I was thinking about de Rham cohomology in characteristic $p$, and in particular the [recent question about Poincare residues](https://mathoverflow.net/questions/349301), 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?



A variant on characteristic $p$ de Rham cohomology
To be able to compute the iterates of the Cartier operator it is convenient to understand how $C$ interacts with the de Rham differential:


Cartier isomorphism induces an isomorphism of complexes $$(\Omega^{\*}\_A,d\_{dR})\simeq (H^{\*}(\Omega^{\bullet}\_A),\beta)$$ where $\beta$ is the Bockstein differential provided by the distinguished triangle $$\Omega^{\bullet}\_A\to R\Gamma\_{cris}(A/W\_2(k))\to \Omega^{\bullet}\_A$$ It shows that for a closed form $\alpha$ the image $C(\alpha)$ is closed iff the class $[\alpha]$ is annihilated by the Bockstein homomorphism which in turn is equivalent to the liftability of $\alpha$ to class in $H^i\_{cris}(A/W\_2(k))$. Passing to cohomology in the above isomorphism, composing it with the Cartier isomorphism and iterating this procedure $(i-1)$ times we get an isomorphism $$(\Omega^{\*}\_A,d\_{dR})\simeq (E\_i^{(1-i)\*,\*},\beta\_i)$$ of the de Rham complex with the complex appearing on the $i$-th page of the Bockstein spectral sequence associated to the crystalline cohomology complex. 


These facts can be seen easily from the following description of the Cartier isomorphism: choose a lift $\tilde{A}$ of $A$ to a complete formally smooth algebra over $W(k)$ equipped with a lift $\widetilde{Fr}$ of the Frobenius endomorphism of $A$(the existence of such lift follows from the vanishing of the relevant obstruction groups which is implied by smoothness of $A$ over $k$). The Cartier operator applied to a form $\omega\in \Omega^i\_A$ is then given by $C(\omega)=\overline{\frac{\widetilde{Fr}^\*(\tilde{\omega})}{p^i}}$ where $\tilde{\omega}$ is any lift of $\omega$ to a form on $\tilde{A}$ and $\overline{\cdot}$ denotes the reduction. 


By tracing through the construction of the Bockstein differentials we get the following



> 
> **Lemma**. For a closed form $\alpha$ the $i$-th iteration of the Cartier operator is defined and gives a closed form if and only if
>  $[\alpha]\in H^j(\Omega\_A^{\bullet})$ lifts to a class
>  $\widetilde{[\alpha]}$ in the crystalline cohomology of $A$ over
>  $W\_{i+1}(k)$. The $(i+1)$-th iteration of the Cartier operator is zero
>  if and only if the class $p^i\widetilde{[\alpha]}\in
>  H^j\_{cris}(A/W\_{i+1}(k))$ vanishes.
> 
> 
> 


Combining these conditions for all $i$ we get that a form is forever closed iff its class is in the image of the map $H^j\_{cris}(A/W(k))\to H^i\_{dR}(A/k)$ and it is eventually exact iff the class is in the image of $H^j\_{cris}(A/W(k))[p^{\infty}]\to H^i\_{dR}(A/k)$.


Crystalline cohomology of $A$ coincides with the cohomology of the $p$-adically completed de Rham complex of any lift of $A$ to $W(k)$ and for the purposes of computing the above invariants we can replace $H^j\_{cris}(A/W(k))$ by the (non-complete) de Rham cohomology $H^j\_{dR}(\widetilde{A}/W(k))$ where $\widetilde{A}$ is any lift. The quotient $H^j\_{dR}(\widetilde{A})/H^j\_{dR}(\widetilde{A}){[p^{\infty}]}$ is a $W(k)$-lattice in the finite-dimensional vector space $H^j\_{dR}(\widetilde{A}[1/p]/W(k)[1/p])$(it is finite-dimensional e.g. by comparison with singular cohomology). 


It indeed seems to follow that $EH^j$ is a finite-dimensional vector space over $k$ with dimension at most the $j$-th rational Betti number of any lift of $A$.


