Purity Reconsidered the Cohomology of Moduli Spaces

Convention: all base schemes will be in characteristic p. Writing k or we will assume k = k F p , respectively = F p. We will write X ! S for a p-divisible group over a base S (and sometimes will say \a family of p-divisible groups"), while we tend to write X when we consider a p-divisible group over a base eld (but we are not strict in this convention). Instead of \p-divisible group" one can also use the equivalent notion \Barsotti-Tate group".