3.3 k-Spaces 

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 aX = X × {a} of X. We denote by kX the set of all constant sections of X; kX is a field isomorphic to k. We often identify kX with k

Definition 3.3.2.k-geometry (or simply geometry) on X is a subset O of S(X) having the following properties: 
(g1) 1X 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 kX. We say O is full if we have the following: 
(g1') kX 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 GX(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 {si | i Î L} Í O, then s = È iÎL (si - si) Î O, so D(s) = ÈiÎL D(si) Î t(O). 
(c) X = D(OX) Î t(O), and Æ = D(1/0X) Î t(O). 

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

Example 3.3.7. Suppose X is a k-space and Y Í X. Let OX|[Y] = {s Ç (Y × k) | s Î OX}. Define OX|Y = GY(OX|[Y]). Then (Y, OX|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, OX) and (Y, OY) 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, 1k): X × k ® Y × k

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

Proposition 3.3.9. A morphism f: X ® Y is continuous. 

Proof. Suppose U Î t(OY) is open, then U = D(s) for some s Î OY. Since f is a morphism, f*(s) Î OX, so f-1(U) = D(f*(s)) Î t(OX) 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 pX: X ×Z Y and pY: X ×Z Y ® Y

Let TZY = pX*(OX) È pY*(OY). Denote by OZY = G(TZY) the k-geometry generated by TZY. Consider the k-space (X ×Z Y, OZY). Since pX*(OX) È pY*(OY) Í OZY, pX: X ×Z Y ® X and pY: X ×Z Y ® Y are morphisms of k-spaces. 

Proposition 3.3.10. (X ×Z Y, OZY) together with the morphism pX and pY 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 pXh = u and pYh = v. It suffices to prove that h is a morphism of k-spaces: 

= h*(G(pX*(OX) Ç pY*(OY))) 
Í G(h*(pX*(OX)) È h*(pY*(OY))) 
= G(u*(OX) È v*(OY)) 

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

Proof. Suppose {Xi} is a collection of k-spaces. Let W = Õ Xi be the product of {Xi} in the category of sets with the projection maps pi: W ® Xi. Let OW be the k-geometry on Õ Xi generated by the union of pi*(OXi). Then clearly (W, OW) is the product of {Xi} in k-Space. Since fibre products exist in k-Space (3.3.10), limits exist in k-Space
For coproducts let T = È Xi be the union of Xi with the injection qi: Xi ® T. Let OT be the set of sections s on T such that qi*(s) Î OXi for all i. Then OT is a k-geometry on T and (T, OT) is obviously the coproduct of {Xi} 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 qX: X ® E and qY: Y ® E. Let OE be the set of sections of E such that qX*(s) Î OX and qY*(s) Î OY. Then OE is a k-geometry on E and (E, OE) 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 pX-1(U) Ç pY-1(V) of X ×S Y (considered as k-spaces over W). 
(b) OWV = OSY|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 si of An by 
si = {(p, ai) Î An × k | p = (a1,...,an) Î An}. 
Let OAn = G({s1,...,sn} È kX). 

Definition 3.3.16. (An, OAn) is called the affine space of dimension n. OAn is the affine geometry on An; t(OAn) 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 Ui (called affine open cover) such that each subspace Ui 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 - Èi=1,...,n (D(si - si+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 Am are two closed subspaces of affine spaces. Then by (3.3.12.c) the product space X × Y is a closed subspace of An × Am = An+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 {Zk} is an affine open cover of Z and {Xi} (resp. {Yj}) is an open affine cover of X (resp. Y) such that f(Xi) Í Zk for some k (resp. g(Yj) Í Zk for some k). These affine varieties Xi ×Zk Yj 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 {Xi}, {Yj} and {Zk} are finite open cover, then {Xi ×Zk Yj} 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 

OAn(F) = G({s1,...,sn} È FX). 

Definition 3.3.22. (An, OAn(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,...,xn) is a point of An we write F[x] for the ring F[x1,...,xn] generated by {x1,...,xn} over F, and F(x) for the field F(x1,...,xn). 

Remark 3.3.23. Any irreducible F-closed set V of An(F) has at least one generic point x = (x1,...,xn) (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 Ui 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. 

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