2.4 Zariski Site  In this section we consider the category Ringop. We shall define a metric topology t on Ringop so that (Ringop, t) is a strict metric site, called the Zariski site.  In the following we shall write Ao for a ring A if it is regarded as an object of Ringop, called an affine ring. If f: A ® B is a homomorphism of rings we write fo: Bo ® Ao for the morphism in Ringop.   Suppose Ao is an affine ring. We write |A| for the set Spec A of prime ideals of A. If jo: Ao ® Bo 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 {Ti} is any collection of subsets of A, then Ç Yi = Z(È Ti).  (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 jo: Ao ® Bo 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: Ringop® Top sending each affine ring Ao to the space |Ao| = Spec A. We now consider the presite (Ringop, 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 = ÈfÎT (|A| - Z(f)) = ÈfÎT 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| = ÈgÎT 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 Ao.  Proposition 2.4.8. (Ringop, t) is a metric site.  Proof. Suppose Ao 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 Ringop is a locally effective presite by (1.2.15) since elementary open sets form a basis for |A| and fibre products exist in Ringop. Consider the morphism jo: Afo d Ao induced by the canonical map j: A ® Af. |j| is bicontinuous with the image D(f) in |A|. Suppose fo: Co ® Ao is a morphism such that |fo(Co)| 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 ® Af, thus f factors through j. This shows that j is effective.   The subsite Fieldop of Ringop is separable (1.3.11) which is a generic subsite of Ringop, thus Ringop is separable.  Proposition 2.4.9. (Ringop, t) is a strict metrict site.  Proof. We have to prove that the identity functor Ringop ® Ringop is a cosheaf. Suppose Ao is an affine ring and {Ui} is an open effective cover of |A|. We prove that Ao is the colimit of {Uij} where Uij = Ui Ç Uj. Since elementary open subsets of |A| form a basis, and |A| is quasi-compact, we may assume that {Ui} is a finite cover consisting of elementary open subsets Ui = |Afi| with fi Î A. Thus it suffices to prove that the ring A is the limit of Afi.  (a) Suppose a Î A such that a/1 = 0 in Afi for all i. We have a/1 = 0 Î Ap 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 ui Î Afi for each i such that ui|D(fifj) = uj|D(fifj). Suppose ui = vi/fin for some n. (Note that {Ui} is a finite cover, so we may choose a single large n.) For each pair (i, j) we have   (fifj)m(vifjn - vjfin) = 0 for some m. Put vjfjm = wj, and m + n = k, we find that  ui = wi/fik, wi/fjk = wjfik.  Since {D(fi}) is a cover of |A|, {f1,...,fn} generates the ideal (1) of A (2.4.6). Thus we have  S fikgi = 1. Put u = S wjgj. Then   fiku = S wjgjfik = S wigjfjk = wi.  It follows that u|D(fi) = wi/fik = ui. This proves that A is the limit of Afi.  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 Ringop/ko. Thus (k-Alg)o is also a strict metric site.    [Next Section][Content][References][Notations][Home]