Decomposability of the de Rham complex in positive characteristic

Deligne and Illusie proved that if a smooth variety over F_p admits a lift over Z/p^2 then the truncation of its de Rham complex in degrees <p is quasi-isomorphic to the direct sum of its cohomology sheaves. As a consequence, the Hodge-to-de Rham spectral sequence of a smooth proper liftable variety degenerates, provided that the dimension of the variety is <=p. It turns out that further truncations of the de Rham complex of a liftable variety need not be decomposable. I will describe the obstruction to decomposing the truncation of the de Rham complex in degrees <=p in terms of other invariants of the variety, and will use this to give an example of a smooth projective variety over F_p that lifts to Z_p but whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. The proof proceeds by first understanding the obstruction to the formality of the de Rham complex as a commutative algebra (in the appropriate derived sense) using an analog of Steenrod operations from algebraic topology.