Page 410 -

§1. Algebraic and Analytic Geometry 385

given by composition of maps and suitable sheaf morphisms. We denote the cate-
gory of (commutative) rings by (rg) with morphisms of rings always preserving the
unit. The cross section functor is a contravariant functor with the following universal
property.
(1.4) Assertion. For each ring R in (rg), there is a local ringed space, denoted
Spec(R), such that there is a natural isomorphism between the morphism sets
Hom (rg) (T, (X, O X )) → Hom (l/rg/sp) (X, Spec(R)).
Moreover, the local ringed space Spec(R) has the set of prime ideals in R as the
underlying set.
Under the above bijection, a ring morphism φ : R → (X, O X ) corresponds to
the morphism f : X → Spec(R) given by the following construction. We compose
the ring morphism φ with its evaluation on the fibre O x to define a ring morphism

φ : R → O x and compose further with the reduction morphism π x to define a
x
second ring morphism φ x : R → K(x). Since φ x : R → K(x) is a ring morphism
into a field, the kernel p(x) is a prime ideal, and the corresponding function f :
X → Spec(R) is defined by the formula f (x) = p(x). The various morphisms
under consideration are displayed as follows
φ
p(x) = ker(φ x ) ⊂ R −−−−→ (X, O X )
 
φ x = π x φ and  
x
π x
R p(x) −−−−→ O x −−−−→ K(x).
φ x ∗
The closed sets Z(E) of Spec(R) are given by subsets E ⊂ R where Z(E) consists
of all p ∈ Spec(R) with E ⊂ p. Thus the open sets of Y = Spec(R) have a basis
consisting of set Y f for f ∈ R where Y f equals the set of all prime ideals p ∈
Spec(R) = Y with f /∈ p. The fibre of the structure sheaf O Spec(R) at p ∈ Spec(R) is
the local ring which is the localization R p at the prime ideal p. Finally, the morphism
∗
f ∗ : O Y → O X on the germs at x ∈ X is the morphism φ : R p(x) → O x in the
x
above diagram.
Since with sheaf theory we can localize on open sets and glue local ringed space
over open sets, we can describe the basic structures in geometry as local ringed
spaces which are locally of certain type.
(1.5) Definition. An affine scheme X is a local ringed space isomorphic to some
Spec(R). A scheme X is a local ringed space locally isomorphic at each point to
an affine scheme, that is, for x ∈ X there exists an open set U with x ∈ U and
(U, O X |U) is an affine scheme. The schemes define a full subcategory (sch) of
(l/rg/sp) called the category of schemes.
Smooth manifolds and complex analytic manifolds are local ringed spaces lo-
n
cally isomorphic to the local ringed spaces of germs of smooth functions on R and
n
analytic functions on open subset of C respectively.