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Dec 11, 2019 at 17:47 comment added Achim Krause That's exactly what the machinery of faithfully flat descent does for you. It tells you precisely what additional structure you need to endow your $C_*$ with in order to find a $D_*$ as desired. I don't think you can turn this into a manageable list of obstructions in general, but if your complex is of finite length, this in principle turns into a finite amount of coherences.
Dec 10, 2019 at 22:16 history edited Johannes Hahn CC BY-SA 4.0
Fixed minor typos
Dec 10, 2019 at 20:06 history asked HenrikRüping CC BY-SA 4.0