Let $X$ be a locally Noetherian scheme and $K^{\bullet}$ a perfect complex of $\mathcal{O}\_X$-modules. We say $K^{\bullet}$ is "formal" if it is quasi-isomorphic to the complex $\bigoplus\_{n}H\_n(K^{\bullet})[n]$.



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> Can formality of a perfect complex of $\mathcal{O}\_X$-modules be checked Zariski-locally on $X$?
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*Remark.* This seems dubious to me, unless there's some natural zigzag between $K^{\bullet}$ and $\bigoplus\_nH\_n(K^{\bullet})[n]$, in which case it'd be better to define formality in terms of that map in $D(\mathcal{O}\_X)$ being an isomorphism. But maybe there's a procedure to build a formal representatives out of local data and I'm just unaware of it.


I'd be interested to know if it's possible. A quick literature search didn't turn up much.



Checking formality of a perfect complex Zariski-locally
Consider two vector bundles $V,W$ and $\alpha \in {\rm Ext}^2(V,W)$, $\alpha \neq 0$.
Then $$C:={\rm cone}(V[1] \to^\alpha W[-1])$$ is a nonsplit complex with $H^0(C) = V$ and $H^1(C) = W$. But $C$ splits locally, because locally $V$ and $W$ are projective.


E.g. we can take $X = \mathbb P^2, V =\mathcal O,$ and $W = \mathcal O(-3)$.


