relation between extensions and $\mathrm{Hom}$ for an abelian subcategory of a triangulated category

Question

Let $T$ be a triangulated category and $A$ and abelian full subcategory of $T$. Consider the Yoneda extension groups $Ext_{A}^n$. For any two objects $X$ and $Y$ of $A$ here is a map $$ Ext^n_A(X, Y) \longrightarrow Hom_T(X, Y[n]). $$

I am looking for a reference for this theorem: assume that $A$ is admissible (meaning that every short exact sequence in $A$ comes from a distinguished triangle in $T$) and stable under extension. Then the above map is an isomorphism for $n=1$ and injective for $n=2$.

To be honest, I am not completely sure that this is true in this generality. I have seen people using this when $A$ is the heart of a t-structure on $T$ (which is an example of full admissible abelian category), but I couldn't find any reference in BBD.

Answer 1

Here are two places in the literature where this is discussed, probably there are more:

On p.3 of

• P. Deligne and A. Goncharov: Groupes fondamentaux motiviques de Tate mixte. Ann. scient. ENS 38 (2005), pp. 1-56.

The proof shows that $Hom$ is a $\delta$ functor and $Ext$ is the universal $\delta$-functor and then invokes a theorem of Buchsbaum.

There is also a proof in Section 1 of

• M. Levine: Tate motives and the vanishing conjectures for algebraic K-theory. In: Algebraic K-theory and algebraic topology (Lake Louise, 1991), 167–188, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer, 1993.