We will work over the ring Wn = 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 Wn 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 Wn, denoted Cris(X/Wn), has as its objects pairs
