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