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If $\mathcal{A}$ is any abelian category, then for all objects $X,Y$ in $\mathcal{A}$ and for all integers $i \geq 0$, there is an natural isomorphism $$\mathrm{Ext}_\mathcal{A}^i(X,Y) \simeq \mathrm{Hom}_{D(\mathcal{A})}(X,Y[i]),$$ where the left-hand side denotes the Yoneda $\mathrm{Ext}$-group in $\mathcal{A}$ and the right-hand side denotes the morphisms between $X$ and the shift of $Y$ by $i$ in the derived category $D(\mathcal{A})$ of $\mathcal{A}$. Note that this holds without any assumption on $\mathcal{A}$ (e.g. existence of enough injectives/projectives), cf. Verdier's thesis III.3.2.

Now, I am interested in an exact category $\mathcal{E}$ (in the sense of Quillen, i.e. a strictly full subcategory closed under taking extensions of an abelian category $\mathcal{A}$). Exact categories encapsulate just enough of the formalism of abelian categories to define the Yoneda $\mathrm{Ext}$-groups (as sequences $E_1E_2...E_i$ of short exact sequences with compatible ends). Moreover, under certain assumptions one can define the derived category $D(\mathcal{E})$ of $\mathcal{E}$. In my situation, $\mathcal{E}$ has kernels so that $D(\mathcal{E})$ can be defined as in [Beilinson-Bernstein-Deligne, Faisceaux pervers, §1.1.4]. In this setup, do we have a natural isomorphism $$\mathrm{Ext}_\mathcal{E}^i(X,Y) \simeq \mathrm{Hom}_{D(\mathcal{E})}(X,Y[i])$$ for all objects $X,Y$ in $\mathcal{E}$ and for all integers $i \geq 0$?

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Firstly, for any Quillen exact category $\mathcal E$, one can define the derived category $D(\mathcal E)$, as well as its bounded versions $D^+(\mathcal E)$, $D^-(\mathcal E)$, and $D^b(\mathcal E)$.

Existence of kernels in $\mathcal E$ is irrelevant. The relevant conditions are idempotent-closedness (Karoubianness) and weak idempotent closedness.

The (bounded or unbounded) derived category of an exact category $\mathcal E$ can be defined without these conditions. But assuming one of them (depending on how bounded is the derived category which you want to define) simplifies some questions related to this definition.

References:

  1. A. Neeman, "The derived category of an exact category", Journ. of Algebra 135 (1990).
  2. B. Keller, "Derived categories and their uses", Handbook of Algebra vol.1 (1995), p.671-701.
  3. T. Buehler, "Exact categories", Expositiones Math. 28 (2010), Section 10.

Secondly, the isomorphism $$ \operatorname{Ext}_{\mathcal E}^i(X,Y)\simeq \operatorname{Hom}_{D(\mathcal E)}(X,Y[i]) $$ for all $X$, $Y\in\mathcal E$ and $i\ge0$ holds for all Quillen exact categories $\mathcal E$.

I would be glad to be informed of an earlier/better reference, but having glanced through the three papers cited above and not found this assertion explicitly formulated there, I can only suggest Proposition A.13 in my paper

  1. L. Positselski, Mixed Artin-Tate motives with finite coefficients, Moscow Math. Journ. 11 #2 (2011), Appendix A.

In the arXiv version, https://arxiv.org/abs/1006.4343 , Appendix A, this is the only (unnumbered) Proposition in Section A.7.

The proof of this proposition in my paper is very short and omits all the details, but the assertion in question is indeed quite straightforward (particularly, if you already know how to prove this for abelian categories). This may also be the reason why this result is not even formulated in some other references.

Further discussion can be found in Section A.8.

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  • $\begingroup$ This is sort of a follow-up question, but what kind of hypotheses on $\mathcal{E}$ ensures that $D(\mathcal{E})$ is locally small? Is Karoubianness enough? $\endgroup$ – Arkandias Apr 26 '17 at 19:22
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    $\begingroup$ No, this is unrelated to Karoubianness, and I am not sure about what might be a good general answer to this question. How would you answer this question in the case of an abelian category? What kind of hypotheses on an abelian category $\mathcal A$ would you use to ensure that $D(\mathcal A)$ is locally small? $\endgroup$ – Leonid Positselski Apr 26 '17 at 21:25
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    $\begingroup$ I was thinking of having enough injectives/projectives, but this only ensures that the bounded below/above derived categories are locally small... $\endgroup$ – Arkandias Apr 26 '17 at 21:58
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    $\begingroup$ Yes, this works for exact categories just as well. You can define what it means for an exact category $\mathcal E$ to have enough projective/injective objects, and then $D^-(\mathcal E)$ is locally small if $\mathcal E$ has enough projectives, and $D^+(\mathcal E)$ is locally small if $\mathcal E$ has enough injectives. $\endgroup$ – Leonid Positselski Apr 27 '17 at 12:38
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    $\begingroup$ Another known result of this kind is that $D(\mathcal A)$ is locally small when $\mathcal A$ is a Grothendieck abelian category. Likely it can be, and perhaps already has been, generalized to locally presentable abelian categories $\mathcal A$. A good (probably not very difficult) project in this direction might be to define what an accessible exact category is and prove that $D(\mathcal E)$ is locally small when $\mathcal E$ is an accessible exact category. $\endgroup$ – Leonid Positselski Apr 27 '17 at 12:43

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