Any local p—ring is, of course, a global p—ring and if A is a global p—ring then
its localization A^ is a local p—ring. (A warning about terminology: what we
call here a local p—ring is called, in Matsurnura's book [103], p. 223, a p—ring.
On the other hand Serre's concept of p—ring considered in his book [120], p. 37,
does not coincide with Matsurnura's! Due to these terminology discrepancies in
the literature we preferred to use a different terminology.) The following basic fact
about local p—rings will be repeatedly used in our book. Let A and B be local
complete p—rings with residue fields kA and &;#. Let Horn denote homomorphisms
in the category of rings. Then the natural map
Hom(A, B) Hom(kAj kB)
is surjective; cf. [103], p. 224. If, in addition, kA and ks are perfect then the
above map is an isomorphism; cf. [120], p. 38. In particular, if A is a complete
local p—ring with residue field kA then the p—power Frobenius endomorphism of
kA lifts to an endomorphism (p of A; if, in addition, kA is perfect then 0 is unique.
(Uniqueness may fail if kA is not perfect: e.g. one can take x a variable, A :=
g G Z[x], and / = (fig : A A, (j)g{x)
Recall also that for any
perfect field k there exists a complete local p—ring with residue field k and this ring
is unique up to a unique isomorphism. In particular
is the unique complete
local p—ring with residue field F£.
1.1.3. Schemes and formal schemes. We are using the standard terminol-
ogy and conventions of algebraic geometry as found, for instance, in [66]. Fur-
thermore, schemes and formal schemes will always be assumed separated. Formal
schemes will always be assumed locally Noetherian. A group scheme over a scheme
X is a group object in the category of schemes over X. It is convenient to introduce
the following:
1.3. A group formal scheme over a formal scheme X is a group
object in the category of formal schemes over X.
A variety over an algebraically closed field k will mean an irreducible reduced
scheme of finite type over /c; we denote, as usual, by k(X) the field of rational
functions on X.
If X is a scheme over a ring R and S is an R—algebra we denote, as usual, by
X(S) :=HomR(SpecS,X)
the set of S—points of X; we use a similar notation for formal schemes. For a
variety X over k we sometimes write X for X(k). If X is a scheme and M is a
locally free sheaf on it we denote by
(1.7) V(M) = Spec Symm{M)
the associated group scheme over X.
By a formal group or formal group law (of dimension n) over a ring R we
will understand, as usual, an n—tuple tuple of formal power series in R[[Ti, ...,Tn]]
satisfying the usual axioms. (So formal groups over R are not the same as group
formal schemes over Spf R ! Formal groups are "local objects" while group formal
schemes are "global objects".)
If R is a ring we denote by
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