Let I: D ® C be
a covariant functor of categories. Suppose X is an object of C.
An object J(X) of D together with a morphism r:
I(J(X)) ® X
(resp. r: X ® I(J(X)))
is called a left
(resp. right) associated
object of X in D if it has the following universal
property:
We say that (J(X), r) is a strict left (resp. right) associated object of X if I(J(X)) = X and r is the identity morphism 1_{X}. An associated object of X is uniquely determined by X and I up to isomorphism. The morphism r is called the counit (resp. unit) of X. If Y Î D and I(Y) has a left (resp. right) associated object J(I(Y)) in D, then the unique morphism s: Y Î J(I(Y)) (resp. s: J(I(Y)) ® Y) such that 1_{I}(Y) = rI(s) (resp. 1_{I}(Y) = I(s)r) is called the unit (resp. counit) of Y. Clearly I: D ® C has a right (resp. left) adjoint J: C ® D if and only if any object X of C has a left (resp. right) associated object J(X). Recall that if D is a subcategory of C and the inclusion functor I: D ® C has a right (resp. left) adjoint, then D is called a coreflective (resp. reflective) subcategory of C. Example 4.1.1. Let D be a coreflective (resp. reflective) subcategory of C. Then any object X of D together with the identity morphism 1_{X}: X ® X is a left (resp. right) associated object of X if and only if D is a full subcategory of C. Example 4.1.2. Suppose D is a coreflective subcategory of a category C, and D' a coreflective subcategory of D. Then D' is a coreflective subcategory of C. Definition 4.1.3. Suppose C is an everywhere effective site. An everywhere effective subsite D of C is called embedded if D is a coreflective subcategory of C such that for any object X e D, the unit s: X ® J(X) is an effective morphism in D. Remark 4.1.4. If D is a full coreflective subsite of C, then J(X) = X for any X Î D and s: X ® X is the identity morphism 1_{X} of X (4.1.1). Thus D is embedded. Proposition 4.1.5. Suppose C
is an everywhere effective site, D an everywhere effective
coreflective subsite of C having the following properties:
Proof. Suppose X ® D
and Y = J(X) is a left associated object of X
in D with the unit s: X ®
Y and counit r: Y ®
X. Then we have
Proposition 4.1.6. (Chevalley, cf. [H] p.68) Any ringed space (X, O) has a left associated object in the category LSp of local ringed spaces. The subsite LSp is an embedded subsite of the site RSp of ringed spaces. Proof. Let Spec X = {(x, z)x Î
X and z is a prime ideal of O_{x}}. For any
(x, z) Î Spec X we
shall write A_{z} for the localization of O_{x}
at z. We take the topology on Spec X generated by the open
subsets U_{g} = <U, g>, where U
is an open subset of X and g Î
O(U), such that <U, g> = {(x, z)x
Î U and g_{x} Ï
z}. For any open subset W of Spec X we define
O_{SpecX}(W) to be the ring of functions s:
W ® È_{(x,z)ÎW}
A_{z} such that s(x, z) Î
A_{z} for each (x, z), and such that for any
(x, z) Î W, there
is a small neighborhood (V, g) of (x, z) contained
in W and an elements a of O(V) such that and
s(x', z') = a/g in A_{z}'
for any (x', z') Î (V,
g). We thus defined a sheaf O_{SpecX} on Spec
X. For each point (x, z) Î
Spec X it is easy to see that the stalk of O_{SpecX}
at (x, z) is isomorphic to A_{z}. Thus (Spec
X, O_{SpecX}) is a local ringed space. Let j:
Spec X ® X be the natural
map sending any (x, z) Î
Spec X to x Î X.
The natural homomorphism O(U) ®
O_{SpecX}(j^{1}(U)) induces a morphism
of ringed spaces (j, j^{#}) from Spec X to
X.
Proposition 4.1.7. GSp is an embedded subsite of LSp. Proof. For any local ringed space X the geometric space X_{red} = (X, O_{X}/I) is a left associated object of X in the category of geometric spaces. We obtain a right adjoint functor red: LSp ® GSp. Gsp is a full subsite of LSp, hence an embedded subsite. Proposition 4.1.8. RSet is an embedded subsite of RSp. LSet and GSet are embedded subsites of LSp. GSet is an embedded subsite of GSp. Proof. Any ringed space X determines a ringed set f(X)
= {O_{x}x Î X}
canonically. If X is a local ringed space, then f(X)
is a local ringed set, and f_{k}(X) = {k_{x}x
Î X} is a geometric set, where
k_{x} is the residue field of O_{x}. Clearly
f: RSp ® RSet, f:
LSp ® LSet, f_{k}:
LSp ® GSet and f_{k}:
GSp ® GSet are right adjoint
functor of the inclusion functors. Also these subsites are full, hence
are embedded.
