3.6. Spectrums

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 P and P. Since P is integral, we have either P = P or 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( V) = C(U)  C(V). 
(b) If U is the intersection of a set {Ui} of strong subobjects of X, then C(U)i C(Ui). 
(c) C(X) and C(0) = Spec(X). 

Proof. (a) If P in C( 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. 
(b) and (c) are obvious. n 

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)). 
(b) Spec(X) is homeomorphic to the space of points of S(X)op
(c) Spec(X) is a sober space for any object of X

Proof. (a) follows easily from (3.6.1.c). 
(b) follows immediately from (a); (c) from (b) as the space of points of a locale is sober.  

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. 
(a) The join of all the primes of X is the radical of X
(d) If V is a strong subobject of X then the join of all the primes in C(X) is the radical of V

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
(b) is a variant of (a).  

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 
(a) C(V) = Spec(X) iff V = X
(b) C(V) iff V = 0
(c) X is integral iff its spectrum is irreducible. 

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
(b) Since A is spatial any non-initial strong subobject V has a prime which is also a prime of X
(c) Suppose U and V are two strong subobjects of X. By (a) and (3.6.2.a) we have C( V) = C(U)  C(V) = Spec(X) iff V = X. The assertion follows from this and (b).  

Proposition 3.6.9. Suppose f: Y ® X is a mono in a spatial analytic geometry. 
(a) If f is coflat then Spec(f) is a topological embedding. 
(b) If f is analytic then Spec(f) is an open embedding. 
(c) If f is strong then Spec(f) is a closed embedding. 

Proof. (a) Suppose f is coflat. Then f-1f+1 is the identity R(Y) --> R(Y) by (1.5.4). If Q is a prime of Y, then f-1f+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-1f+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. 
(b) Suppose u: U = Vc --> X is a coflat complement of a strong mono v: V --> X. Clearly the image of Spec(u) is disjoint with C(V). If W is a prime not contained in C(V), then V is a proper strong subobject of the reduced object W. Since Vc is the complement of V in W, it is non-initial, and is integral by (3.2.2.c). Its strong image under Spec(u) is W as V --> W is epic. This shows that the image of Spec(u) is the complement of the closed subset C(V), thus is open. 
(c) The proof is similar to (a) as for a strong mono f, the composite f-1f+1 is the identity R(Y) --> R(Y).

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).  

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