On the crystalline cohornology of Deligne-Lusztig varieties

Let X -> Y-0 be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic p > 0. Let Y be a smooth compactification of Y-0 such that Y - Y-0 is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y whose rational crystalline cohomology is the rigid cobomology of X, in particular provides a natural W[F]-lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, Y-0 and Y then our construction is G-equivariant. As an example we apply it to Deligne-Lusztig varieties. For a finite field k, if G is a connected reductive algebraic group defined over k and L a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant W[F]-lattice in the cohomology (L-adic or rigid) of the corresponding Deligne-Lusztig variety and an expression of its reduction modulo p in terms of equivariant Hodge cohomology groups. (c) 2006 Elsevier Inc. All rights reserved.