Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of differential forms on $X$. On the other hand, it is intuitively clear, that $\Omega^\ast(E)$ contains all the information on $E$ one needs to describe all the rational invariants of $E$, including its characteristic classes. So, I wonder, if there is a homotopy-invariant way to extract this information? That is, I need a method, non-involving the usual Chern-Weil formalism, non-appealing to the general linear group etc., a method, roughly speaking, applicable to any differential graded algebra or even a dg module over $\Omega^\ast(X)$. In addition, it is interesting, how much of the actual structure of $\Omega^\ast(E)$ one actually needs: is it enough to consider the structure of differential graded module over $\Omega^\ast(X)$ on it?
This brings us to a more general question: is there a good definition of (algebraic) K-theory of differential graded modules/algebras over a differential graded algebra, which would reproduce the usual K-theory of a smooth manifold, if applied to its algebra of differential forms? Of course, we assume, that the class of the module $\Omega^\ast(E)$ should correspond to the class of $E$ in $K^0(X)$. The theory we ask for should be invariant under the homotopy equivalencies of algebras. I believe, the answer to this question should exist somewhere, so, if you know a reference, please, do send it to me.
