NOTATION AND TERMINOLOGY 11

Verschiebung homomorphism for S-flat N of finite presentation denoted

VerN/S : N

(p)

→ N see [30, VIIA, 4.2–4.3]). If S is understood from

context then we may denote these as FrN and VerN respectively.

For n 1, the

pn-fold

relative Frobenius and Verschiebung homomor-

phisms N → N

(pn)

and N

(pn)

→ N are respectively denoted FrN/S,pn

and VerN/S,pn .

– For a perfect field k with char(k) = p 0 and the unique lift σ : W (k) →

W (k) of the Frobenius automorphism y → yp of k, a Dieudonn´ e module

over k is a W (k)-module M equipped with additive endomorphisms F :

M → M and V : M → M such that F ◦ V = [p]M = V ◦ F, F(c m) =

σ(c) F(m), and c V(m) = V(σ(c) m) for all c ∈ W (k) and m ∈ M; these

are the left modules over the Dieudonn´ e ring Dk (see 1.4.3.1).

– The semilinear operators F and V on a Dieudonn´ e module M corre-

spond to respective W (k)-linear maps M (p) → M and M → M (p), where

M

(p)

:= W (k) ⊗σ,W

(k)

M.