  We describe the obstruction to decomposing in degrees $\leq p$ the de Rham
complex of a smooth variety over a perfect field $k$ of characteristic $p$ that
lifts over $W_2(k)$, and show that there exist liftable smooth projective
varieties of dimension $p+1$ whose Hodge-to-de Rham spectral sequence does not
degenerate at the first page. We also describe the action of the Sen operator
on the de Rham complex in degrees $\leq p$ and give examples of varieties with
a non-semisimple Sen operator.
  Our methods rely on the commutative algebra structure on de Rham and
Hodge-Tate cohomology, and are inspired by the properties of Steenrod
operations on cohomology of cosimplicial commutative algebras. The example of a
non-degenerate Hodge-to-de Rham spectral sequence relies on a non-vanishing
result on cohomology of groups of Lie type. We give applications to other
situations such as describing extensions in the canonical filtration on de
Rham, Hodge, and \'etale cohomology of an abelian variety equipped with a group
action. We also show that the de Rham complex of a smooth variety over $k$ is
formal as an $E_{\infty}$-algebra if and only if the variety lifts to $W_2(k)$
together with its Frobenius endomorphism.
