Theorem 2.4 (0MUB) · ScalingStacks

0MUB

Theorem 2.4. Let $\mathcal{T}$ be a triangulated category with set indexed coproducts. Suppose $S$ is a compact simply connected corigid 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 co- $t$ -structure on $\mathcal{T}$ :

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

Moreover, its coheart $\mathcal{C}=\mathcal{A}\cap\mathcal{B}$ is an abelian subcategory of $\mathcal{T}$ , and the functor

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

is an equivalence of categories.