Checking formality of a perfect complex Zariski-locally

Question

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.

Answer 1

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