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386 20. Families of Elliptic Curves

(1.6) Remark. If A is an algebra over a ring R, then the morphism R → A defines
a morphism Spec(A) → Spec(R). By definition, a scheme X over a ring R is a
scheme together with a morphism X → Spec(R).Ascheme X over R is sometimes
denoted X/R, and in this case, every ring (U, O X ) will have a given structure as
an R-algebra. For schemes over R, the sheaf O X and the rings (U, O X ) of sections
are R-algebras.
(1.7) Remark. The natural extension of the ideas in the previous remark is to con-
sider the category (Sch/S) of schemes X with a morphism X → S to a fixed scheme
S. This is a general categorical construction also denoted (Sch) /S where the mor-
phisms X → Y are morphisms in (Sch) giving a commutative triangle with the two
structure morphisms over S. For an object X → S the ring (X, O X ) is an algebra
over (S, O S ). This category has the fundamental property that the product, called
the fibre product and denoted X × S Y exists for two objects X → S and Y → S in
(Sch/S). For affine schemes, we have Spec(A) × Spec(R) Spec(B) = Spec(A ⊗ R B),
and this leads to the general existence theorem for products in (Sch/S) which are
fibre products of schemes.
For two fields F and K, the scheme Spec(F) has only one point, and a morphism
Spec(K) → Spec(F) is equivalent to a morphism of fields F → K;itisalways
injective. The morphisms Spec(F) → X are given by point x ∈ X together with a
morphism O x → F which factors as K(x) → F. The geometric points of a scheme
are by definition the points x ∈ X where there is an algebraically closed field F and
a morphism Spec(F) → X with image x.
(1.8) Remark. In remark (1.7), we pointed out that a morphism p : X → B of
schemes as giving a (V, O B )-algebra structure on the rings (p −1 (V ), O X ) in a
functorial manner. But there is another basic interpretation of p : X → B as a family
of schemes X b ={b}× B X given by the fibres of the morphism p : X → B. This is
the picture of p : X → B as the family of the schemes X b for b ∈ B.

One way to study special types of varieties or schemes, like curves, surfaces, 3-
folds, or abelian varieties, is to analyze how they form families. Before doing this,
we consider morphisms given by line bundles and divisors in the next section.
We conclude with a few further remarks about schemes. One very clear advantage
of working in the concept of scheme theory is that questions of families are well
defined. For example, the moduli space of elliptic curves should be a family p : X →
B where each elliptic curve under consideration should be isomorphic to exactly one
fibre X b in the family. The isomorphism classes of elliptic curves are no longer a
discrete set, but they have the structure of a scheme on a set of representatives of
the isomorphism classes. In previous chapters various families of elliptic curves had
this structure of a scheme given in terms of coefficients of the equations defining the
curve.

(1.9) Remark. Most of the ideas and results in commutative algebra have an ana-
logue in scheme theory through local calculations in affine open neighborhoods. For