Let A be an analytic geometry. Denote by Spec(X) the set of primes of an object X. Suppose f: Y --> X is a map. Since the strong image of an integral object is integral, f induces a mapping Spec(f): Spec(Y) --> Spec(X), sending each prime of Y to its strong image in X. If V is any strong subobject of X we denote by C(V) the set of all the primes of X contained in V. Proposition 3.6.1. If P is a prime of X, which is contained in the join W of two strong subobjects U and V of X, then P is contained in U or V. Proof. By (3.5.4) P is the join of U P and V P. Since P is integral, we have either U P = P or V P = P by (3.2.7.b), so P is in U or V. –– Proposition 3.6.2. (a) If U and
V are two strong subobjects of an object X, then C(U
V) = C(U)
C(V).
Proof. (a) If P in C(U
V), then P is contained in U or V by (3.6.1),
so P is in C(U) or C(V). The other direction
is obvious.
It follows from (3.6.2) that the subsets of the form C(U) are the closed subsets of a topology on Spec(X). We shall regard Spec(X) as a topological space with this topology, called the spectrum of X. Proposition 3.6.3. Suppose f: X -->. Y is a map. Then Spec(f): Spec(X) --> Spec(Y) is a continuous mapping of topological spaces. Proof. If V is any strong subobject of Y, then C(f^{-1}(V)) = Spec(f)^{-1}(C(V)), so Spec(f) is continuous. Proposition 3.6.4. (a) A reduced strong
subobject is integral if and only if it is a prime element of the locale
S(X)^{op} (see (3.6.7)).
Proof. (a) follows easily from (3.6.1.c).
Definition 3.6.5. An analytic category is called spatial if any non-initial object has a prime (or equivalently, the class of integral objects is uni-dense). Proposition 3.6.6. An analytic geometry is spatial iff S(X)^{op} is a spatial locale for each object X. Proof. The condition is necessary by (3.6.4.a), and is sufficient by (2.6.5.a). –– Proposition 3.6.7. Suppose X
is a non-initial object in a spatial A.
Proof. (a) Consider the join W of all the primes of X.
Then W is reduced by (3.1.3.d).
We prove that W is unipotent. Consider a non-initial maps t:
T --> X. We have to show that t is not disjoint with W.
Since A is spatial we may assume T is
integral. Then the prime t^{+1}(T) is in W. Thus
t factors through W. This shows that W is the reduced
model of X.
Proposition 3.6.8. Suppose X
is a non-initial reduced object in a spatial A.
Suppose V is a strong subobject of X. Then
Proof. (a) If C(V) = Spec(X) then
V contains all the primes of X, thus contains the join of
these primes W, which is the reduced model of X by (3.6.7.a).
But X is reduced means that W = X. Thus V = X.
Proposition 3.6.9. Suppose f:
Y ® X is a mono in a spatial
analytic geometry.
Proof. (a) Suppose f is coflat. Then f^{-1}f^{+1}
is the identity R(Y) --> R(Y) by (1.5.4).
If Q is a prime of Y, then f^{-1}f^{+1}(Q)
= Q implies that Spec(f) is injective. If Q is
a prime and V is a strong subobject of Y such that f^{+1}(Q)
f^{+1}(V), then Q = f^{-1}f^{+1}(Q)
f^{-1}((f^{+1}(V)) = V, thus the
closed subset C(V) of Spec(Y) is induced from
the closed subset C(f^{+1}(V)) of Spec(X),
so Spec(f) is a topological embedding.
Definition 3.6.10. An open subset U of Spec(X) is called an affine open subset if U = Spec(f)(Y) for an analytic mono f: Y --> X. Proposition 3.6.11. The class of affine open subsets of an object X is a base for the topology on Spec(X), which is closed under intersection and pullbacks. Proof. This follows from the definition of the topology on Spec(X) and (3.6.10). ––– |