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Tag 01J5

25.13. Points of schemes

Given a scheme $X$ we can define a functor $$ h_X : \textit{Sch}^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \mathop{\rm Mor}\nolimits(T, X). $$ See Categories, Example 4.3.4. This is called the functor of points of $X$. A fun part of scheme theory is to find descriptions of the internal geometry of $X$ in terms of this functor $h_X$. In this section we find a simple way to describe points of $X$.

Let $X$ be a scheme. Let $R$ be a local ring with maximal ideal $\mathfrak m \subset R$. Suppose that $f : \mathop{\rm Spec}(R) \to X$ is a morphism of schemes. Let $x \in X$ be the image of the closed point $\mathfrak m \in \mathop{\rm Spec}(R)$. Then we obtain a local homomorphism of local rings $$ f^\sharp : \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{\mathop{\rm Spec}(R), \mathfrak m} = R. $$

Lemma 25.13.1. Let $X$ be a scheme. Let $R$ be a local ring. The construction above gives a bijective correspondence between morphisms $\mathop{\rm Spec}(R) \to X$ and pairs $(x, \varphi)$ consisting of a point $x \in X$ and a local homomorphism of local rings $\varphi : \mathcal{O}_{X, x} \to R$.

Proof. Let $A$ be a ring. For any ring homomorphism $\psi : A \to R$ there exists a unique prime ideal $\mathfrak p \subset A$ and a factorization $A \to A_{\mathfrak p} \to R$ where the last map is a local homomorphism of local rings. Namely, $\mathfrak p = \psi^{-1}(\mathfrak m)$. Via Lemma 25.6.4 this proves that the lemma holds if $X$ is an affine scheme.

Let $X$ be a general scheme. Any $x \in X$ is contained in an open affine $U \subset X$. By the affine case we conclude that every pair $(x, \varphi)$ occurs as the end product of the construction above the lemma.

To finish the proof it suffices to show that any morphism $f : \mathop{\rm Spec}(R) \to X$ has image contained in any affine open containing the image $x$ of the closed point of $\mathop{\rm Spec}(R)$. In fact, let $x \in V \subset X$ be any open neighbourhood containing $x$. Then $f^{-1}(V) \subset \mathop{\rm Spec}(R)$ is an open containing the unique closed point and hence equal to $\mathop{\rm Spec}(R)$. $\square$

As a special case of the lemma above we obtain for every point $x$ of a scheme $X$ a canonical morphism \begin{equation} \tag{25.13.1.1} \mathop{\rm Spec}(\mathcal{O}_{X, x}) \longrightarrow X \end{equation} corresponding to the identity map on the local ring of $X$ at $x$. We may reformulate the lemma above as saying that for any morphism $f : \mathop{\rm Spec}(R) \to X$ there exists a unique point $x \in X$ such that $f$ factors as $\mathop{\rm Spec}(R) \to \mathop{\rm Spec}(\mathcal{O}_{X, x}) \to X$ where the first map comes from a local homomorphism $\mathcal{O}_{X, x} \to R$.

In case we have a morphism of schemes $f : X \to S$, and a point $x$ mapping to a point $s \in S$ we obtain a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(\mathcal{O}_{X, x}) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\rm Spec}(\mathcal{O}_{S, s}) \ar[r] & S } $$ where the left vertical map corresponds to the local ring map $f^\sharp_x : \mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$.

Lemma 25.13.2. Let $X$ be a scheme. Let $x, x' \in X$ be points of $X$. Then $x' \in X$ is a generalization of $x$ if and only if $x'$ is in the image of the canonical morphism $\mathop{\rm Spec}(\mathcal{O}_{X, x}) \to X$.

Proof. A continuous map preserves the relation of specialization/generalization. Since every point of $\mathop{\rm Spec}(\mathcal{O}_{X, x})$ is a generalization of the closed point we see every point in the image of $\mathop{\rm Spec}(\mathcal{O}_{X, x}) \to X$ is a generalization of $x$. Conversely, suppose that $x'$ is a generalization of $x$. Choose an affine open neighbourhood $U = \mathop{\rm Spec}(R)$ of $x$. Then $x' \in U$. Say $\mathfrak p \subset R$ and $\mathfrak p' \subset R$ are the primes corresponding to $x$ and $x'$. Since $x'$ is a generalization of $x$ we see that $\mathfrak p' \subset \mathfrak p$. This means that $\mathfrak p'$ is in the image of the morphism $\mathop{\rm Spec}(\mathcal{O}_{X, x}) = \mathop{\rm Spec}(R_{\mathfrak p}) \to \mathop{\rm Spec}(R) = U \subset X$ as desired. $\square$

Now, let us discuss morphisms from spectra of fields. Let $(R, \mathfrak m, \kappa)$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa$. Let $K$ be a field. A local homomorphism $R \to K$ by definition factors as $R \to \kappa \to K$, i.e., is the same thing as a morphism $\kappa \to K$. Thus we see that morphisms $$ \mathop{\rm Spec}(K) \longrightarrow X $$ correspond to pairs $(x, \kappa(x) \to K)$. We may define a preorder on morphisms of spectra of fields to $X$ by saying that $\mathop{\rm Spec}(K) \to X$ dominates $\mathop{\rm Spec}(L) \to X$ if $\mathop{\rm Spec}(K) \to X$ factors through $\mathop{\rm Spec}(L) \to X$. This suggests the following notion: Let us temporarily say that two morphisms $p : \mathop{\rm Spec}(K) \to X$ and $q : \mathop{\rm Spec}(L) \to X$ are equivalent if there exists a third field $\Omega$ and a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(\Omega) \ar[r] \ar[d] & \mathop{\rm Spec}(L) \ar[d]^q \\ \mathop{\rm Spec}(K) \ar[r]^p & X } $$ Of course this immediately implies that the unique points of all three of the schemes $\mathop{\rm Spec}(K)$, $\mathop{\rm Spec}(L)$, and $\mathop{\rm Spec}(\Omega)$ map to the same $x \in X$. Thus a diagram (by the remarks above) corresponds to a point $x \in X$ and a commutative diagram $$ \xymatrix{ \Omega & L \ar[l] \\ K \ar[u] & \kappa(x) \ar[l] \ar[u] } $$ of fields. This defines an equivalence relation, because given any set of extensions $\kappa \subset K_i$ there exists some field extension $\kappa \subset \Omega$ such that all the field extensions $K_i$ are contained in the extension $\Omega$.

Lemma 25.13.3. Let $X$ be a scheme. Points of $X$ correspond bijectively to equivalence classes of morphisms from spectra of fields into $X$. Moreover, each equivalence class contains a (unique up to unique isomorphism) smallest element $\mathop{\rm Spec}(\kappa(x)) \to X$.

Proof. Follows from the discussion above. $\square$

Of course the morphisms $\mathop{\rm Spec}(\kappa(x)) \to X$ factor through the canonical morphisms $\mathop{\rm Spec}(\mathcal{O}_{X, x}) \to X$. And the content of Lemma 25.13.2 is in this setting that the morphism $\mathop{\rm Spec}(\kappa(x')) \to X$ factors as $\mathop{\rm Spec}(\kappa(x')) \to \mathop{\rm Spec}(\mathcal{O}_{X, x}) \to X$ whenever $x'$ is a generalization of $x$. In case we have a morphism of schemes $f : X \to S$, and a point $x$ mapping to a point $s \in S$ we obtain a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(\kappa(x)) \ar[r] \ar[d] & \mathop{\rm Spec}(\mathcal{O}_{X, x}) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\rm Spec}(\kappa(s)) \ar[r] & \mathop{\rm Spec}(\mathcal{O}_{S, s}) \ar[r] & S. } $$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 2145–2352 (see updates for more information).

    \section{Points of schemes}
    \label{section-points}
    
    \noindent
    Given a scheme $X$ we can define a functor
    $$
    h_X : \Sch^{opp}
    \longrightarrow
    \textit{Sets}, \quad
    T \longmapsto \Mor(T, X).
    $$
    See Categories, Example \ref{categories-example-hom-functor}.
    This is called the {\it functor of points of $X$}.
    A fun part of scheme theory is to find descriptions of
    the internal geometry of $X$ in terms of this functor $h_X$.
    In this section we find a simple way to describe
    points of $X$.
    
    \medskip\noindent
    Let $X$ be a scheme. Let $R$ be a local ring with maximal ideal
    $\mathfrak m \subset R$. Suppose that $f : \Spec(R) \to X$
    is a morphism of schemes. Let $x \in X$ be the image of the closed point
    $\mathfrak m \in \Spec(R)$. Then we obtain a local homomorphism
    of local rings
    $$
    f^\sharp :
    \mathcal{O}_{X, x}
    \longrightarrow
    \mathcal{O}_{\Spec(R), \mathfrak m} = R.
    $$
    
    \begin{lemma}
    \label{lemma-morphism-from-spec-local-ring}
    Let $X$ be a scheme. Let $R$ be a local ring.
    The construction above gives a bijective correspondence
    between morphisms $\Spec(R) \to X$ and pairs
    $(x, \varphi)$ consisting of a point $x \in X$ and
    a local homomorphism of local rings $\varphi : \mathcal{O}_{X, x} \to R$.
    \end{lemma}
    
    \begin{proof}
    Let $A$ be a ring. For any ring homomorphism $\psi : A \to R$
    there exists a unique prime ideal $\mathfrak p \subset A$
    and a factorization $A \to A_{\mathfrak p} \to R$ where the
    last map is a local homomorphism of local rings. Namely,
    $\mathfrak p = \psi^{-1}(\mathfrak m)$. Via
    Lemma \ref{lemma-morphism-into-affine}
    this proves that the lemma holds if $X$ is an affine scheme.
    
    \medskip\noindent
    Let $X$ be a general scheme. Any $x \in X$ is contained in
    an open affine $U \subset X$. By the affine case we conclude that every pair
    $(x, \varphi)$ occurs as the end product of the construction
    above the lemma.
    
    \medskip\noindent
    To finish the proof it suffices to show that any morphism
    $f : \Spec(R) \to X$ has image contained in any affine
    open containing the image $x$ of the closed
    point of $\Spec(R)$. In fact, let $x \in V \subset X$
    be any open neighbourhood containing $x$. Then
    $f^{-1}(V) \subset \Spec(R)$ is an open containing
    the unique closed point and hence equal to $\Spec(R)$.
    \end{proof}
    
    \noindent
    As a special case of the lemma above we obtain for every
    point $x$ of a scheme $X$ a canonical morphism
    \begin{equation}
    \label{equation-canonical-morphism}
    \Spec(\mathcal{O}_{X, x}) \longrightarrow X
    \end{equation}
    corresponding to the identity map on the local ring of $X$ at $x$.
    We may reformulate the lemma above as saying that for any
    morphism $f : \Spec(R) \to X$ there exists a unique point
    $x \in X$ such that $f$ factors as
    $\Spec(R) \to \Spec(\mathcal{O}_{X, x}) \to X$
    where the first map comes from a local homomorphism
    $\mathcal{O}_{X, x} \to R$.
    
    \medskip\noindent
    In case we have a morphism of schemes $f : X \to S$,
    and a point $x$ mapping to a point $s \in S$ we obtain
    a commutative diagram
    $$
    \xymatrix{
    \Spec(\mathcal{O}_{X, x}) \ar[r] \ar[d] & X \ar[d] \\
    \Spec(\mathcal{O}_{S, s}) \ar[r] & S
    }
    $$
    where the left vertical map corresponds to the local ring map
    $f^\sharp_x : \mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$.
    
    \begin{lemma}
    \label{lemma-specialize-points}
    Let $X$ be a scheme.
    Let $x, x' \in X$ be points of $X$.
    Then $x' \in X$ is a generalization of $x$ if and only if
    $x'$ is in the image of the canonical morphism
    $\Spec(\mathcal{O}_{X, x}) \to X$.
    \end{lemma}
    
    \begin{proof}
    A continuous map preserves the relation of specialization/generalization.
    Since every point of $\Spec(\mathcal{O}_{X, x})$ is a
    generalization of the closed point we see every point in the image
    of $\Spec(\mathcal{O}_{X, x}) \to X$ is a generalization of $x$.
    Conversely, suppose that $x'$ is a generalization of $x$.
    Choose an affine open neighbourhood $U = \Spec(R)$ of
    $x$. Then $x' \in U$. Say $\mathfrak p \subset R$ and
    $\mathfrak p' \subset R$ are the primes corresponding
    to $x$ and $x'$. Since $x'$ is a generalization of $x$
    we see that $\mathfrak p' \subset \mathfrak p$. This means
    that $\mathfrak p'$ is in the image of the morphism
    $\Spec(\mathcal{O}_{X, x}) = \Spec(R_{\mathfrak p})
    \to \Spec(R) = U \subset X$ as desired.
    \end{proof}
    
    \noindent
    Now, let us discuss morphisms from spectra of fields.
    Let $(R, \mathfrak m, \kappa)$ be a local ring
    with maximal ideal $\mathfrak m$ and residue field $\kappa$.
    Let $K$ be a field. A local homomorphism $R \to K$ by definition
    factors as $R \to \kappa \to K$, i.e., is the same thing as
    a morphism $\kappa \to K$. Thus we see that morphisms
    $$
    \Spec(K) \longrightarrow X
    $$
    correspond to pairs $(x, \kappa(x) \to K)$. We may define
    a preorder on morphisms of spectra of fields to $X$
    by saying that $\Spec(K) \to X$ dominates
    $\Spec(L) \to X$ if $\Spec(K) \to X$
    factors through $\Spec(L) \to X$. This suggests
    the following notion: Let us temporarily
    say that two morphisms $p : \Spec(K) \to X$ and
    $q : \Spec(L) \to X$ are {\it equivalent} if there exists
    a third field $\Omega$ and a commutative diagram
    $$
    \xymatrix{
    \Spec(\Omega) \ar[r] \ar[d] &
    \Spec(L) \ar[d]^q \\
    \Spec(K) \ar[r]^p &
    X
    }
    $$
    Of course this immediately implies that the unique points of
    all three of the schemes $\Spec(K)$,
    $\Spec(L)$, and $\Spec(\Omega)$
    map to the same $x \in X$. Thus a diagram (by the remarks above)
    corresponds to a point $x \in X$ and a commutative diagram
    $$
    \xymatrix{
    \Omega &
    L \ar[l] \\
    K \ar[u] &
    \kappa(x) \ar[l] \ar[u]
    }
    $$
    of fields. This defines an equivalence relation, because given
    any set of extensions $\kappa \subset K_i$ there exists
    some field extension $\kappa \subset \Omega$ such that all
    the field extensions $K_i$ are contained in the extension $\Omega$.
    
    \begin{lemma}
    \label{lemma-characterize-points}
    Let $X$ be a scheme. Points of $X$ correspond bijectively
    to equivalence classes of morphisms from spectra of
    fields into $X$. Moreover, each equivalence class contains
    a (unique up to unique isomorphism) smallest element
    $\Spec(\kappa(x)) \to X$.
    \end{lemma}
    
    \begin{proof}
    Follows from the discussion above.
    \end{proof}
    
    \noindent
    Of course the morphisms $\Spec(\kappa(x)) \to X$
    factor through the canonical morphisms
    $\Spec(\mathcal{O}_{X, x}) \to X$.
    And the content of Lemma \ref{lemma-specialize-points} is in
    this setting that the morphism $\Spec(\kappa(x')) \to X$
    factors as
    $\Spec(\kappa(x')) \to \Spec(\mathcal{O}_{X, x}) \to X$
    whenever $x'$ is a generalization of $x$.
    In case we have a morphism of schemes $f : X \to S$,
    and a point $x$ mapping to a point $s \in S$ we obtain
    a commutative diagram
    $$
    \xymatrix{
    \Spec(\kappa(x)) \ar[r] \ar[d] &
    \Spec(\mathcal{O}_{X, x}) \ar[r] \ar[d] &
    X \ar[d] \\
    \Spec(\kappa(s)) \ar[r] &
    \Spec(\mathcal{O}_{S, s}) \ar[r] &
    S.
    }
    $$

    Comments (2)

    Comment #2645 by Manuel Hoff on July 12, 2017 a 2:25 pm UTC

    In 25.13.1, the last sentence mentions a map of local rings, but this map should go in the other direction right? Namely $O_{S,s} \rightarrow O_{X,x}$.

    Comment #2665 by Johan (site) on July 28, 2017 a 5:16 pm UTC

    Indeed, it does. Thanks. Fixed here.

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