NOTATION AND TERMINOLOGY 11
Verschiebung homomorphism for S-flat N of finite presentation denoted
VerN/S : N
→ 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
relative Frobenius and Verschiebung homomor-
phisms N → N
→ 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 188.8.131.52).
– 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
:= W (k) ⊗σ,W