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.
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