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 t_U Î G(U). Since O is a sheaf, we can glue these sections t_U 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 X_s È X_1-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 X_s if and only if the germ s_x of s at x is a unit of the stalk P_x. 

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 s_x or (1 - s)_x is a unit of the stalk P_x of O at x, since P_x is a local ring. Thus we have X = X_s È X_1-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 X_s. Then for any x Î U the germ s_x of s at x is a unit. Since P is a sheaf, t = {1/s_x| x Î U} is a section of U, and t.(s|_U) = 1_U, thus s|_U is a unit of G(U). If U is not contained in X_s, we can find a point x Î U such that s_x is not a unit in P_x, 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 P_x of O at x is the limit of nonzero rings O(U) for all effective open neighborhood U of x. Thus P_x ¹ 0. We have x Î X_s È X_1-s by (3.1.3.b). Thus either s_x or (1 - s)_x is a unit of P_x (3.1.6). Since any element of P_x is the germ of a section, P_x 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, f_y^#: P_f(y) ® Q_y is a local morphism of local rings. Thus the following assertions are equivalent: 
(a) f(Y) Í X_s. 
(b) Each s_f(y) is a unit in P_f(y) for any y Î Y. 
(c) Each f^#(s)_y is a unit in Q_y 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 Ring^op is a ringed site with the identity functor (Ring^op)^op = Ring ® Ring as the structure sheaf. 

Example 3.1.14. Denote by RedRing the category of reduced ring. Then RedRing^op is a ringed subsite of Ring^op consisting of reduced affine rings. 

Proposition 3.1.15. (a) Ring^op is a local ringed site. 
(b) RedRing^op is a geometric site. 

Proof. (a) Suppose A^o Î Ring^op is an affine ring with G(A^o) = A. Then |A| = Æ if and only if A = 0. For any s Î G(A^o) = 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 Ring^op. 
Suppose j^o: B^o ® A^o is a morphism in Ring^op induced by a homomorphism j: A ® B of rings. Then |j^o(B^o)| 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) = (j^o)^#(s) and B = G(B^o), (3.1.3.c) holds for Ring^op. This proves that Ring^op is a local ringed site. 
(b) Suppose A^o is a reduced affine ring and s Î G(A^o) = 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 RedRing^op 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, Ring^op 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 X_f#_(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 X_f#_(1-s) È X_f#_(s) = Y, |A| is the union of A_s and A_1-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 X_f#_(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 Ring^op is a local ringed site, (Ring^op)^{^} and (Ring^op)^~ are local ringed sites.  
(b) (RedRing^op)^{^} and (RdeRing^op)^~ 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) Ring^op ® LSp and (RedRing)^op ® GSp; these are isometric embeddings (see (3.2.14)). 
(b) (Ring^op)^~ ® LSp and (RedRing^op)^~ ® GSp; 
(c) (Ring^op)^{^} ® LSp and (RedRing^op)^{^} ® GSp.  

Remark 3.1.24. (a) The image of the functor Ring^op ® LSp is equivalent to ASch. Since Sch is the Cauchy-completion of ASch in LSp, Sch is a Cauchy-completion of the Zariski site Ring^op. 
(b) Similarly GSch is a Cauchy-completion of the site RedRing^op.