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The following is stated in Luc Illusie, "Frobenius and Hodge degeneration", part 4.6.

Let $L$ be a bounded chain complex. There is a sequence of obstructions, first $c_i\in \mathrm{Ext}^2(H^iL, H^{i-1}L)$, if these vanish then secondary obstructions $c'_i\in \mathrm{Ext}^3(H^iL, H^{i-2}L)$ are defined, etc. $L$ is decomposable if and only if all these obstructions vanish.

(Decomposability here means that in the derived category it is isomorphic to a chain complex with $0$ differential.)

If the homology is concentrated in two subsequent dimensions, then this is proved, but in general, no proof (or construction of the obstructions) is given.

Question: Is there a reference for the general case?

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