Some refinements of the Deligne–Illusie theorem

We extend the results of Deligne and Illusie on liftings modulo $p^2$ decompositions de Rham complex in several ways. show that for a smooth scheme $X$ over perfect field $k$ characteristic $p>0$, truncations $\max(p-1, 2)$ consecutive degrees can be reconstructed as objects derived category terms its truncation at most one (or, equivalently, obstruction class to lifting $p^2$). Consequently, these are decomposable if admits $W_2(k)$, which case first nonzero differential conjugate spectral sequence appears no earlier than page $\max(p,3)$ (these corollaries have been recently strengthened by Drinfeld, Bhatt-Lurie, Li-Mondal). Without assuming existence lifting, we describe gerbes splittings two-term differentials second sequence, answering question Katz. The main technical result used $p>2$ belongs purely homological algebra. It concerns certain commutative graded algebras whose cohomology algebra is exterior algebra, dubbed us "abstract Koszul complexes", $p$ an example. In appendix, use aforementioned stronger decomposition prove Kodaira-Akizuki-Nakano vanishing Hodge-de degeneration both hold $F$-split $(p+1)$-folds.