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:
Proposition 3.1.2. (a) X_{s}
= X if and only if s is a 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.
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):
Remark 3.1.4. Let (C, O)
be a local ringed site.
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_{1s} 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.
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:
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
Suppose R is a ring. If (C, O) is a ringed site and O is a sheaf of Ralgebras, 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.
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.
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}.
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.
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}#_{(1s)} È X_{f}#_{(s)} = Y, A is the union of A_{s} and A_{1s} 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).
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.
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:
Remark 3.1.24.
(a) The image of the functor Ring^{op} ®
LSp is equivalent to ASch. Since Sch is the Cauchycompletion
of ASch in LSp, Sch is a Cauchycompletion of the
Zariski site Ring^{op}.
