For the following discussion we suppose is a smooth map of schemes. Let be the standard differential. It is an integrable connection on . We’ll define the (relative) de Rham cohomology to be the hyper-derived functor pushforward applied to the de Rham complex .
We will denote the relative Frobenius map where is the pullback of the structure map and the absolute Frobenius , i.e. . If is a complex of sheaves, then we denote by the sheaf that is obtained by taking cohomology with respect to the maps of the complex.
Suppose that is an -scheme and is smooth, then there is a unique map of graded -algebras that satisfies:
and in .
The inverse of this isomorphism is the traditional Cartier isomorphism.
The construction is quite simple. First note that we can immediately reduce to constructing for . This is because if and is -linear it is determined for . Likewise, if is determined for , then the case is determined from the second property.
Now to construct for we just note that such a map is equivalent to a -linear derivation . This is equivalent to defining a map on local sections that is biadditive and satisfies the extra three properties
Now define the map explicitly by . It can be checked that this map satisfies all the properties listed and is indeed an isomorphism. This is , the inverse of the Cartier isomorphism.
For this discussion let’s assume that 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 .
Now notice that if we form the complex which is in degree and everywhere, then the left side of the inverse Cartier isomorphism is exactly . Likewise, the right side is of the complex . We can think of both of these complexes as living in .
We can ask whether or not there is some map in the derived category with the property that for all . It turns out this is a sufficient condition for convergence of the HdR SS. This is just because we get a string of isomorphisms
. Thus the dimensions of the -vector spaces at the term match the dimensions at the term. Since everything is a -vector space this is all that is needed for degeneration (there can be no non-trivial quotients without dimension decreasing).
See Kaledin Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie on the arxiv
Nilpotent Connections and the Monodromy Theorem by Nicholas M. Katz
Relevements modulo et decomposition du complexe de de Rham by Deligne and Illusie