I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.

Let $k$ be a perfect field of characteristic $p$ and let $A$ be a regular $k$-algebra. Let $\Omega^j$ be the Kahler $j$-forms, let $Z^j$ be the closed $j$-forms, $B^j$ the exact $j$-forms and $H^j = Z^j/B^j$. The inverse Cartier operator is the unique isomorphism $C^{-1} : \Omega^j \to H^j$ satisfying $$C^{-1}(\alpha \wedge \beta) = C^{-1}(\alpha) \wedge C^{-1}(\beta) \quad C^{-1}(f) = f^p \quad C^{-1}(df) = f^{p-1} df$$ for $f \in A$. (It is easy to see that there is at most one such map, a nice exercise to see that it is well defined, and not at all clear that it is an isomorphism.)

The inverse operator is an isomorphism $Z^j/B^j \to \Omega^j$, which we can also consider as a surjection $Z^j \to \Omega^j$. By abuse of notation, I'll write $C$ for the surjection $Z^j \to \Omega^j$ as well. We thus have two maps $Z^j \to \Omega^j$: The surjection $C$, and the obvious inclusion.

Define a differential form $\alpha \in \Omega^j$ to be *forever closed* if, for all $i$, we have $C^i(\alpha) \in Z^j$. Note that we must have $C^{i-1}(\alpha) \in Z^j$ for it to make sense to define $C^i(\alpha)$, so this condition spells out as "we impose that $\alpha$ is closed, and therefore $C(\alpha)$ is defined, and we impose that $C(\alpha)$ is closed, and therefore $C^2(\alpha)$ is defined, etcetera."

Define a forever closed form $\alpha$ to be "eventually exact" if $C^k(\alpha)$ is $0$ for $k$ sufficiently large. Note that exact forms are eventually exact, since the exact forms are the kernel of $C$. Define the eventual cohomology, $EH^j$, to be the forever closed forms modulo the eventually exact forms.

It looks like $EH^{\bullet}$ is always finite dimensional, and forms a graded ring. It does not appear that the dimension of $EH^j$ gives topological betti numbers -- it appears to give something like the multiplicity of the highest weight part of the cohomology.

Is this some object people have studied before?