4.1 Embedded Subsites  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:  For any morphism f: I(Z) ® X (resp. f: X ® I(Z)) in C with Z Î D, there exists a unique morphism g: Z ® J(X) (resp. g: J(X) ® Z) such that f = rI(g) (resp. f = I(g)r).  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 1X.  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 1I(Y) = rI(s) (resp. 1I(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 1X: 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 1X 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:  (a) For any object X Î D and any subset U of |X|, the inclusion U ® X (in C) is in D.  (b) Suppose X, Y are objects of D. Then f: X ® Y is in D if and only if for any x Î |X|, the morphism fx: x ® f(x) of effective points is a morphism of D.  (c) Suppose f: X ® Y, g: Y ® Z are morphisms of C and f is surjective. Suppose f and gf are in D. Then g is in D.  Then D is an embedded subsite of C.  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  (d) r.s = 1X.  (e) If h, h': Z ® Y are any two morphisms in D and rh = rh', then h = h'. We shall prove the following three facts:  (i) For any morphism h: Z ® Y in D with |h(Z)| Í |s(X)| we have rh Î D.  (ii) s: X ® Y is effective in D.  (iii) |s(X)| = {y Î |Y| ½ ry: y ® r(y) is in D}.  For any x Î |X| we have (rs)x = rs(x)sx = 1x Î D and sx: x ® s(x) is in D by (b). It follows that rs(x) Î D by (c). Suppose h: Z ® Y is a morphism in D with |h(Z)| Í |s(X)|. For any z Î |Z| we have h(z) Î |s(X)|, thus rh(z) Î D. Since h Î D, hz Î D by (b). Thus (rh)z = rh(z)hz Î D for any z Î |Z|. This means that rh Î D by (b), hence (i) holds.  The morphism s: X ® Y is a bicontinuous monomorphism by (d). Suppose h: Z ® Y is a morphism in D and |h(Z)| Í |s(X)|. We have rh Î D by (i), hence srh Î D. Then rsrh = 1Xrh = rh implies that srh = h by (e). This proves that h factors through s by the morphism rh Î D (the uniqueness of rh is obvious since s is monomorphic). This proves (ii).  Suppose y Î |Y| and i: y ® Y is the inclusion morphism in D. If y Î |s(X)| then ry = ri: y ® X is in D by (i). If y Ï |s(X)| then we have sri(y) ¹ i(y) but rsri(y) = ri(y) as rs = 1X. Since i Î D we must have sri Î D by (e), which implies that ry = ri Î D because s e D. Thus (iii) holds.  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 Ox}. For any (x, z) Î Spec X we shall write Az for the localization of Ox at z. We take the topology on Spec X generated by the open subsets Ug = , where U is an open subset of X and g Î O(U), such that = {(x, z)|x Î U and gx Ï z}. For any open subset W of Spec X we define OSpecX(W) to be the ring of functions s: W ® È(x,z)ÎW Az such that s(x, z) Î Az 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 Az' for any (x', z') Î (V, g). We thus defined a sheaf OSpecX on Spec X. For each point (x, z) Î Spec X it is easy to see that the stalk of OSpecX at (x, z) is isomorphic to Az. Thus (Spec X, OSpecX) 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) ® OSpecX(j-1(U)) induces a morphism of ringed spaces (j, j#) from Spec X to X.  One can easily prove that (Spec X, OSpecX) together with the morphism (j, j#) has the universal property required. This proves that LSp is a coreflective subsite of RSp.  We verify the conditions of (4.1.5.) for LSp. First (4.1.5.a & b) follow directly from the definition of morphisms of local ringed spaces. For (4.1.5c) note that if f#: Oy ® Ox, g#: Oz ® Oy are morphisms of local rings, and if f# and f#g# are local, then g# is local. This proves that LSp is an embedded subsite of RSp.  Proposition 4.1.7. GSp is an embedded subsite of LSp.  Proof. For any local ringed space X the geometric space Xred = (X, OX/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) = {Ox|x Î X} canonically. If X is a local ringed space, then f(X) is a local ringed set, and fk(X) = {kx|x Î X} is a geometric set, where kx is the residue field of Ox. Clearly f: RSp ® RSet, f: LSp ® LSet, fk: LSp ® GSet and fk: GSp ® GSet are right adjoint functor of the inclusion functors. Also these subsites are full, hence are embedded.      [Next Section][Content][References][Notations][Home]