1.1. ALGEBRO-GEOMETRIC TERMINOLOGY

5

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 :=

(Z[x](p))/V,

g G Z[x], and / = (fig : A — A, (j)g{x) —

xp+pg.)

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

Z^r

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:

DEFINITION

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

(1.8)

AN

-SpecRlT^.^Tn]