3.1 Ringed Sites  Definition 3.1.1. A ringed site (C, O) consists of a site C and a sheaf O of rings on C such that O(X) = 0 for any empty object X (i.e., |X| = Æ); C and O are called the underlying site and the structure sheaf of (C, O) respectively.  Suppose (C, O) is a ringed site. If X is an object and f: Y ® X a morphism in C, we often write G(X) for the ring O(X), and f# for the homomorphism O(X) ® O(Y) of rings.  Suppose U is an effective subset of |X|. An element s of the ring G(U) is called a section over U. If s is a section over U and V an effective subset of U with the inclusion i: V ® U, we shall write s|V for the element i#(s) Î G(V), called the restriction of s on V.  For any section s Î G(X) we define an open subset |X|s of |X|:  |X|s = {x Î |X| ½ s|U is a unit for some effective open neighborhood of x}.  Proposition 3.1.2. (a) |X|s = |X| if and only if s is a unit of G(X).  (b) |X|s is empty if s is a nilpotent of G(X).  (c) If X is a spot, then |X|s is empty if and only if s is a non-unit of G(X).  Proof. (a) If s is invertible, clearly we have |X|s = |X|. Conversely, suppose |X|s = |X|. For any x Î |X| there is an effective open subset U of x such that s|U is a unit with an inverse tU Î G(U). Since O is a sheaf, we can glue these sections tU to obtain an inverse of s, so s is a unit.  (b) and (c) are obvious.  Definition 3.1.3. A ringed site (C, O) is called a local ringed site if the following conditions are satisfied for any object X Î C and any s Î G(X):  (a) G(X) = 0 if and only if X is empty.  (b) |X|s È |X|1-s = |X|.  (c) Suppose f: Y ® X is a morphism in C. Then |f(Y)| Í |X|s if and only if f#(s) is a unit in G(Y).  Remark 3.1.4. Let (C, O) be a local ringed site.  (a) Suppose X is a spot. If s Î G(X), we have Xs È X1-s = X (3.1.3.b), so either |X|s = |X| or |X|1-s = |X|, hence either s or 1 - s is a unit in G(X) by (3.1.2.a). Thus G(X) ¹ 0 is a local ring.  (b) Suppose f: Y ® X is a morphism of spots. If s Î G(X) is a non-unit in G(X), then |X|s is empty by (3.1.2.c), so f#(s) is a non-unit of G(Y) by (3.1.3.c). Hence f#: G(X) ® G(Y) is a local homomorphism of local rings.  Example 3.1.5. Suppose B is a subsite of a ringed site (C, O). Then (B, O|B) is a ringed site, called a ringed subsite of C. For simplicity we often write B for the ringed subsite (B, O|B). If (C, O) is a local ringed site, then so is (B, O|B).  Example 3.1.6. Any ringed space (X, P) is a ringed site with the underlying site W(X) and the structure sheaf P. For any x Î X and s Î G(X), we have x Î Xs if and only if the germ sx of s at x is a unit of the stalk Px.  Proposition 3.1.7. A ringed space (X, P) is a local ringed site if and only if it is a local ringed space.  Proof. Suppose (X, P) is a local ringed space. Then G(U) ¹ 0 for any nonempty open subset U. For any x Î X and s Î G(X) either sx or (1 - s)x is a unit of the stalk Px of O at x, since Px is a local ring. Thus we have X = Xs È X1-s by (3.1.6). This proves (3.1.3.b). Next we verify (3.1.3.c). Suppose U is an open subset of Xs. Then for any x Î U the germ sx of s at x is a unit. Since P is a sheaf, t = {1/sx| x Î U} is a section of U, and t.(s|U) = 1U, thus s|U is a unit of G(U). If U is not contained in Xs, we can find a point x Î U such that sx is not a unit in Px, so s|U is not a unit in G(U). This proves that (X, P) is a local ringed site.  Conversely, suppose (X, P) is local ringed site. Suppose x is any point of X and s a section of G(X). The stalk Px of O at x is the limit of nonzero rings O(U) for all effective open neighborhood U of x. Thus Px ¹ 0. We have x Î Xs È X1-s by (3.1.3.b). Thus either sx or (1 - s)x is a unit of Px (3.1.6). Since any element of Px is the germ of a section, Px is a local ring for any x Î X. Therefore (X, P) is a local ringed space.  Example 3.1.8. The site RSp of ringed spaces is a ringed site with the structure sheaf G sending a ringed space (X, P) to the ring G(X) of global sections on X. The site LSp of local ringed spaces is a ringed subsite of RSp.  Proposition 3.1.9. LSp is a local ringed site.  Proof. We have seen that (3.1.3.b) holds for a local ringed space (X, P) (3.1.7). Thus we only need to verify (3.1.3.c) for a morphism (f, f#): (Y, Q) ® (X, P) of local ringed spaces. Suppose s Î G(X) is a section of X. Since f is a morphism of local ringed spaces, fy#: Pf(y) ® Qy is a local morphism of local rings. Thus the following assertions are equivalent:  (a) f(Y) Í Xs.  (b) Each sf(y) is a unit in Pf(y) for any y Î Y.  (c) Each f#(s)y is a unit in Qy for any y Î Y.  (d) f#(s) is a unit of G(Y).  The equivalence of (a) and (d) implies that (3.1.3.c) holds for LSp.  Definition 3.1.10. A geometric site is a local ringed site in which any section s over an object X with |X|s = Æ is zero.  Remark 3.1.11. Suppose (C, O) is a geometric site and X Î C. Then  (a) G(X) is a reduced ring for any X Î C. So the structure sheaf of a geometric site is a sheaf of reduced rings.  (b) If X is a spot then G(X) ¹ 0 is a field because for any nonunit s Î G(X), |X|s is empty (3.1.2.c), hence s = 0 (3.1.10).  Suppose R is a ring. If (C, O) is a ringed site and O is a sheaf of R-algebras, then we say that (C, O) is a ringed site over R. Similarly we define a local ringed site or a geometric site over R.  Example 3.1.12. (a) Sch and ASch are local ringed sites.  (b) GSp, GSch, GASch are geometric sites.  (c) PVar/k and AVar/k are geometric sites over k.  (d) RPot is a ringed site. LPot is a local ringed site. GPot is a geometric site.  Example 3.1.13. The Zariski site Ringop is a ringed site with the identity functor (Ringop)op = Ring ® Ring as the structure sheaf.  Example 3.1.14. Denote by RedRing the category of reduced ring. Then RedRingop is a ringed subsite of Ringop consisting of reduced affine rings.  Proposition 3.1.15. (a) Ringop is a local ringed site.  (b) RedRingop is a geometric site.  Proof. (a) Suppose Ao Î Ringop is an affine ring with G(Ao) = A. Then |A| = Æ if and only if A = 0. For any s Î G(Ao) = A, |A|s = D(s) consists of prime ideals p of A such that s Ï p. For any p Î |A| we have either s Ï p or 1 - s Ï p, so p Î D(s) È D(1 - s). Thus (3.1.3.b) holds for Ringop.  Suppose jo: Bo ® Ao is a morphism in Ringop induced by a homomorphism j: A ® B of rings. Then |jo(Bo)| Í |A|s if and only if s Ï j-1(p) for any prime ideal p of B. This is equivalent to that j(s) is a unit in B. Since j(s) = (jo)#(s) and B = G(Bo), (3.1.3.c) holds for Ringop. This proves that Ringop is a local ringed site.  (b) Suppose Ao is a reduced affine ring and s Î G(Ao) = A such that |A|s is empty. Then s is contained in all the prime ideals of A, so s is nilpotent. Since A is reduced, s = 0. Thus RedRingop is a geometric site.  3.1.16 Suppose (C, O) is a ringed site. Consider the site C^ of presheaves of sets on C. Since Ring has limits (or equivalent, Ringop has colimits), O has a Kan extension on C^, denoted by O^. O^ is a sheaf on C^ by (2.2.9). Thus we obtain a ringed site (C^, O^). Also we have a ringed subsite (C~, O~) with O~ = O^|C~.  Remark 3.1.17. Suppose s Î G(A) for some A Î C^. Suppose f: X ® A is a morphism with X Î C.  (a) |A|s consists of the points x of |A| such that f-1(x) Î Xf#(s) for any f: X ® A.  (b) f(X) Í |A|s if and only if f#(s) is a unit of G(X).  (c) A section s Î G(A) is a unit if and only if f#(s) is a unit of G(X) for any f: X ® A.  (d) If B ® A is a morphism in C^, then |g(B)| Î |A|s if and only if |gh|(X) Í |B|s for any h: X ® A.   Proposition 3.1.18. Suppose (C, O) is a local ringed site (resp. geometric site). Then (C^, O^) is a local ringed site (resp. geometric site).   Proof. Suppose s Î G(A) for some A Î C^. Suppose f: X ® A is a morphism with X Î C. Since Xf#(1-s) È Xf#(s) = Y, |A| is the union of As and A1-s by (3.1.17.a). Thus (3.1.3.b) holds.   Next consider a morphism g: B ® A in C^. Suppose |g(B)| Í |A|s. Then for any X Î C and h: X ® B we have gh(X) Í |A|s. Thus (gh)#(s) = h#(g#(s)) is a unit in G(X). Hence g#(s) is a unit of G(B) by (3.1.17.c). Conversely, suppose g#(s) is a unit of G(B). Since (gh)#(s) = h#(g#(s)) is a unit of G(X), we have (gh)(X)Í |A|s by (3.1.17.b), hence |g(B)| Í |A|s by (3.1.17.d).   Now suppose C is a geometric site. If |A|s is empty, then Xf#(s) is empty, thus f#(s) = 0. Since this is true for any f: X ® A, we conclude that s = 0. Thus (C^, O^) is a geometric site.  Corollary 3.1.19. Suppose (C, O) is a local ringed site (resp. geometric site). Then C~ is a local ringed site (resp. geometric site). If C is strict then C is a local ringed subsite (resp. geometric subsite) of C~.  Example 3.1.20. (a) Since Ringop is a local ringed site, (Ringop)^ and (Ringop)~ are local ringed sites.   (b) (RedRingop)^ and (RdeRingop)~ are geometric sites.   Example 3.1.21. Suppose X is a topological space. Suppose B is an open basis of X which is closed under intersection. Then B is a metric site. Suppose O is a sheaf of rings on B. O has a Kan extension on W(X), called the sheaf on X generated by O, denoted also by O. Thus we obtain a ringed space (X, O). For any open subset U of X, G(X) is simply the limit of rings G(V) for all V Î B contained in U. If (B, O) is a local ringed space, then so is (X, O) because it is a subsite of (B^, O).   Remark 3.1.22. Suppose (C, O) is a ringed sites. Any object X Î C determines a ringed space (X, O|X) where O|X is the sheaf on X generated by the restriction of O on the subsite of effective open subsets of X. We obtain a functor b: (C, O) ® RSp, which is an isometry of sites. Clearly (C, O) is a local ringed site (resp. geometric site) if and only if the image b(C) of C is contained in the subsite LSp (resp. GSp) of RSp.  Example 3.1.23. Applying (3.1.22) we obtain six important isometries:   (a) Ringop ® LSp and (RedRing)op ® GSp; these are isometric embeddings (see (3.2.14)).  (b) (Ringop)~ ® LSp and (RedRingop)~ ® GSp;  (c) (Ringop)^ ® LSp and (RedRingop)^ ® GSp.   Remark 3.1.24. (a) The image of the functor Ringop ® LSp is equivalent to ASch. Since Sch is the Cauchy-completion of ASch in LSp, Sch is a Cauchy-completion of the Zariski site Ringop.  (b) Similarly GSch is a Cauchy-completion of the site RedRingop.    [Next Section][Content][References][Notations][Home]