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Consider an abelian category $\mathcal{A}$ (or more specifically, $R$-Mod). Suppose $C_1$ and $C_2$ are chain complexes with componentwise isomorphic homology. What conditions must be imposed upon $\mathcal{A}$ (or $R$), or $C_1$ and $C_2$, for there to exist a quasi-isomorphism $\alpha: C_1 \to C_2$? Or, under what conditions, if any, is there necessarily an isomorphism $C_1 \cong C_2$ in $\mathcal{D}(\mathcal{A})$?

I certainly expect existing conditions to be rare. If the answer is "there basically are none" then I'd like to see some good examples highlighting this too!

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    $\begingroup$ If $R$ is a field, then in fact it happens always. $\endgroup$
    – Zhen Lin
    Commented Mar 30, 2022 at 23:20
  • $\begingroup$ Indeed, as this answer states. I'm hoping for possibly more general statements though, if they exist. $\endgroup$
    – Sam K
    Commented Mar 30, 2022 at 23:34
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    $\begingroup$ You're probably aware of this, but if $\mathcal{A}$ has Ext-dimension 1, then every bounded complex splits in the derived category, see e.g. stacks.math.columbia.edu/tag/0GM4 $\endgroup$
    – Grisha Papayanov
    Commented Apr 1, 2022 at 15:19
  • $\begingroup$ @ Zhen Lin, is it true thought for sheaves of fields $\mathbb{F}_X$? $\endgroup$
    – Flavius Aetius
    Commented Jul 21, 2023 at 12:58

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