The purpose of this section is to show that all the basic sites and basic algebraic sites have fibre products. Let C be a category. Recall that the product of two objects X and Y in C is a triple (X × Y, p, q), where X × Y is an object, p: X × Y ® X and q: X × Y ® Y are morphisms, such that given any object P and any morphisms u: P ® X and v: P ® Y, there exists a unique morphism h: P ® X × Y such that ph = u and qh = v; the morphism h is also denoted by (u, v). If X and Y are two objects over an object S, then the fibre product of X any Y over S is the product of Sobjects X and Y in the category C_{/S} of Sobjects, denoted X ×_{S} Y. The fibre product X ×_{S} Y can also be regarded as an object over Y, via the projection q, denoted X_{Y}, which is called the result of extending the base object of X from S to Y. If f: X ® X' is an Smorphism, we shall write f_{Y} for the induced morphism X_{Y} to X'_{Y}. Example 4.2.1. Fibre products exist in W(X) and w(X) for any topological spaces X. Example 4.2.2. In Set fibre products exist. Suppose f: X ® S and g: Y ® S are two maps of sets. Denote by W the subset of the cartesian product X × Y consisting of the pairs (x, y) such that f(x) = g(x). Let p, q be the restrictions of the projections X × Y ® X and X × Y ® Y on W respectively. Then (W, p, q) is clearly a fibre product of X and Y over S. Example 4.2.3. In Top fibre products exist. Suppose f: X ® S and g: Y ® S are two continuous maps of topological spaces. Let (X ×_{S} Y, p, q) be the fibre product of X and Y over S in the category of sets. Give X ×_{S} Y the smallest topology such that p and q are continuous maps. Then the topological space X ×_{S} Y together with the continuous maps p and q is a fibre product of X and Y over S in Top. Example 4.2.4. In RPot fibre products exist. Suppose f: spot A ® spot C and g: spot B ® spot C are two morphisms of ringed points, where A, B and C are rings. Then spot A Ä_{C} B is a fibre product of spot A and spot B over spot C. This is because the category RPot is equivalent to the opposite of Ring, and A Ä_{C} B is a fibre sum of A and B over C in Ring. Example 4.2.5. Similarly fibre products exist in the site AVar/k of affine varieties over an algebraically closed field k, since AVar/k is equivalent to the opposite of the category of finitely generated reduced algebras over k. Let f: X' ® X, g: Y' ® Y be morphisms over an object S. Let (X' ×_{S} Y', p', q') and (X ×_{S} Y, p, q) be the fibre products over S. Then fp': X' ×_{S} Y' ® X and gq': X' ×_{S} Y' ® Y induces a morphism (fp', gq')_{S}: X' ×_{S} Y' ® X ×_{S} Y, denoted simply by f ×_{S} g. (note that if Y = Y' = S', and g = 1_{S}', then f ×_{S} 1_{S}' = f_{S}'). Proposition 4.2.6. Suppose fibre
products exist in C. Let X, Y, Z, S' be Sobjects. Then
Proof. Note that (f) follows from (d) and (e). The other assertions can be verified easily. We now consider fibre products in a metric site C. Proposition 4.2.7. Suppose i: X ® S and j: Y ® S are two morphisms in a site C. Suppose U Í X, V Í Y, and W Í S are effective subsets of X, Y and S respectively such that i(U) Í W and j(V) Í W. Suppose the fibre product X ×_{S} Y of X and Y over S exists. Let E = p^{1}(U) Ç q^{1}(V). Suppose E is effective. The subobject E together with the morphism p_{E} and q_{E} is the fibre product U ×_{W} V of subobjects U and V over W, which is isomorphic to U ×_{S} V. If U and V are open (resp. closed) subobjects of X and Y respectively, then E is an open (resp. closed) subobject of Z. Proof. Let P be an object over W and let u: P ® U and v: P ® V be morphisms over W. Composing u, v with the effective morphisms: U ® X, V ® Y, we obtain two morphisms u': P ® X and v': P ® Y over S, which determines a morphism h = (u', v')_{S}: P ® X ×_{S} Y, with ph(P) = u(P) Í U and qh(P) = v(P) Í V, thus h(P) Í p^{1}(U) Ç q^{1}(V) = E. The morphism h induces a morphism from P to the effective subobject E having the required properties. On the other hand, p_{E} and q_{E} are clearly morphisms over W. Thus (E, p_{E}, q_{E}) is the fibre product of U and V. Now taking W = S, one has the isomorphism E = U ×_{S} V. The last assertion is obvious. Proposition 4.2.8. Suppose I: D ® C is a covariant functor of categories and J: C ® D is a right adjoint of I. Suppose X, Y are Sobjects with the structure morphisms i: X ® S and j: Y ® S in C. Suppose (Z, p, q) is the fibre product of X and Y over S in C. Then (J(Z), J(p), J(q)) is the fibre product of J(X) and J(Y) over J(S) in D, i.e, J preserves fibre products. Proof. Let W be an object of D over J(S). Then Theorem 4.2.9. Let C be an everywhere effective site and D an embedded subsite of C. Suppose fibre products exist in C. Then fibre products exist in D. Proof. Let J: C ®
D be the right adjoint of the inclusion functor D ®
C. Suppose X and Y are objects over S
in D with the structural morphisms i: X ®
S and j: Y ® S.
Let (Z, p, q) be the fibre product of X and
Y over S in C.
If D satisfies the conditions of (4.1.5), we can describe the fibre product of X and Y over S in D more explicitly: Theorem 4.2.10. Let D be a coreflective everywhere effective subsite of an everywhere effective site C satisfying the conditions of (4.1.5). Suppose X and Y are objects over S in D with the structural morphisms i: X ® S and j: Y ® S. Let (Z, p, q) be the fibre product of X and Y over S in C. Let J(Z) be a left associate object of Z in D and let r: J(Z) ® Z be the counit. Let W = {z Î J(Z) ½ (pr)_{z} and (qr)_{z} are in D}. Let p', q' be the restrictions of pj and qj on the subobject W of J(Z) respectively. Then (W, p', q') is the fibre product of X and Y over S in D. Proof. Note that p' and q' are morphisms in D by (4.1.5.b). Suppose P is an Sobject of D and u: P ® X, v: P ® Y are morphisms of D over S. Since (Z, p, q) is the fibre product of X, Y over S in C, we obtain a morphism h = (u, v)_{S}: P ® Z. Since P is an object of D, by the universal property of J(Z) there exists a unique morphism h': P ® J(Z) such that rh' = h. But u, v are morphisms of D, and h'(P) Í W, thus we may regard h' as a morphism from P to W in D, which has the property that p'h' = u and q'h' = v. This proves that (W, p', q') is the fibre product of X and Y over S in D. Proposition 4.2.11. (a) Fibre products
exist in the site RSp of ringed spaces.
Proof. (a) Suppose (X, O_{X}) and (Y,
O_{Y}) are ringed spaces over a ringed space (S,
O_{S}) with the structural morphisms f: X
® S and g: Y ®
S. Let (Z, p, q) be the fibre product of the
spaces X and Y over S in the category of topological
spaces. Let O_{Z} be the tensor product of p^{1}(O_{X})
and q^{1}(O_{Y}) over (fp)^{1}(O_{S})
(these are sheaves over Z). For each z = (x, y)
Î Z we have O_{Z}_{,z}
= O_{X},_{x} Ä_{OS},_{s}
O_{Y},_{y} where s = p(x) =
q(y). The canonical maps O_{X}_{,x}
® O_{Z},z determines a morphism
p#: O_{X} ®
O_{Z}, thus we obtain a projection morphism (p, p#):
(Z, O_{Z}) ®
(X, O_{X}). Similarly we define the projection morphism
(q, q#): (Z,
O_{Z}) ®
(Y, O_{Y}). Using the universal property of tensor
products it is easy to see that ((Z, O_{Z}), (p,
p#), (q, q#))
is the fibre product of X and Y over S.
Corollary 4.2.12.
(a) The forgetful functor RSp d Top preserves fibre
products.
Proof. These assertions are obvious by the explicit construction of X ×_{S} Y given in (4.2.11). Note that (d) follows from (c). Theorem 4.2.13. Suppose C' is a metric site with fibre products. Suppose C is a full, subsite of C' such that for any objects X, Y over S in C, the fibre product X ×_{S} Y in C' is an object of C. Then the Cauchycompletion E(C) of C in C' also has this property. Fibre products exist in E(C). Proof. Suppose X, Y and S are objects in E(C), and f: X ® S, g: Y ® S are two morphisms. Let (X ×_{S} Y, p, q) be the fibre product of X and Y over S in C'. For any z in X ×_{S} Y let W be an open neighborhood of fp(z) in S such that W Î C. Since f^{1}(W) and g^{1}(W) are open subsets of X and Y respectively, we can find sufficiently small open neighborhoods U Í f^{1}(W) of p(z) and V Í g^{1}(W) of p(z) such that U and V are objects of C. Then the fibre product U ×_{W} V of U and V over W is an object of C by assumption, which is an open subobject of X ×_{S} Y containing z (4.2.7). This proves that X ×_{S} Y Î E(C), which means that X ×_{S} Y is a fibre product of X and Y over S in E(C). Theorem 4.2.14. If C is a standard site with fibre products (resp. products), then any Cauchycompletion C' of C has fibre products (resp. products). Proof. We give two proofs for fibre products. (The
case of products is similar.) Suppose C has fibre products.
Theorem 4.2.15.
(a) Suppose X = Spec A, Y = Spec B are affine schemes over an affine
scheme S = Spec C. Then Spec A Ä_{C}
B is the fibre product of X and Y over S in
the category of local ringed spaces.
Proof. (a) Denote by Ring^{op} the
opposite of the category of rings (commutative with unit). Then the functor
J: Ring^{op} ®
LSp given by A ®
Spec A for any ring A is a right adjoint of the functor LSp
® Ring^{op}
given by X ®
G(X) for any local ringed space X. Thus the assertion
follows from (4.2.8).
Similarly we can prove the following theorem concerning prevarieties over an algebraically closed fields k (using (4.2.5)). Theorem 4.2.16. Suppose X and Y are affine kvarieties (resp. kprevarieties) over an affine kvariety (resp. kprevariety) S. Then the fibre product of X and Y over S in the site of geometric spaces over k is an affine kvariety (resp. kprevariety). Remark 4.2.17. (4.2.15.c) was proved in [EGA I, Ch. I, (3.2.1)] by a different method, since the existence of fibre products of local ringed space was not established in [EGA]. Proposition 4.2.18.
Suppose D is a generic subsite of a presite C.
Proof. (a) The proof of (a) is similar to the proof
of (4.2.7).
In a standard site if {U_{i}} is a disjoint open cover of an object X, then X is a sum of the subobjects U_{i}. If {X_{i}} is any collection of objects in a complete site C, the disjoint union of the X_{i} exists. This implies that in a complete site arbitrary sum exist. Remark 4.2.19. Suppose D is a generic subsite of a complete site C. Then any Dset may be viewed as a discrete object of C as a disjoint union of spots in D. Thus D/Set is an isogenous coreflective subsite of C. (This is the reason behind the facts that RSet, LSet, and GSet are coreflective subsite of RSp, LSp and GSp respectively.) Remark 4.2.20. If C is a Cauchycomplete site and D a generic subsite of C then (4.2.18a & b) follows easily from the fact that D/Set is naturally an embedded subsite of C (4.2.19). We see that D/Set has fibre products if C does by (4.2.9) and the coreflector J = f: C ® D/Set preserves fibre products. Example 4.2.21. Fibre products exist in RSet, LSet and GSet. The functors f: RSp ® RSet, f: LSp ® LSet, and f_{k}: LSp ® GSet preserve fibre products. Note that for LSet this has been proved in (3.4.4). Example 4.2.22.
If D is a category has fibre products, then D/Set
is a Cauchycomplete site with fibre products defined pointwise, as in
the case of RSet. On the other hand, even fibre products doest not
exist in D, D/Set may still have fibre products, as
in the case of D = GPot and D/Set = GSet.
