  We introduce a version of the Cartier isomorphism for de Rham cohomology
valid for associative, not necessarily commutative algebras over a field of
positive characteristic. Using this, we imitate the well-known argument of P.
Deligne and L. Iluusie and prove, in some cases, a conjecture of M. Kontsevich
which claims that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral
sequence degenerates.
