In this section we consider the category Ring^{op}. We shall define a metric topology t on Ring^{op} so that (Ring^{op}, t) is a strict metric site, called the Zariski site. In the following we shall write A^{o} for a ring A if it is regarded as an object of Ring^{op}, called an affine ring. If f: A ® B is a homomorphism of rings we write f^{o}: B^{o} ® A^{o} for the morphism in Ring^{op}. Suppose A^{o} is an affine ring. We write A for the set Spec A of prime ideals of A. If j^{o}: A^{o} ® B^{o} is a morphism of affine rings, then j induces a map j: A ® B sending each prime ideal p Î A to the prime ideal j^{1}(p) Î B. If T is any subset of A we define the subset Z(T) to be the set of all prime ideals of A which contain T. Proposition 2.4.1. (a)
If T_{1} and T_{2} are subsets of A, then Z(T_{1})
È Z(T_{2})
= Z(T_{1}T_{2}).
Proof. If p Î Z(T_{1}) È Z(T_{2}), then either p Î Z(T_{1}) or p Î Z(T_{2}), thus for any f Î T_{1} and g Î T_{2}, either f Î p or g Î p; hence fg Î p. This shows that p Î Z(T_{1}T_{2}). Conversely, if p Î Z(T_{1}T_{2}), and p  Z(T_{1}) say, then there is an f Î T_{1} such that f Ï p. Now for any g Î T_{2}, fg Î p implies that g Î p, so that p Î Z(T_{2}). This proves (a). The assertions (b) and (c) are obvious. Definition 2.4.2. We introduce the Zariski topology on A: a subset Y of A is closed if there exists a subset T of A such that Y = Z(T). (This is a topology by (2.4.1).) Proposition 2.4.3. If j^{o}: A^{o} ® B^{o} is a morphism of rings, then the map j: A ® B is continuous with respect to the Zariski topologies. Proof. If V = Z(T) is a closed subset of B with T Í B, then j^{1}(V) = Z(j(T)), so it is a closed subset of A. Thus we obtain a metric functor t: Ring^{op}® Top sending each affine ring A^{o} to the space A^{o} = Spec A. We now consider the presite (Ring^{op}, t). For any f Î A let D(f) = A  Z(f); D(f) is an open subset of A, called an elementary open subset of A. Proposition 2.4.4. Elementary open sets form an open basis for the Zariski topology on A. Proof. If U is an open subset of A with U = A  Z(T) for some T Í A, then U = È_{fÎT} (A  Z(f)) = È_{fÎT} D(f). Remark 2.4.5.
For any subset Y of A let I(Y) = {f
Î A f
Î p for all
p Î Y}.
Then I(Y) is a radical ideal of A (i.e., an ideal
which is equal to its own radical).
Lemma 2.4.6.
Suppose T is a subset of a ring A; then the following conditions are
equivalent:
Proof. A = È_{gÎT} D(g) if and only if I(Z(T)) = A, i.e., Ö(T) = A. Thus (T) = (1). So (a) implies (b). Similarly (c) implies (d). The equivalence of (b) and (c) is obvious. Finally (d) implies (a) trivially. Corollary 2.4.7. A is quasicompact for any affine ring A^{o}. Proposition 2.4.8. (Ring^{op}, t) is a metric site. Proof. Suppose A^{o} is an affine ring. For any f Î A, we show that the elementary open set D(f) Í A is effective. This would imply that Ring^{op} is a locally effective presite by (1.2.15) since elementary open sets form a basis for A and fibre products exist in Ring^{op}. Consider the morphism j^{o}: A_{f}^{o} d A^{o} induced by the canonical map j: A ® A_{f}. j is bicontinuous with the image D(f) in A. Suppose f^{o}: C^{o} ® A^{o} is a morphism such that f^{o}(C^{o}) Í D(f). Then f(f) is invertible in C. It follows by the universal property of localization that f: A ® C factors through the canonical map j: A ® A_{f}, thus f factors through j. This shows that j is effective. The subsite Field^{op} of Ring^{op} is separable (1.3.11) which is a generic subsite of Ring^{op}, thus Ring^{op} is separable. Proposition 2.4.9. (Ring^{op}, t) is a strict metrict site. Proof. We have to prove that the identity functor
Ring^{op} ®
Ring^{op} is a cosheaf. Suppose A^{o} is
an affine ring and {U_{i}} is an open effective cover of
A. We prove that A^{o} is the colimit of {U_{ij}}
where U_{ij} = U_{i} Ç
U_{j}. Since elementary open subsets of A form
a basis, and A is quasicompact, we may assume that {U_{i}}
is a finite cover consisting of elementary open subsets U_{i}
= A_{fi} with f_{i} Î
A. Thus it suffices to prove that the ring A is the limit
of A_{fi}.
Remark 2.4.10.
For any ring k let kAlg be the category of commutative
algebras over k. Then (kAlg)^{op} is isomorphic
to Ring^{op}_{/ko}. Thus (kAlg)^{o}
is also a strict metric site.
