Realization functors in algebraic triangulated categories

Abstract

Let $\mathcal{T}$ be an algebraic triangulated category and $\mathcal{C}$ an extension-closed subcategory with $\mathrm{Hom}(\mathcal{C},\hat{I}{\pounds }^{<0}\mathcal{C})=0$. Then $\mathcal{C}$ has an exact structure induced from exact triangles in $\mathcal{T}$. Keller and Vossieck say that there exists a triangle functor $D^b(\mathcal{C})\to \mathcal{T}$ extending the inclusion $\mathcal{C}\subseteq \mathcal{T}$. We provide the missing details for a complete proof.

10 pages; v2: add Example 2.13 and small corrections; to appear in Abh. Math. Semin. Univ. Hambg