# Checking formality of a perfect complex Zariski-locally

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 $$\bigoplus_{n}H_n(K^{\bullet})[n]$$.

Can formality of a perfect complex of $$\mathcal{O}_X$$-modules be checked Zariski-locally on $$X$$?

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.

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 $$W = \mathcal O(-3)$$.