fiber products of schemes

One may easily see that the definition of the fiber product is the ``opposite'' of the property of the tensor products shown in the previous subsection. In more accurate terms,

LEMMA 6.4   Fiber products always exists in the category (Affine Schemes) of affine schemes.

Namely, let $X=\operatorname{Spec}(A_1),Y=\operatorname{Spec}(A_2),Z=\operatorname{Spec}(B)$ be affine schemes. Assume that morphisms $f: X\to Z$ and $g:Y\to Z$ are given. Then we have

$$\operatorname{Spec}(A_1)\times_{f,\operatorname{Spec}(B),g}\opera... ...Spec}(A_2) \cong \operatorname{Spec}(A_1 \otimes_{\Gamma(f),B,\Gamma(g)} A_2)$$

(If the morphisms and homomorphisms involved are clear from the context, we often abbreviate the above equation as:

$$\operatorname{Spec}(A_1)\times_{\operatorname{Spec}(B)}\operatorname{Spec}(A_2) \cong \operatorname{Spec}(A_1 \otimes_{B} A_2)$$

by the abuse of language.)

REMARK 6.5   By using ``gluing lemma'' for schemes, we may also prove that fiber products always exists in the category of schemes. We omit the proof. See for example [11] for details. (The author (Tsuchimoto) has often forgot to say (sorry), but of course, Grothendieck's enormous works including EGA1[4],EGA2[5],EGA3[6][7] EGA4[8][9][10] and SGA are the primary source for the whole of this talk.)

So far, we have not developed enough theory of a general schemes except for the affine case. In local theories, the affine case suffices and the generalization to general schemes is fairly easy. Due to the lack of time, we omit detailed arguments. For more detailed account, see EGA or Iitaka [11]

Note that the universality of the fiber product may be interpreted as the following way.

LEMMA 6.6   Let $\mathcal{C}$ be a category. Let $X,Y,Z \in \operatorname{Ob}(\mathcal{C})$ , $f\in \operatorname{Hom}_{\mathcal{C}}(X,Z), g\in \operatorname{Hom}_{\mathcal{C}}(Y,Z)$ . Assume that the fiber product $X\times_Z Y$ exists. Then we have

$$\operatorname{Hom}(W_1, X)\times_{\operatorname{Hom}(W_1,Z)}\operatorname{Hom}(W_1,Y) \cong \operatorname{Hom}(W_1,X\times_Z Y)$$

COROLLARY 6.7   Let $X,Y,Z$ be schemes. Let $f: X\to Z, g: Y\to Z$ be morphisms. Then we have

$$X(K)\times_{ Z(K)}Y(K) \cong (X\times_Z Y)(K)$$

for any field $K$ . (Recall that for a scheme $X$ , $X(K)$ denotes the set of $K$ -valued points of $K$ .)