In this section we fix a field k. We introduce the notion of a kspace 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 kspaces. 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 ksection (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 kgeometry
(or simply geometry) on X is
a subset O of S(X) having the following properties:
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:
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.
Definition 3.3.6. A kspace is a pair (X, O_{X}) consisting of a set X and a kgeometry 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 kspace 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 kspace into a category we need the
notion of a morphism of kspaces. Suppose (X, O_{X})
and (Y, O_{Y}) are two kspaces, and f:
X ® Y is a map. Then f
induces a map f^{*}: S(Y) ®
S(X) given by
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 kSpace the category of kspaces. It is a Cauchycomplete geometric site. We prove that limits and colimits exist in kSpace. Suppose X and Y are two kspaces over a kspace 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 kgeometry generated by T_{X×ZY}. Consider the kspace (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 kspaces. 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 kspaces. Proof. Suppose T is a kspace 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_{X}h = u and p_{Y}h = v. It suffices to prove that h is a morphism of kspaces: h^{*}(O_{X×ZY})
Proposition 3.3.11. Limits and colimits exist in the category of kspaces. Proof. Suppose {X_{i}} is a collection of kspaces.
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 kgeometry
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 kSpace. Since fibre products exist in kSpace
(3.3.10), limits exist in kSpace.
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)):
Definition 3.3.13. A kspace 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.
3.3.15 We now use the notion of kspaces
to define an algebraic variety in the sense of Serre. First we define affine
spaces over k. For any n > 0 let A^{n} = k
× ... × k be the set of all ntuples of elements
of k. For each i = 1,...,n we define a section s_{i}
of A^{n} by
Definition 3.3.16. (A^{n}, O_{A}n) is called the affine space of dimension n. O_{A}n is the affine geometry on A^{n}; t(O_{A}n) is the Zariski topology on A^{n}. Definition 3.3.17. An affine variety is a kspace which is isomorphic to a closed subspace of an affine space over k. A prevariety is a kspace which has an open cover U_{i} (called affine open cover) such that each subspace U_{i} is an affine kvariety. A variety (in the sense of Serre) is a kprevariety 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 A^{n} is separated. Clearly we have A^{n} × A^{n} = A^{2n} as kspaces. The diagonal is then the closed set A^{2n}  È_{i=1,...,n} (D(s_{i}  s_{i+n})). This shows that A^{n} 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 kspaces. 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.
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_{A}n(F) = G({s_{1},...,s_{n}} È F_{X}). Definition 3.3.22. (A^{n}, O_{A}n(F)) is called the Faffine space of dimension n, denoted simply by A^{n}(F). A closed subset V of A^{n}(F) is called an Fclosed set. If x = (x_{1},...,x_{n}) is a point of A^{n} we write F[x] for the ring F[x_{1},...,x_{n}] generated by {x_{1},...,x_{n}} over F, and F(x) for the field F(x_{1},...,x_{n}). Remark 3.3.23. Any irreducible Fclosed set V of A^{n}(F) has at least one generic point x = (x_{1},...,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 Faffine variety is a kspace which is isomorphic to an irreducible closed subspace V of an Faffine space A^{n}(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 kspace X having a finite affine open cover U_{i} consisting of Faffine 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.
