On higher order analogues of de Rham cohomology

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.