The Stacks project

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25.17 Fibre products of schemes

Here is a review of the general definition, even though we have already shown that fibre products of schemes exist.

Definition 25.17.1. 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

\[ \xymatrix{ X \times _ S Y \ar[r]_ q \ar[d]_ p & Y \ar[d]^ g \\ X \ar[r]^ f & S } \]

which is universal among all diagrams of this sort, see Categories, Definition 4.6.1.

In other words, given any solid commutative diagram of morphisms of schemes

\[ \xymatrix{ T \ar[rrrd] \ar@{-->}[rrd] \ar[rrdd] & & \\ & & X \times _ S Y \ar[d] \ar[r] & Y \ar[d] \\ & & X \ar[r] & S } \]

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 25.17.2. 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 25.6.7 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 25.17.3. 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.

slogan

Lemma 25.17.4. 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

\[ X \times _ S Y = \bigcup \nolimits _{i \in I} \bigcup \nolimits _{j \in J_ i, \ k \in K_ i} V_ j \times _{U_ i} W_ k \]

is an affine open covering of $X \times _ S Y$.

Proof. See discussion above the lemma. $\square$

In other words, we might have used the previous lemma as a way of construction the fibre product directly by glueing the affine schemes. (Which is of course exactly what we did in the proof of Lemma 25.16.1 anyway.) Here is a way to describe the set of points of a fibre product of schemes.

Lemma 25.17.5. 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

\[ (x, y, s, \mathfrak p) \]

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 $\mathfrak p$.

Proof. Let $z$ be a point of $X \times _ S Y$ and let us construct a triple as above. Recall that we may think of $z$ as a morphism $\mathop{\mathrm{Spec}}(\kappa (z)) \to X \times _ S Y$, see Lemma 25.13.3. 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

\[ \xymatrix{ X \times _ S Y \ar@/_/[dddr] \ar@/^/[rrrd] & & & \\ & \mathop{\mathrm{Spec}}(\kappa (x) \otimes _{\kappa (s)} \kappa (y)/\mathfrak p) \ar[r] \ar[d] \ar@{-->}[lu] & \mathop{\mathrm{Spec}}(\kappa (y)) \ar[d] \ar[r] & Y \ar[dd] \\ & \mathop{\mathrm{Spec}}(\kappa (x)) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\kappa (s)) \ar[rd] & \\ & X \ar[rr] & & S } \]

see the discussion in Section 25.13. 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 25.17.6. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target.

  1. 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 $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$.

  2. If $f : X \to S$ is an open immersion, then $X \times _ S Y \to Y$ is an open immersion.

  3. 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 25.4.7 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 25.10.1 $Z$ is a scheme. Hence $Z = X \times _ S Y$ and the first statement follows. The second follows from Lemma 25.17.3 for example. The third is a combination of the first two. $\square$

Definition 25.17.7. 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 25.17.6 above.

We may occasionally also use this terminology with locally closed and open subschemes.


Comments (2)

Comment #1388 by Jackson Morrow on

In Lemma 25.17.4, the line above the displayed equation should say ``let be an affine open cover of ". At the moment it says, an affine open cover of .


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