In this section we fix a field k. We introduce the notion of a k-space which can be used to define manifolds, (reduced) complex analytic spaces, or algebraic varieties in the sense of Serre or Weil. The main advantage of our approach is that it is very easy to define limits and colimits of k-spaces. Later in (3.4) we will develop the theory of local ringed models following the same line. The notation and the results of these two sections will not be used elsewhere in these notes. 

Definition 3.3.1. Suppose X is a set. A k-section (or simply section) of X is a subset s of the cartesian products X × k such that for any x Î X the set s(x) = s Ç ({x} × k) consists of at most one element of s; the subset D(s) = {x Î X | s(x) ¹ Æ} of X is called the domain of s. 

Denote by S(X) the set of all sections of X. 

We introduce the rational operations +, -, ×, ÷ on S(X). For any s, t Î S(X), let 

s ± t = {(x, a) | x Î D(s) Ç D(t), a = s(x) ± t(x)}; 

s.t = {(x, a) | x Î D(s) Ç D(t), a = s(x).t(x)}; 

s/t = {(x, a) | x Î D(s) Ç D(t), t(x) ¹ 0, a = s(x)/t(x)}; 

these are sections of X. 

Any element a Î k determines a constant section a_X = X × {a} of X. We denote by k_X the set of all constant sections of X; k_X is a field isomorphic to k. We often identify k_X with k. 

Definition 3.3.2. A k-geometry (or simply geometry) on X is a subset O of S(X) having the following properties: 
(g1) 1_X Í O. where 1 Î k is the identity of k. 
(g2) O is closed under the rational operations (i.e., if s, t Î O, then s ± t, s.t and s/t are in O). 
(g3) Any section of X which is the union of a subcollection of O is in O. 

Remark 3.3.3. From (g1) and (g2) it follows that a geometry O contains the prime field of k_X. We say O is full if we have the following: 
(g1') k_X Í O. 
Note that (g1) is essential for the definition of a variety in the sense of Weil. Besides this application, all the other results in this section are valued if we assume (g1') for O. 

Definition 3.3.4. 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 3.3.5. t(O) is a topology on X for any geometry O. 

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 Æ = D(1/0_X) Î t(O). 

Definition 3.3.6. A k-space is a pair (X, O_X) consisting of a set X and a k-geometry O_X on X; the topological space (X, t(O_X)) is the underlying topological space of (X, O_X); any s Î O_X is called a regular section or function. For simplicity we often write X for (X, O_X). 

Example 3.3.7. Suppose X is a k-space and Y Í X. Let O_X|_[Y] = {s Ç (Y × k) | s Î O_X}. Define O_X|_Y = G_Y(O_X|_[Y]). Then (Y, O_X|_Y) (or simply Y) is called a subspace of X. 

To make the collection of k-space into a category we need the notion of a morphism of k-spaces. Suppose (X, O_X) and (Y, O_Y) are two k-spaces, and f: X ® Y is a map. Then f induces a map f^*: S(Y) ® S(X) given by 
f^*(s) = f^-1(s) for any s Î S(Y), here f = (f, 1_k): X × k ® Y × k. 

Definition 3.3.8. A morphism from (X, O_X) to (Y, O_Y) is a map f: X ® Y such that f^*(O_Y) Í O_X. 

Proposition 3.3.9. A morphism f: X ® Y 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, so f^-1(U) = D(f^*(s)) Î t(O_X) is open. Hence f is continuous. 

Denote by k-Space the category of k-spaces. It is a Cauchy-complete geometric site. We prove that limits and colimits exist in k-Space. 

Suppose X and Y are two k-spaces over a k-space Z, with the structure morphisms f: X ® Z and g: Y ® Z. Let X ×_Z Y be the product of X and Y over Z in the category of sets with the projection p_X: X ×_Z Y and p_Y: X ×_Z Y ® Y. 

Let T_X×ZY = p_X^*(O_X) È p_Y^*(O_Y). Denote by O_X×ZY = G(T_X×ZY) the k-geometry generated by T_X×ZY. Consider the k-space (X ×_Z Y, O_X×ZY). Since p_X^*(O_X) È p_Y^*(O_Y) Í O_X×ZY, p_X: X ×_Z Y ® X and p_Y: X ×_Z Y ® Y are morphisms of k-spaces. 

Proposition 3.3.10. (X ×_Z Y, O_X×ZY) together with the morphism p_X and p_Y is the fibre product of X and Y over Z in the category of k-spaces. 

Proof. Suppose T is a k-space and u: T ® X, v: T ® Y are two morphisms such that fu = gv. Since X ×_Z Y is the fibre product of X and Y over Z in the category of sets, there is a unique map h: T ® X ×_Z Y such that p_Xh = u and p_Yh = v. It suffices to prove that h is a morphism of k-spaces: 

h^*(O_X×ZY) 
= h^*(G(p_X^*(O_X) Ç p_Y^*(O_Y))) 
Í G(h^*(p_X^*(O_X)) È h^*(p_Y^*(O_Y))) 
= G(u*(O_X) È v*(O_Y)) 
Í O_T. 

Proposition 3.3.11. Limits and colimits exist in the category of k-spaces. 

Proof. Suppose {X_i} is a collection of k-spaces. Let W = Õ X_i be the product of {X_i} in the category of sets with the projection maps p_i: W ® X_i. Let O_W be the k-geometry on Õ X_i generated by the union of p_i^*(O_Xi). Then clearly (W, O_W) is the product of {X_i} in k-Space. Since fibre products exist in k-Space (3.3.10), limits exist in k-Space. 
For coproducts let T = È X_i be the union of X_i with the injection q_i: X_i ® T. Let O_T be the set of sections s on T such that q_i^*(s) Î O_Xi for all i. Then O_T is a k-geometry on T and (T, O_T) is obviously the coproduct of {X_i} in k-Space. 
It remains to prove that fibre coproducts exist in k-Space. Suppose i: Z ® X, j: Z ® Y are two morphisms of k-spaces. Let E be the fibre coproduct of X and Y over Z in the category of sets with the canonical maps q_X: X ® E and q_Y: Y ® E. Let O_E be the set of sections of E such that q_X^*(s) Î O_X and q_Y^*(s) Î O_Y. Then O_E is a k-geometry on E and (E, O_E) is the fibre coproduct of X and Y over Z. 

Remark 3.3.12. Suppose U, V and W are subspaces of X, Y and S respectively and f(U) Í W and g(V) Í W. One can verify the following assertions directly (or using (4.2.7)): 
(a) The fibre product U ×_W V is identified canonical as the subset p_X^-1(U) Ç p_Y^-1(V) of X ×_S Y (considered as k-spaces over W). 
(b) O_U×WV = O_X×SY|_U×WV. 
(c) If U and V are open (resp. closed) then U ×_W V is naturally an open (resp. closed) subspace of X ×_S Y. 

Definition 3.3.13. A k-space X is separated if the image of the morphism D = (1, 1): X ® X × X (called the diagonal) is a closed subset of the product space X × X. 

Remark 3.3.14. (see (5.3)). (a) If the underlying topology of X is Hausdorff, then X is separated. 
(b) Subspaces of a separated space are separated. 
(c) The fibre product of separated spaces is separated. 

3.3.15 We now use the notion of k-spaces to define an algebraic variety in the sense of Serre. First we define affine spaces over k. For any n > 0 let Aⁿ = k × ... × k be the set of all n-tuples of elements of k. For each i = 1,...,n we define a section s_i of Aⁿ by 
s_i = {(p, a_i) Î Aⁿ × k | p = (a₁,...,a_n) Î Aⁿ}. 
Let O_An = G({s₁,...,s_n} È k_X). 

Definition 3.3.16. (Aⁿ, O_An) is called the affine space of dimension n. O_An is the affine geometry on Aⁿ; t(O_An) is the Zariski topology on Aⁿ. 

Definition 3.3.17. An affine variety is a k-space which is isomorphic to a closed subspace of an affine space over k. A prevariety is a k-space which has an open cover U_i (called affine open cover) such that each subspace U_i is an affine k-variety. A variety (in the sense of Serre) is a k-prevariety which is separated and has a finite affine open cover. 

Remark 3.3.18. Any open or closed subspace of an affine variety (resp. prevariety, resp. variety) is an affine variety (resp. prevariety, resp. variety.) 

Remark 3.3.19. An affine variety is a variety. To see this it is sufficient, by remark (3.3.14.b), to check that any affine space Aⁿ is separated. Clearly we have Aⁿ × Aⁿ = A²ⁿ as k-spaces. The diagonal is then the closed set A²ⁿ - È_i=1,...,n (D(s_i - s_i+n)). This shows that Aⁿ is separated. 

Theorem 3.3.20. The fibre product of affine varieties (resp. prevarieties, resp. varieties) is an affine variety (resp. prevariety, resp. variety) 

Proof. Suppose f: X ® Z and g: Y ® Z are two morphisms of k-spaces. Then X ×_S Y is also the fibre product of Z and Y × X over Z × Z with the structure morphism D: X ® X × X and f × g: X × Y ® Z × Z. If D: X ® X × X is a closed embedding, then the base extension X ×_Z Y ® Z × Y is also a closed embedding. 
(a) Suppose X, Y and Z are affine varieties. We may assume that X Í Aⁿ and Y Í A^m are two closed subspaces of affine spaces. Then by (3.3.12.c) the product space X × Y is a closed subspace of Aⁿ × A^m = A^n+m. Thus X × Y is an affine variety. Since X ×_Z Y is a closed subspace of X × Y, X ×_Z Y is an affine variety. 
(b) Suppose X, Y and Z are prevarieties. Suppose {Z_k} is an affine open cover of Z and {X_i} (resp. {Y_j}) is an open affine cover of X (resp. Y) such that f(X_i) Í Z_k for some k (resp. g(Y_j) Í Z_k for some k). These affine varieties X_i ×_Zk Y_j form an open cover of X ×_Z Y. This shows that X ×_Z Y is a prevariety. 
(c) If the X, Y and Z in part (b) are varieties and {X_i}, {Y_j} and {Z_k} are finite open cover, then {X_i ×_Zk Y_j} is a finite affine open cover of X ×_Z Y. Since X ×_Z Y is separated by (3.3.14.c), it is a variety. This finishes the proof. 

3.3.21 Finally we define an abstract variety in the sense of Weil. A subfield F of k is called a base field of k if k/F has infinite transcendence degree. We say k is a universal field if it is algebraically closed and the prime field of k is a base field. (Note that k is not a base field of k.) 

Suppose k is a universal filed. For any base field F of k we define 

O_An(F) = G({s₁,...,s_n} È F_X). 

Definition 3.3.22. (Aⁿ, O_An(F)) is called the F-affine space of dimension n, denoted simply by Aⁿ(F). A closed subset V of Aⁿ(F) is called an F-closed set. 

If x = (x₁,...,x_n) is a point of Aⁿ we write F[x] for the ring F[x₁,...,x_n] generated by {x₁,...,x_n} over F, and F(x) for the field F(x₁,...,x_n). 

Remark 3.3.23. Any irreducible F-closed set V of Aⁿ(F) has at least one generic point x = (x₁,...,x_n) (i.e., V is the closure of x). The ring F[x] and the field K(x) are uniquely determined by V up to isomorphism over k. 

Definition 3.3.24. An F-affine variety is a k-space which is isomorphic to an irreducible closed subspace V of an F-affine space Aⁿ(F) with a generic point x e V such that F(x)/F is a regular extension. An abstract variety (defined over a base field F of k) in the sense of Weil is an irreducible, separated k-space X having a finite affine open cover U_i consisting of F-affine varieties. 

Remark 3.3.25. (a) Suppose X is an abstract variety of dim X > 0 in the sense of Weil. Then X has infinitely may generic points. 
(b) Suppose X is a variety of dim X > 0 in the sense of Serre. Then X has no generic point because any point of X is closed. 
(c) Suppose X is a variety in the sense of [H, p.105] (i.e., an integral separated scheme of finite type over k). Then X has a unique generic point.