Cohomological Theory of Crystals over Function Fields

Tannakian properties of unit Frobenius-modules

“…Step (1) shows that the base change of Z to Spec Λ × Fq Spec R along 1 Λ × Fq f has the form Z f × Fq Spec R for a certain Z f ⊂ Spec Λ. Therefore Z x = Z f = Z x ′ .…”

“…For infinite Λ the connection of U (X, Λ) with theétale fundamental group is lost. Nevertheless in the spirit of Böckle and Pink [1] we prove the following: Theorem 3.6. Suppose that Λ is a field, X is connected, and locally noetherian.…”

“…Define a functor ε : µ(X, Λ) → Sh(Xé t , Λ) by ε(M, ϕ)(u : U → X) = Hom µ(U,Λ) (O Λ U , 1), (u * M, u * ϕ) . Since ε transforms nil-isomorphisms to isomorphisms ( [1] propostion 10.1.7 (b)) one gets a functor ε : Crys(X, Λ) → Sh(Xé t , Λ).…”

“…The interest in infinite coefficient algebras Λ comes from a construction of Drinfeld (see e.g. section 3.5 of [1]) which produces an O Λ F,X -module, and a fortiori a Λ-crystal out of a sufficiently good Drinfeld module ϕ : Λ → End Fq (L), L a line bundle on X. So one has a natural source of Λ-crystals which have arithmetic significance.…”

“…For an F q -scheme X, and a commutative F q -algebra Λ, Böckle and Pink [1] introduced the category of Λ-crystals on X which is in many ways analogous to the category ofétale constructible sheaves of Λ-modules. Let us recall the definitions.…”