6 1. PRELIMINARIES FROM ALGEBRAIC GEOMETR Y
the affine space over R and by
(1.9) G
0
: = ( A \ + ) , G
m
:=(5peci?[T,T-
1
],-)
the additive and multiplicative group schemes over R respectively. An elliptic curve
over R is a smooth proper group scheme over R of relative dimension one with
irreducible geometric fibers. (See the discussion below for a review of smoothness.)
If X is a Noetherian scheme and P G X is a closed point we denote by
(1.10) X, Xfor
the p—adic completion of X and the completion of X at P respectively.
Recall Grothendieck's existence theorem saying in particular that if X and Y
are projective schemes over a p—adically complete ring S then the map
Homs(X,Y) - Homs{X,Y)
is a bijection.
Let X be a scheme (resp. a formal scheme) and let G X be a group scheme
over X (resp. a group formal scheme over X). A principal homogeneous space
under G is a morphism Y X of schemes (resp. a formal schemes) together with
an action m : G xx Y Y such that the induced morphism
m x pr2 : G xxY -+Y xxY
is an isomorphism. A principal homogeneous space is said to be trivial if it has a
section; equivalently if it is isomorphic to the principal homogeneous space G X.
By a Galois cover of schemes we understand a finite morphism of schemes
n : Y - X such that
{ir*Oy)T
= Ox, where V := Aut(Y/X) is the group of
X—automorphisms of Y; T is referred to as the Galois group of the cover.
1.1.4. Derivations and smoothness. If B is an A—algebra and C is a
B—algebra with structure homomorphism / : B C we denote by DerA(B,C)
the B—module of A—derivations D : B C relative to / , by which we mean
A—module homomorphisms satisfying
D(xy) = f{x) D(y) + f(y) D(x),
for x,y G B. If A is a sheaf of rings on a topological space, B is a sheaf of
A—algebras, and C is a sheaf of B—algebras we denote by Der^(B,C) the group of
.4—derivations B C and by Ver^(B, C) the sheaf of C—modules of A—derivations
B^C.
We recall here the basic definitions and properties of smoothness we are using
in this book; cf. [10], Chapter 2. Let A be a ring and B a finitely generated
A—algebra. One says B is smooth (resp. etale) over A if for any A—algebra C and
any ideal I C C with
I2
= 0 the canonical map
HomA(B,C) - HomA(B,C/I)
is surjective (resp. bijective). A scheme X over A is called smooth (resp. etale) if
it can be covered with open affine schemes Spec B with B finitely generated and
smooth (resp. etale) over A. A morphism / : X Y of schemes is called smooth
(resp. etale) if Y can be covered by affine open schemes U Spec A such that
f~x(U)
are smooth (resp. etale) over A. A morphism X Y of finite type of
Noetherian schemes is smooth if and only if it is flat and all geometric fibers are
regular. If Spec B Spec A is of finite type and smooth (resp. etale) then B is
smooth (resp. etale) over A. If a scheme X over A is smooth then each point in X
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