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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.

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  • $\begingroup$ Maybe this is part of admissible and I'm misreading it, but one needs to ask for some sort extension closure to have any hope of an isomorphism for $n=1$. For instance if E is an object with endomorphism ring a division algebra, then the full subcategory on sums of copies of E is semisimple abelian, but E could have self extensions. $\endgroup$ Jan 27 '17 at 20:08
  • $\begingroup$ Dear Greg, you are of course right! I forgot to ask that $A$ is stable under extension. This is not part of the definition of admissible. I will edit it now. $\endgroup$
    – triang
    Jan 27 '17 at 20:18
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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.
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