Here is a review of the general definition, even though we have already shown that fibre products of schemes exist.
Definition 26.17.1. Given morphisms of schemes $f : X \to S$ and $g : Y \to S$ the fibre product is a scheme $X \times _ S Y$ together with projection morphisms $p : X \times _ S Y \to X$ and $q : X \times _ S Y \to Y$ sitting into the following commutative diagram which is universal among all diagrams of this sort, see Categories, Definition 4.6.1.
\[ \xymatrix{ X \times _ S Y \ar[r]_ q \ar[d]_ p & Y \ar[d]^ g \\ X \ar[r]^ f & S } \]
In other words, given any solid commutative diagram of morphisms of schemes
there exists a unique dotted arrow making the diagram commute. We will prove some lemmas which will tell us how to think about fibre products.
Lemma 26.17.2. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. If $X, Y, S$ are all affine then $X \times _ S Y$ is affine.
Proof. Suppose that $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(B)$ and $S = \mathop{\mathrm{Spec}}(R)$. By Lemma 26.6.7 the affine scheme $\mathop{\mathrm{Spec}}(A \otimes _ R B)$ is the fibre product $X \times _ S Y$ in the category of locally ringed spaces. Hence it is a fortiori the fibre product in the category of schemes. $\square$
Lemma 26.17.3. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $X \times _ S Y$, $p$, $q$ be the fibre product. Suppose that $U \subset S$, $V \subset X$, $W \subset Y$ are open subschemes such that $f(V) \subset U$ and $g(W) \subset U$. Then the canonical morphism $V \times _ U W \to X \times _ S Y$ is an open immersion which identifies $V \times _ U W$ with $p^{-1}(V) \cap q^{-1}(W)$.
Proof. Let $T$ be a scheme. Suppose $a : T \to V$ and $b : T \to W$ are morphisms such that $f \circ a = g \circ b$ as morphisms into $U$. Then they agree as morphisms into $S$. By the universal property of the fibre product we get a unique morphism $T \to X \times _ S Y$. Of course this morphism has image contained in the open $p^{-1}(V) \cap q^{-1}(W)$. Thus $p^{-1}(V) \cap q^{-1}(W)$ is a fibre product of $V$ and $W$ over $U$. The result follows from the uniqueness of fibre products, see Categories, Section 4.6. $\square$
In particular this shows that $V \times _ U W = V \times _ S W$ in the situation of the lemma. Moreover, if $U, V, W$ are all affine, then we know that $V \times _ U W$ is affine. And of course we may cover $X \times _ S Y$ by such affine opens $V \times _ U W$. We formulate this as a lemma.
Lemma 26.17.4. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $S = \bigcup U_ i$ be any affine open covering of $S$. For each $i \in I$, let $f^{-1}(U_ i) = \bigcup _{j \in J_ i} V_ j$ be an affine open covering of $f^{-1}(U_ i)$ and let $g^{-1}(U_ i) = \bigcup _{k \in K_ i} W_ k$ be an affine open covering of $g^{-1}(U_ i)$. Then is an affine open covering of $X \times _ S Y$.
\[ X \times _ S Y = \bigcup \nolimits _{i \in I} \bigcup \nolimits _{j \in J_ i, \ k \in K_ i} V_ j \times _{U_ i} W_ k \]
Proof. See discussion above the lemma. $\square$
In other words, we might have used the previous lemma to construct the fibre product directly by glueing the affine schemes. (Which is of course exactly what we did in the proof of Lemma 26.16.1 anyway.) Here is a way to describe the set of points of a fibre product of schemes.
Lemma 26.17.5. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Points $z$ of $X \times _ S Y$ are in bijective correspondence to quadruples where $x \in X$, $y \in Y$, $s \in S$ are points with $f(x) = s$, $g(y) = s$ and $\mathfrak p$ is a prime ideal of the ring $\kappa (x) \otimes _{\kappa (s)} \kappa (y)$. The residue field of $z$ corresponds to the residue field of the prime $\mathfrak p$.
\[ (x, y, s, \mathfrak p) \]
Proof. Let $z$ be a point of $X \times _ S Y$ and let us construct a quadruple as above. Recall that we may think of $z$ as a morphism $\mathop{\mathrm{Spec}}(\kappa (z)) \to X \times _ S Y$, see Lemma 26.13.3. This morphism corresponds to morphisms $a : \mathop{\mathrm{Spec}}(\kappa (z)) \to X$ and $b : \mathop{\mathrm{Spec}}(\kappa (z)) \to Y$ such that $f \circ a = g \circ b$. By the same lemma again we get points $x \in X$, $y \in Y$ lying over the same point $s \in S$ as well as field maps $\kappa (x) \to \kappa (z)$, $\kappa (y) \to \kappa (z)$ such that the compositions $\kappa (s) \to \kappa (x) \to \kappa (z)$ and $\kappa (s) \to \kappa (y) \to \kappa (z)$ are the same. In other words we get a ring map $\kappa (x) \otimes _{\kappa (s)} \kappa (y) \to \kappa (z)$. We let $\mathfrak p$ be the kernel of this map.
Conversely, given a quadruple $(x, y, s, \mathfrak p)$ we get a commutative solid diagram
see the discussion in Section 26.13. Thus we get the dotted arrow. The corresponding point $z$ of $X \times _ S Y$ is the image of the generic point of $\mathop{\mathrm{Spec}}(\kappa (x) \otimes _{\kappa (s)} \kappa (y)/\mathfrak p)$. We omit the verification that the two constructions are inverse to each other. $\square$
Lemma 26.17.6. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target.
If $f : X \to S$ is a closed immersion, then $X \times _ S Y \to Y$ is a closed immersion. Moreover, if $X \to S$ corresponds to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ S$, then $X \times _ S Y \to Y$ corresponds to the sheaf of ideals $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$.
If $f : X \to S$ is an open immersion, then $X \times _ S Y \to Y$ is an open immersion.
If $f : X \to S$ is an immersion, then $X \times _ S Y \to Y$ is an immersion.
Proof. Assume that $X \to S$ is a closed immersion corresponding to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ S$. By Lemma 26.4.7 the closed subspace $Z \subset Y$ defined by the sheaf of ideals $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$ is the fibre product in the category of locally ringed spaces. By Lemma 26.10.1 $Z$ is a scheme. Hence $Z = X \times _ S Y$ and the first statement follows. The second follows from Lemma 26.17.3 for example. The third is a combination of the first two. $\square$
Definition 26.17.7. Let $f : X \to Y$ be a morphism of schemes. Let $Z \subset Y$ be a closed subscheme of $Y$. The inverse image $f^{-1}(Z)$ of the closed subscheme $Z$ is the closed subscheme $Z \times _ Y X$ of $X$. See Lemma 26.17.6 above.
We may occasionally also use this terminology with locally closed and open subschemes.
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