On the crystalline cohomology of Deligne–Lusztig varieties

Abstract

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 cohomology 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.