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Cartier operator

Notation

For the following discussion we suppose $\pi :X\to S$ is a smooth map of schemes. Let $d:{𝒪}_X\to {\Omega }_{X/S}^1$ be the standard differential. It is an integrable connection on ${𝒪}_X$. We’ll define the (relative) de Rham cohomology to be the hyper-derived functor pushforward applied to the de Rham complex $R^q{\pi }_{*}({\Omega }_{X/S}^{•})$.

We will denote the relative Frobenius map $F:X\to X^{(p)}$ where $X^{(p)}$ is the pullback of the structure map and the absolute Frobenius $F_{\mathrm{ab}}:S\to S$, i.e. $X^{(p)}=X{\otimes }_{{\pi }^{-1}({𝒪}_S)}{\pi }^{-1}({𝒪}_S)$. If ${𝒜}^{•}$ is a complex of sheaves, then we denote by ${\mathscr{H}}^q({𝒜}^{•})$ the sheaf that is obtained by taking cohomology with respect to the maps of the complex.

The Cartier Isomorphism

Suppose that $S$ is an ${𝔽}_p$-scheme and $X/S$ is smooth, then there is a unique map of graded ${𝒪}_{X^{(p)}}$-algebras $C^{-1}:{\Omega }_{X^{(p)}/S}^i\stackrel{\sim }{\to }{\mathscr{H}}^i(F_{*}{\Omega }_{X/S}^{•})$ that satisfies:

$C^{-1}(1)=1$,

$C^{-1}(\omega \wedge \tau )=C^{-1}(\omega )\wedge C^{-1}(\tau )$

and $C^{-1}({\mathrm{dF}}_{\mathrm{ab}}^{-1}(f))=[f^{p-1}\mathrm{df}]$ in ${\mathscr{H}}^1(F_{*}{\Omega }_{X/S}^{•})$.

The inverse of this isomorphism is the traditional Cartier isomorphism.

The construction is quite simple. First note that we can immediately reduce to constructing $C^{-1}$ for $i=1$. This is because if $C^{-1}(1)=1$ and $C^{-1}$ is ${𝒪}_{X^{(p)}}$-linear it is determined for $i=0$. Likewise, if $C^{-1}$ is determined for $i=1$, then the case $i\ge 1$ is determined from the second property.

Now to construct for $i=1$ we just note that such a map is equivalent to a $({\pi }^{(p)}{)}^{-1}$-linear derivation ${𝒪}_{X^{(p)}}\to {\mathscr{H}}^1(F_{*}{\Omega }_{X/S}^{•})$. This is equivalent to defining a map on local sections $\delta :{𝒪}_X\times {\pi }^{-1}({𝒪}_S)\to {\mathscr{H}}^1({\Omega }_{X/S}^{•})$ that is biadditive and satisfies the extra three properties

$\delta (\mathrm{fs},s\prime )=\delta (f,s^{\mathrm{ps}}\prime )$,

$\delta (\mathrm{gf},s)=g^p\delta (f,s)+f^p\delta (g,s)$ and

$\delta (f,1)=[f^{p-1}\mathrm{df}]$.

Now define the map explicitly by $\delta (f,s)=[{\mathrm{sf}}^{p-1}\mathrm{df}]$. It can be checked that this map satisfies all the properties listed and is indeed an isomorphism. This is $C^{-1}$, the inverse of the Cartier isomorphism.

Relation to the Hodge-de Rham Spectral Sequence

For this discussion let’s assume that $X/k$ is proper and smooth. Deligne and Illusie had an insight that the degeneration of the Hodge-de Rham spectral sequence (HdR SS) is closely related to the Cartier isomorphism. Recall that the HdR SS is formed by taking the spectral sequence associated to hypercohomology $E_1^{p,q}=H^q(X,{\Omega }_{X/k}^p)\Rightarrow H_{\mathrm{dR}}^{p+q}(X/k)$.

Now notice that if we form the complex ${⨁}_i{\Omega }_{X^{(p)}}^i[-i]$ which is ${\Omega }^i$ in degree $i$ and $d=0$ everywhere, then the left side of the inverse Cartier isomorphism is exactly ${\mathscr{H}}^i({⨁}_i{\Omega }_{X^{(p)}}^i[-i])$. Likewise, the right side is ${\mathscr{H}}^i$ of the complex $F_{*}{\Omega }_{X/k}^{•}$. We can think of both of these complexes as living in $D(X^{(p)}):=D_{\mathrm{qCoh}}^b(X^{(p)})$.

We can ask whether or not there is some map in the derived category $\varphi :{⨁}_i{\Omega }_{X^{(p)}}^i[-i]\to F_{*}{\Omega }_{X/k}^{•}$ with the property that ${\mathscr{H}}^i(\varphi )=C^{-1}$ for all $i$. It turns out this is a sufficient condition for convergence of the HdR SS. This is just because we get a string of isomorphisms

$H^n(X,{\Omega }_X^{•})=H^n(X^{(p)},F_{*}{\Omega }_X^{•})\simeq {⨁}_i{H}^{n-i}(X^{(p)},{\Omega }_X^i)$. Thus the dimensions of the $k$-vector spaces at the $E_1$ term match the dimensions at the $E_{\mathrm{\infty }}$ term. Since everything is a $k$-vector space this is all that is needed for degeneration (there can be no non-trivial quotients without dimension decreasing).

The Generalization to Non-commutative algebra

See Kaledin Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie on the arxiv

References

Nilpotent Connections and the Monodromy Theorem by Nicholas M. Katz

Relevements modulo $p^2$ et decomposition du complexe de de Rham by Deligne and Illusie