0}$. Then we give a way to determine the distance of two t-structure if we have known that their distance is finite.In addition, if we set a t-structure $D_{1}$ whose heart $H_{1} \neq 0$ and that $H_{1}$ has a non-trivial torsion pair, then for any integer $n$, we can construct a t-structure $D_{2}$ such that the distance between $D_{1}$ and $D_{2}$ is $n$.