Theorem 2.3 (0MUA) · ScalingStacks

0MUA

Theorem 2.3 ([11], Theorem 1.3). Let $\mathcal{T}$ be a triangulated category with set indexed coproducts. Suppose $S$ is a compact rigid object of $\mathcal{T}$ and assume that $\{\Sigma^{i}S\ |\ i\in\mathbb{Z}\}$ is a generating set for $\mathcal{T}$ . Then the following forms a non-degenerate $t$ -structure on $\mathcal{T}$ :

	$\displaystyle\mathcal{X}$	$\displaystyle=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}_{\mathcal{T}}(S,\Sigma^{i}% X)=0\textnormal{ for }i>0\},$
	$\displaystyle\mathcal{Y}$	$\displaystyle=$	$\displaystyle\{X\in\mathcal{T}\ \|\ \textnormal{Hom}_{\mathcal{T}}(S,\Sigma^{i}% X)=0\textnormal{ for }i<0\}.$

Moreover, its heart $\mathcal{H}=\mathcal{X}\cap\mathcal{Y}$ is an admissible abelian subcategory of $\mathcal{T}$ in the sense of [3], and the functor

\textnormal{Hom}(S,-):\mathcal{H}\rightarrow\textnormal{Mod}(\textnormal{End}(% S)^{\textnormal{op}})

is an equivalence of categories.