  We compute the moduli of endomorphisms of the de Rham and crystalline
cohomology functors, viewed as a cohomology theory on smooth schemes over
truncated Witt vectors. As applications of our result, we deduce Drinfeld's
refinement of the classical Deligne--Illusie decomposition result for de Rham
cohomology of varieties in characteristic $p>0$ that are liftable to $W_2$, and
prove further functorial improvements.
