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Tag 01JO

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.

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.

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 2975–3224 (see updates for more information).

    \section{Fibre products of schemes}
    \label{section-fibre-products}
    
    \noindent
    Here is a review of the general definition, even though
    we have already shown that fibre products of schemes exist.
    
    \begin{definition}
    \label{definition-fibre-product}
    Given morphisms of schemes $f : X \to S$ and $g : Y \to S$
    the {\it 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 \ref{categories-definition-fibre-products}.
    \end{definition}
    
    \noindent
    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.
    
    \begin{lemma}
    \label{lemma-fibre-product-affines}
    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.
    \end{lemma}
    
    \begin{proof}
    Suppose that $X = \Spec(A)$, $Y = \Spec(B)$
    and $S = \Spec(R)$. By Lemma \ref{lemma-fibre-product-affine-schemes}
    the affine scheme $\Spec(A \otimes_R B)$
    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.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-open-fibre-product}
    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 $p^{-1}(V) \cap q^{-1}(W)$.
    \end{lemma}
    
    \begin{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
    \ref{categories-section-fibre-products}.
    \end{proof}
    
    \noindent
    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.
    
    \begin{lemma}
    \label{lemma-affine-covering-fibre-product}
    \begin{slogan}
    Bare-hands construction of fiber products: an affine open cover of a
    fiber product of schemes can be assembled from compatible
    affine open covers of the pieces.
    \end{slogan}
    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 $X \times_S Y$.
    \end{lemma}
    
    \begin{proof}
    See discussion above the lemma.
    \end{proof}
    
    \noindent
    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 \ref{lemma-fibre-products} anyway.)
    Here is a way to describe the set of points of a fibre product of schemes.
    
    \begin{lemma}
    \label{lemma-points-fibre-product}
    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 $\mathfrak p$.
    \end{lemma}
    
    \begin{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
    $\Spec(\kappa(z)) \to X \times_S Y$, see
    Lemma \ref{lemma-characterize-points}. This morphism corresponds
    to morphisms $a : \Spec(\kappa(z)) \to X$
    and $b : \Spec(\kappa(z)) \to Y$ 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.
    
    \medskip\noindent
    Conversely, given a quadruple $(x, y, s, \mathfrak p)$ we get a
    commutative solid diagram
    $$
    \xymatrix{
    X \times_S Y
    \ar@/_/[dddr] \ar@/^/[rrrd]
    & & & \\
    &
    \Spec(\kappa(x) \otimes_{\kappa(s)} \kappa(y)/\mathfrak p)
    \ar[r] \ar[d] \ar@{-->}[lu]
    &
    \Spec(\kappa(y)) \ar[d] \ar[r] &
    Y \ar[dd] \\
    &
    \Spec(\kappa(x)) \ar[r] \ar[d] &
    \Spec(\kappa(s)) \ar[rd] &
    \\
    &
    X \ar[rr] &
    &
    S
    }
    $$
    see the discussion in Section \ref{section-points}. Thus we get the
    dotted arrow. The corresponding point $z$ of $X \times_S Y$ is the
    image of the generic point of
    $\Spec(\kappa(x) \otimes_{\kappa(s)} \kappa(y)/\mathfrak p)$.
    We omit the verification that the two constructions are inverse
    to each other.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-fibre-product-immersion}
    Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes
    with the same target.
    \begin{enumerate}
    \item 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
    $\Im(g^*\mathcal{I} \to \mathcal{O}_Y)$.
    \item If $f : X \to S$ is an open immersion,
    then $X \times_S Y \to Y$ is an open immersion.
    \item If $f : X \to S$ is an immersion,
    then $X \times_S Y \to Y$ is an immersion.
    \end{enumerate}
    \end{lemma}
    
    \begin{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 \ref{lemma-restrict-map-to-closed} the closed subspace
    $Z \subset Y$ defined by the sheaf of ideals
    $\Im(g^*\mathcal{I} \to \mathcal{O}_Y)$ is the fibre product
    in the category of locally ringed spaces.
    By Lemma \ref{lemma-closed-subspace-scheme} $Z$ is a scheme.
    Hence $Z = X \times_S Y$ and the first statement follows.
    The second follows from Lemma \ref{lemma-open-fibre-product}
    for example. The third is a combination of
    the first two.
    \end{proof}
    
    \begin{definition}
    \label{definition-inverse-image-closed-subscheme}
    Let $f : X \to Y$ be a morphism of schemes. Let $Z \subset Y$ be a
    closed subscheme of $Y$. The {\it inverse image $f^{-1}(Z)$ of the
    closed subscheme $Z$} is the closed subscheme $Z \times_Y X$ of
    $X$. See Lemma \ref{lemma-fibre-product-immersion} above.
    \end{definition}
    
    \noindent
    We may occasionally also use this terminology with locally closed and
    open subschemes.

    Comments (2)

    Comment #1388 by Jackson Morrow on April 2, 2015 a 9:35 pm UTC

    In Lemma 25.17.4, the line above the displayed equation should say ``let $g^{-1}(U_i) = \bigcup_{k \in K_i}W_k$ be an affine open cover of $g^{-1}(U_i)$". At the moment it says, an affine open cover of $f^{-1}(U_i)$.

    Comment #1410 by Johan (site) on April 15, 2015 a 6:17 pm UTC

    Many thanks. See here.

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