Crystalline Cohomology - Crystalline Cohomology

Crystalline Cohomology

In characteristic p the most obvious analogue of the crystalline site defined above in characteristic 0 does not work. The reason is roughly that in order to prove exactness of the de Rham complex, one needs some sort of Poincaré lemma, whose proof in turn uses integration, and integration requires various divided powers, which exist in characteristic 0 but not always in characteristic p. Grothendieck solved this problem by defining objects of the crystalline site of X to be roughly infinitesimal thickenings of Zariski open subsets of X, together with a divided power structure giving the needed divided powers.

We will work over the ring W_n = W/pnW of Witt vectors of length n over a perfect field k of characteristic p>0. For example, k could be the finite field of order p, and W_n is then the ring Z/pnZ. (More generally one can work over a base scheme S which has a fixed sheaf of ideals I with a divided power structure.) If X is a scheme over k, then the crystalline site of X relative to W_n, denoted Cris(X/W_n), has as its objects pairs U→T consisting of a closed immersion of a Zariski open subset U of X into some W_n-scheme T defined by a sheaf of ideals J, together with a divided power structure on J compatible with the one on W_n.

Crystalline cohomology of a scheme X over k is defined to be the inverse limit

where

is the cohomology of the crystalline site of X/W_n with values in the sheaf of rings O = O_X/Wn.

A key point of the theory is that the crystalline cohomology of a smooth scheme X over k can often be calculated in terms of the algebraic de Rham cohomology of a proper and smooth lifting of X to a scheme Z over W. There is a canonical isomorphism

of the crystalline cohomology of X with the de Rham cohomology of Z over the formal scheme of W (an inverse limit of the hypercohomology of the complexes of differential forms). Conversely the de Rham cohomology of X can be recovered as the reduction mod p of its crystalline cohomology (after taking higher Tors into account).