In this section we assume D is a strict metric site with colimits. We present an axiomatic approach to the theory of local ringed sites, replacing Ring^{op} by D. If D is a T_{0}site this approach also provides a general method to construct a Cauchycompletion for D, using the notion of Dspaces (see 3.2.16). This method also works for an arbitrary strict T_{0}site (which may not have colimits); see (3.2.18). Definition 3.2.1. A site over D (or simply Dsite) is a pair (C, O) consisting of a metric site C and a cosheaf O: C ® D on C with values in D sending any empty object of C to an empty object of D; C and O are called the underlying site and the structure cosheaf of (C, O) respectively. Suppose (C, O) is a Dsite. For any
object X Î
C and any open subset V of O(X) we define
an open subset X_{V} of X:
Definition 3.2.2.
A local site over D (or
local Dsite, or simply
local site) is a Dsite (C,
O) satisfying the following conditions for any X Î
C:
Remark 3.2.3. If B is a basis for D, then in (3.2.2) we may assume that V_{i} and V are in B. Moreover, if the space of any object of D is quasicompact, we may assume that {V_{i}} is a finite cover. Example 3.2.4. Suppose (C, O) is a Dsite and B a subsite of a C. Then (B, O_{B}) is a Dsite, denoted simply by B. If C is a local site, then so is B. Example 3.2.5. D is a local Dsite with the identity functor D ® D as the structure cosheaf. Example 3.2.6. A ringed site (C, O) is a site over Ring^{op} with the structure cosheaf O: C ® Ring^{op}. It is easy to see that (C, O) is a local ringed site if and only if it is a local site over Ring^{op}. (The equivalence of (3.2.2.a) and (3.1.3.a) can be established by an inductive argument based on (3.2.3).) Next we define Dspaces and local Dspaces, generalizing the concepts of ringed spaces and local ringed spaces respectively. Definition 3.2.7. A space over D (or simply Dspace) is a pair (X, P) consisting of a topological space X and a cosheaf P: W(X) ® D on X such that P(Æ) ® D is an empty object. Thus a Dspace is a Dsite whose underlying site consists of open sets of a topological space. Suppose (X, P) and (Y, Q) are two Dspaces. A morphism from (Y, Q) to (X, P) is a pair (f, f) consisting of a continuous map f: Y ® X, and a morphism f (i.e., natural transformation) from Qf^{1}: W(X) ® D to P: W(X) ® D, where f^{1} = W(X) ® W(Y) is the functor sending each open set U of X to f^{1}(U) Í Y. We obtain a category of Dspaces, denoted DSp. Remark 3.2.8. DSp is a metric site with the obvious metric topology. As in the case of ringed spaces, DSp is strict and Cauchycomplete (here we need the assumption that D has colimits). It is a Dsite with the structure cosheaf sending each Dspace (X, P) to P(X). Definition 3.2.9. A local space over D (or local Dspace, or simply local space) is a Dspace (X, P) which is a local Dsite. A morphism of local spaces over D is a morphism (f, f): (Y, Q) ® (X, P) of Dspaces such that for any open subsets V Í P(X) and U Í Y, f(U) Í X_{V} if and only if the morphism Q(U) ® P(X) induced by f: Q(Y) ® P(X) sends O(U) into V. Remark 3.2.10. Denote by DLSp the category of local Dspaces. Then DLSp is a Cauchycomplete site. It is a local Dsite. If D = Ring^{op} we obtain the site of local ringed spaces. Suppose (C, O) is a Dsite. Since D has colimits, as in (3.1) we can define the Kan extensions (C^{^}, O^{^}) and (C^{~}, O^{~}) which are again Dsites. We have the following generalization of (3.1.18) which can be proved in a similar way: Proposition 3.2.11. Suppose (C, O) is a local Dsite. Then (C^{^}, O^{^}) and (C^{~}, O^{~}) are local Dsites. 3.2.12 Suppose (C, O) is a Dsite. Any object X determines a space b(X) = (X, O_{W}(X)) over D where O_{W}(X) is the restriction of the structure sheaf O: C^{^} ® D sending each open subset U of X to O(U). We obtain a functor b: (C, O) ® DSp which is an isometry of metric sites. Note that the structure cosheaf O: C ® D is isomorphic to the pullback of the structure cosheaf on DSp along b. Also (C, O) is a local Dsite if and only if its image b(C) in DSp is in DLSp. Since D is a local Dsite (3.2.5), we obtain a canonical functor b: D d DLSp by (3.2.12). Proposition 3.2.13. Suppose X, Y are two objects of D and X is a T_{0}space. Then any morphism from b(Y) to b(X) in DLSp is induced by a morphism from Y to X in D. Proof. Suppose (f, f): b(Y) ® b(X) is a morphism of local spaces. Since b(X) = (X, O_{W}(X)), b(Y) = (Y, O_{W}(Y)) and O is the identity functor of D, we have O(Y) = Y and O(X) = X. We prove that (f, f) is induced by the morphism f_{X}: Y = O(Y) = O(f^{1}(X)) ® O(X) = X. Since f is uniquely determined by f_{X}, it suffices to prove that f_{X} coincides with f. Assume there is a point y Î Y such that f_{X}(y) / f(y). Since X is a T_{0}space, we can find an open neighborhood U of f_{X}(y) such that f(y) Ï U, or an open neighborhood V of f(y) such that f_{X}(y) Ï V. In the first case there is an effective open neighborhood W of y in O(Y) = Y such that f_{X}(W) Í U. Then f(W) Í X_{U} = U by (3.2.9), so f(y) Î U which is a contradiction. Similarly we can prove that the second case is impossible. This shows that f(y) = f_{X}(y). Thus f_{X} coincides with f. Theorem 3.2.14. Suppose D is a T_{0}site (i.e., the space of any object of D is a T_{0}space). Then the canonical morphism b: D ® DLSp is an isometric embedding. Proof. Clearly b: D ® DLSp is faithful. It is full by (3.2.13). Definition 3.2.15. Suppose D is a T_{0}site. An affine Dscheme is a local Dspace which is isomorphic to b(X) for an object X e D. A Dscheme is a local Dspace (X, P) in which any point x has an open neighborhood U such that the Dsubspace (U, P_{U}) of X is an affine Dscheme. Denote by DSch and DASch the sites of Dschemes and affine Dschemes respectively. Theorem 3.2.16.
Assume D is a T_{0}site. Then
Proof. (a) follows from (3.2.14). (b) follows from (a) and (2.1.10) as DLSp is Cauchycomplete and DSch is the completion of DASch in DLSp. Example 3.2.17. If D = Ring^{op} we obtain the ordinary schemes and affine schemes. Remark 3.2.18. If C is any standard T_{0}site we first embed C isometrically into a standard T_{0}site D with colimits. Define an affine Cscheme to be an affine Dscheme which is isomorphic to b(X) for an object X e C. Define a Cscheme to be a Dscheme (X, P) in which any point x has an open neighborhood U such that the Dsubspace (U, P_{U}) of X is an affine Cscheme. Then the site CASch of affine Cschemes is equivalent to C and the site CSch of Cschemes is a Cauchycompletion of C. Example 3.2.19. If C is the opposite of the category kAlg_{f} of finitely generated reduced algebras over an algebraically closed field k, and D is the opposite of the category kAlg of kalgebras with the Zariski topology, then a Cscheme (resp. affine Cscheme) in the sense of (3.2.18) is precisely a prevariety (resp. affine variety) over k in the sense of (1.1.19). Thus the site PVAr/k of prevarieties is a Cauchycompletion of the T_{0}site kAlg_{f}^{op}. A topological space is called sober if every closed irreducible set has a unique generic point. A site D is called sober if the space of any object of D is sober. Proposition 3.2.20.
Suppose D is a sober site. A Dsite
(C, O) is a local site if and only if for any X Î
C there is a continuous map X ®
O(X) satisfying the following conditions:
Proof. First assume (C, O) is a local
site. Consider an object X Î
C. For any point x Î
X let W be the union of open subsets V of O(X)
such that x is not contained in X_{V}. Then
O(X)  W is an irreducible closed set. Since O(X)
is sober, O(X)  W has a unique generic point, denoted
by t(x). We obtain a map t: X ®
O(X) sending each x Î
X to t(x). This is a continuous map satisfying the
conditions (a) and (b).
Remark 3.2.21.
Suppose (C, O) is a Dsite. For any X Î
C let W^{o}(X)
be the subsite of C consisting of effective open subobjects of X.
We say X is a local object of
C if the Dsubsite W^{o}(X)
of C is a local site. Suppose X, Z are two local objects.
A morphism f: Z ®
X is called local if (3.2.2.b)
holds for any object Y of W^{o}(Z).
Denote by Loc(C) the subsite of local objects of C
with local morphisms. It is the largest local Dsubsite of C.
Clearly we have Loc(DSp) = DLSp.
