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 T₁ and T₂ are subsets of A, then Z(T₁) È Z(T₂) = Z(T₁T₂). 
(b) If {T_i} is any collection of subsets of A, then Ç Y_i = Z(È T_i). 
(c) X = Z(0) and Æ = Z(1). 

Proof. If p Î Z(T₁) È Z(T₂), then either p Î Z(T₁) or p Î Z(T₂), thus for any f Î T₁ and g Î T₂, either f Î p or g Î p; hence fg Î p. This shows that p Î Z(T₁T₂). Conversely, if p Î Z(T₁T₂), and p - Z(T₁) say, then there is an f Î T₁ such that f Ï p. Now for any g Î T₂, fg Î p implies that g Î p, so that p Î Z(T₂). 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). 
(a) For any ideal p of a ring A we have I(Z(p)) = Öp, which is the radical of p.  
(b) There is a one-to-one inclusion-reversing correspondence between closed sets in |A| and radical ideals in A, given by Y ® I(Y) and p ® Z(p). 
(c) If A is a noetherian ring then |A| is a noetherian space. Thus any subspace of |A| is quasi-compact. 
(d) The dimension of the topological space |A| is equal to the Krull dimension of A. 

Lemma 2.4.6. Suppose T is a subset of a ring A; then the following conditions are equivalent: 
(a) {D(g) | g Î T} is an open covering of |A|.  
(b) 1 Î (T), the ideal generated by the set T. 
(c) 1 Î (T₁), where T₁ is a finite subset of T. 
(d) There is a finite subset S of T such that {D(g) | g Î S} is a finite open covering of |A|. 

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 quasi-compact 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 quasi-compact, 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. 
(a) Suppose a Î A such that a/1 = 0 in A_fi for all i. We have a/1 = 0 Î A_p for any prime ideal p Î |A|. Hence there exists an element h - p such that ha = 0 in A. Let a be the annihilator of a. Then h Î a, and h - p so a - p. The ideal a is not contained in any prime ideal of A, so a = (1). Hence a.1 = a = 0. 
(b) Now suppose we have u_i Î A_fi for each i such that u_i|_D(fifj) = u_j|_D(fifj). Suppose u_i = v_i/f_iⁿ for some n. (Note that {U_i} is a finite cover, so we may choose a single large n.) For each pair (i, j) we have  

Remark 2.4.10. For any ring k let k-Alg be the category of commutative algebras over k. Then (k-Alg)^op is isomorphic to Ring^op_/ko. Thus (k-Alg)^o is also a strict metric site.