  If K is a commutative ring and A is a K-algebra, for any sequence $\sigma $
of positive integers there exists an higher order analogue dR($\sigma $) of the
standard de Rham complex dR(1,...,1,...), which can also be defined starting
from suitable ("differentially closed") subcategories of (A-mod). The main
result of this paper is that the cohomology of dR($\sigma $) does not depend on
$\sigma $, under some smoothness assumptions on the ambient category. Before
proving the main theorem we give a rather detailed exposition of all relevant
(to our present purposes) functors of differential calculus on commutative
algebras. This part can be also of an independent interest.
