The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

25.13 Points of schemes

Given a scheme $X$ we can define a functor

\[ h_ X : \mathit{Sch}^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \mathop{Mor}\nolimits (T, X). \]

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 $f : \mathop{\mathrm{Spec}}(R) \to X$ is a morphism of schemes. Let $x \in X$ be the image of the closed point $\mathfrak m \in \mathop{\mathrm{Spec}}(R)$. Then we obtain a local homomorphism of local rings

\[ f^\sharp : \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{\mathop{\mathrm{Spec}}(R), \mathfrak m} = R. \]

Lemma 25.13.1. Let $X$ be a scheme. Let $R$ be a local ring. The construction above gives a bijective correspondence between morphisms $\mathop{\mathrm{Spec}}(R) \to X$ 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 $f : \mathop{\mathrm{Spec}}(R) \to X$ has image contained in any affine open containing the image $x$ of the closed point of $\mathop{\mathrm{Spec}}(R)$. In fact, let $x \in V \subset X$ be any open neighbourhood containing $x$. Then $f^{-1}(V) \subset \mathop{\mathrm{Spec}}(R)$ is an open containing the unique closed point and hence equal to $\mathop{\mathrm{Spec}}(R)$. $\square$

As a special case of the lemma above we obtain for every point $x$ of a scheme $X$ a canonical morphism

25.13.1.1
\begin{equation} \label{schemes-equation-canonical-morphism} \mathop{\mathrm{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 $f : \mathop{\mathrm{Spec}}(R) \to X$ there exists a unique point $x \in X$ such that $f$ factors as $\mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to X$ 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

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \ar[r] & S } \]

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 $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to X$.

Proof. A continuous map preserves the relation of specialization/generalization. Since every point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is a generalization of the closed point we see every point in the image of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to X$ is a generalization of $x$. Conversely, suppose that $x'$ is a generalization of $x$. Choose an affine open neighbourhood $U = \mathop{\mathrm{Spec}}(R)$ 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 $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) = \mathop{\mathrm{Spec}}(R_{\mathfrak p}) \to \mathop{\mathrm{Spec}}(R) = U \subset X$ 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

\[ \mathop{\mathrm{Spec}}(K) \longrightarrow X \]

correspond to pairs $(x, \kappa (x) \to K)$. We may define a preorder on morphisms of spectra of fields to $X$ by saying that $\mathop{\mathrm{Spec}}(K) \to X$ dominates $\mathop{\mathrm{Spec}}(L) \to X$ if $\mathop{\mathrm{Spec}}(K) \to X$ factors through $\mathop{\mathrm{Spec}}(L) \to X$. This suggests the following notion: Let us temporarily say that two morphisms $p : \mathop{\mathrm{Spec}}(K) \to X$ and $q : \mathop{\mathrm{Spec}}(L) \to X$ are equivalent if there exists a third field $\Omega $ and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\Omega ) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(L) \ar[d]^ q \\ \mathop{\mathrm{Spec}}(K) \ar[r]^ p & X } \]

Of course this immediately implies that the unique points of all three of the schemes $\mathop{\mathrm{Spec}}(K)$, $\mathop{\mathrm{Spec}}(L)$, and $\mathop{\mathrm{Spec}}(\Omega )$ 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 $\mathop{\mathrm{Spec}}(\kappa (x)) \to X$.

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

Of course the morphisms $\mathop{\mathrm{Spec}}(\kappa (x)) \to X$ factor through the canonical morphisms $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to X$. And the content of Lemma 25.13.2 is in this setting that the morphism $\mathop{\mathrm{Spec}}(\kappa (x')) \to X$ factors as $\mathop{\mathrm{Spec}}(\kappa (x')) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to X$ 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

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\kappa (x)) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\kappa (s)) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \ar[r] & S. } \]

Comments (2)

Comment #2645 by Manuel Hoff on

In 25.13.1, the last sentence mentions a map of local rings, but this map should go in the other direction right? Namely .


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01J5. Beware of the difference between the letter 'O' and the digit '0'.