In this section we define the category of local ringed models, which is a Cauchy-complete site containing the metric site LSp of local ringed spaces as a subsite. 

Recall that a local ringed set is a discrete local ringed space X. If X and Y are two local ringed sets, a morphism (f, f^#) from X to Y is a morphism as local ringed space. The category of local ringed sets is denoted by LSet. 

Remark 3.4.1. Alternatively, one can define a local ringed set as a set X such that for any x Î X a local ring O_x is attached. A morphism of local ringed sets consists of a map f: X ® Y, and for each x Î X, a local homomorphism f_x^#: O_f(x) ® O_x. 

Remark 3.4.2. LSet is an isogenous coreflective subsite of LSp with the coreflector j sneding each local ringed space to the local ringed set b(X) = {O_x} consisting of the local rings of the points of X. 

Example 3.4.3. If A is a ring we write Spec A for the local ringed set {A_p | p e Spec A}. 

Proposition 3.4.4. Limits and colimits exist in LSet. 

Proof. Suppose f: X ® Z and g: Y ® Z are two morphisms of local ringed sets. For any x Î X, y Î Y and z Î Z such that f(x) = z = g(y), let x ×_z y be the set of points w of Spec O_x Ä_Oz O_y over x e Spec O_x and y e Spec O_y by the projections. Then x ×_z y is the fibre product of x and y over z in LSet. The union X ×_Z Y of all x ×_z y is then the fibre product of X and Y over Z in LSet. Thus fibre products exist in LSet. Similarly we can prove that arbitrary products exist in LSet using the notion of (infinite) tensor product of rings. Hence limits exist in LSet. 
Clearly the union of a set of local ringed sets is the coproduct of the local ringed sets. Thus to prove that colimits exist in LSet it suffices to prove that cokernels exist in LSet. Suppose f, g: Z ® Y are two morphisms of local ringed sets. Let X be the cokernel of f and g in the category of sets with the canonical projection p: Y ® X. For each x e X let U = p^-1(x) and V = f^-1(U) = g^-1(U). Let O_x be the kernel of the maps f^#, g^#: G(V) = P_zÎZ O_z ® G(U) = Õ_yÎY O_y. We obtain a ringed set X. That O_x is a local ring follows from the following assertion (b): 
(a) If w, w' Î U and there is a z Î Z such that f(z) = w, g(z) = w', then the fibre product of f_z^#: O_w ® O_z and g_z^#: O_w' ® O_z is local. Thus for any a Î U, a_w is a unit of O_w if and only if a_w' is a unit of O_w'. 
(b) For any two points y, y' of U there is a finite set y = w₀, w₁,..., y' = w_n such that for each pair (w_i, w_i+1) there is a z_i Î Z such that f(z_i) = w_i and g(z_i) = w_i+1. Then (a) implies that for any a Î U, a_y is a unit of O_y if and only if a_y' is a unit of O_y'. 
Thus we obtain a local ringed set X which is clearly the cokernel of f and g in LSet. 

Let X be a local ringed set and let X^ = È_xÎX O_x be the disjoint union of the sets O_x. A section of X is a subset s of X such that s(x) = O_x Ç s consists of at most one element for any x e X; the set D(s) = {x Î X | s(x) ¹ Æ} is called the domain of s. Denote S(X) the set of sections of X. 

We define the rational operations on S(X): for any s, t Î S(T) let 
s ± t = {s(x) ± t(x) | x Î D(s) Ç D(t)}; 
s.t = {s(x).t(x) | x Î D(s) Ç D(t)}; 
s/t = {s(x)/t(x) | x Î D(s) Ç D(t) and t(x) Ï m_x}; 
these are sections of X. 

Definition 3.4.5. A geometry on a local ringed set X is a subset O of S(X) having the following properties: 
(G1) 1_X = {1_x Î O_x|x Î X} is in O. 
(G2) O is closed under rational operations. 
(G3) Any section of X which is the union of the elements of a subcollection of O is in O. 
If furthermore we have 
(G4) s Ç t Î O for any s, t Î O and X^ = È_sÎO s, 
then we say that O is exact. 

Definition 3.4.6. Suppose T is any subset of S(X). The intersection of all the geometries on X containing T is a geometry on X, denoted by G_X(T) (or simply G(T)), called the geometry on X generated by T. 

For any subset T of S(X) let t(T) = {D(s) | s Î T}. 

Proposition 3.4.7. t(O) is a topology on X for any geometry O on X. 

Proof. (a) If s, t Î O, then s + t Î O, so D(s + t) = D(s) Ç D(t) is in O. 
(b) If {s_i | i Î L} Í O, then s = È_iÎL (s_i - s_i) Î O, so D(s) = È_iÎL D(s_i) Î t(O). 
(c) X = D(O_X) Î t(O), and f = D(1/O_X) Î t(O) where 0_X = 1_X - 1_X. 

Definition 3.4.8. A local ringed model is a pair (X, O_X) consisting of a set X and a geometry O_X on X; the topological space (X, t(O_X)) is the underlying topological space of (X, O_X); any s e O_X is called a regular section. For simplicity we often write X for (X, O_X). We say (X, O_X) is exact if the geometry O_X is exact. 

Example 3.4.9. Suppose X is a local ringed model and Y Í X. Let O_X|_[Y] = {s Ç Y^ | s Î O_X}. Define O_X|_Y = G_Y(O_X|_[Y]). Then (Y, O_X|_Y) (or simply Y) is called a submodel of X. 

To make the collection of local ringed models into a category we need the notion of a morphism of local ringed models. Suppose (X, O_X) and (Y, O_Y) are two local ringed models. 

Definition 3.4.10. A morphism of local ringed models is a morphism of local ringed sets f: X ® Y such that for any regular section s of Y, the section 
f^*(s) = {f_x^*(s(f(x))) | x Î X and f(x) Î D(s)} 
is a regular section of X. 

Proposition 3.4.11. A morphism f: X ® Y of local ringed models is continuous. 

Proof. Suppose U Î t(O_Y) is open. Then U = D(s) for some s Î O_Y. Since f is a morphism, f ^*(s) Î O_X. Thus f^-1(U) = D(f ^*(s)) Î t(O_X) is open. Hence f is continuous. 

We obtain the category of local ringed models, denoted by LMod. LMod is a Cauchy-complete metric site. 

Proposition 3.4.12. Limits and colimits exist in LMod. 

Proof. The proof is similar to that for k-spaces (see (3.3.11), replacing the category of sets by LSet, as we have proved that limits and colimits exist in LSet (3.4.4). 

Remark 3.4.13. Suppose X is a local ringed model. For any open set U of X let O(U) = {s Î O_X | D(s) = U}. Then O(U) is a ring. The function U ® O(U) is a sheaf of rings on X, called the structure sheaf of X, denoted by O. Thus we obtain a ringed space b(X) = (X, O). 
(a) b(X) is a local ringed space by (3.4.5,G2). 
(b) For each x Î X, O_x is isomorphic to the local ring of b(X) at x if and only if (X, O_X) is exact. 
(c) The subcategory of exact local ringed models is equivalent to the category of local ringed spaces. Thus we may regard LSp as a subcategory of LMod. More precisely, we have 

Proposition 3.4.14. LSp is a reflective isometric subsite of LMod with the reflector b: LMod ® LSp. LSp has limits and colimits. 

Proof. The first assertion is obvious. That colimits exist in LSp follows from the fact that colimits exist in LMod and b: LMod d LSp preserves colimits as a left adjoint functor. To see that limits exist in LSp it is sufficient to prove that it is closed under fibre products and infinite products in LMod. These can be done by verifying the condition (3.3.5, G4). We omit the details because we shall give another proof of the existence of fibre products of local ringed spaces later in (4.3). 

Example 3.4.15. Using the notion of local ringed models we can give a very simper definition of affine schemes, similar to the early definition of affine spaces (3.3.16). Let A be a ring. Let Spec A = {A_p | p Î Spec A}. Define a map s: A ® S(Spec A) by s(a) = {a/1 Î A_p | p e Spec A} for any a Î A. Let O_A = G(s(A)); O_A is an exact geometry on spec A. We obtain an exact local ringed model, called the affine model of A. An affine scheme is then a local ringed model which is isomorphic to the affine model of some ring. A scheme is a local ringed model X in which every point has an open neighborhood (called affine neighborhood) such that the submodel U is an affine scheme. 

Remark 3.4.16. Since LMod is Cauchy-complete, it can be used to define the completion of strict metric sites which can be isometrically embedding into LMod, such as the Zariski site Ring^op. 

Remark 3.4.17. Suppose X is a local ringed models. For any open subset U of X a regular section s Î O(U) may be viewed as a morphism form the submodel U to the affine model Spec Z[T] of the ring Z[T] of one variable over the ring Z of integers. Hence s is also called a function. Thus a local ringed model may also be defined as a local ringed set X equipped with a topology and a sheaf O of functions on the space X such that (X, O) is a local ringed spaces. A morphism of local ringed models is then a continuous morphism of local ringed sets f: X d Y such that the pull back of regular functions is a regular functions. In this sense the notion of local ringed models is a direct generalization of the usual notion of function spaces defined over a field. On the other hand, our definitions of local ringed models and morphisms between them do not involve the underlying topology. In fact, the underlying topology is derived as an invariant of the regular functions. This makes it easy to define new objects. For instance to define the fibre products of k-spaces or local ringed models and to verify the universal property one only need to consider regular functions. The whole procedure is then completely straightforward and conceptional. 

Remark 3.4.18. Another advantage of LMod is that it provides a more "faithful" representation of algebraic spaces or algebraic functors. To see this first note that we have natural embeddings: 

Example 3.4.19 (cf. [K, p9-10]). Let C be the complex numbers and let U be the scheme obtained by taking two copies of the affine line L₁ = Spec C[s], L₂ = Spec C[t] and identifying the points s = 0 and t = 0, dented by p. Let R consists of one copy of U and a scheme U' obtained from U by deleting the point p. We obtain two maps p₁, p₂: R ® U: 

The coequalizer Z of h_R ® h_U induced by p₁ and p₂ in the category of sheaves on Ring^op = ASch in the étale topology is an algebraic space whose representation is X in LSp and X ' in LMod. We see that by switching to LMod we are able to get more information at "bad points" of algebraic spaces.