The Stacks Project


Tag 01JD

Chapter 25: Schemes > Section 25.14: Glueing schemes

Example 25.14.3 (Affine space with zero doubled). Let $k$ be a field. Let $n \geq 1$. Let $X_1 = \mathop{\rm Spec}(k[x_1, \ldots, x_n])$, let $X_2 = \mathop{\rm Spec}(k[y_1, \ldots, y_n])$. Let $0_1 \in X_1$ be the point corresponding to the maximal ideal $(x_1, \ldots, x_n) \subset k[x_1, \ldots, x_n]$. Let $0_2 \in X_2$ be the point corresponding to the maximal ideal $(y_1, \ldots, y_n) \subset k[y_1, \ldots, y_n]$. Let $U_{12} = X_1 \setminus \{0_1\}$ and let $U_{21} = X_2 \setminus \{0_2\}$. Let $\varphi_{12} : U_{12} \to U_{21}$ be the isomorphism coming from the isomorphism of $k$-algebras $k[y_1, \ldots, y_n] \to k[x_1, \ldots, x_n]$ mapping $y_i$ to $x_i$ (which induces $X_1 \cong X_2$ mapping $0_1$ to $0_2$). Let $X$ be the scheme obtained from the glueing data $(X_1, X_2, U_{12}, U_{21}, \varphi_{12}, \varphi_{21} = \varphi_{12}^{-1})$. Via the slight abuse of notation introduced above the example we think of $X_i \subset X$ as open subschemes. There is a morphism $f : X \to \mathop{\rm Spec}(k[t_1, \ldots, t_n])$ which on $X_i$ corresponds to $k$ algebra map $k[t_1, \ldots, t_n] \to k[x_1, \ldots, x_n]$ (resp. $k[t_1, \ldots, t_n] \to k[y_1, \ldots, y_n]$) mapping $t_i$ to $x_i$ (resp.  $t_i$ to $y_i$). It is easy to see that this morphism identifies $k[t_1, \ldots, t_n]$ with $\Gamma(X, \mathcal{O}_X)$. Since $f(0_1) = f(0_2)$ we see that $X$ is not affine.

Note that $X_1$ and $X_2$ are affine opens of $X$. But, if $n = 2$, then $X_1 \cap X_2$ is the scheme described in Example 25.9.3 and hence not affine. Thus in general the intersection of affine opens of a scheme is not affine. (This fact holds more generally for any $n > 1$.)

Another curious feature of this example is the following. If $n > 1$ there are many irreducible closed subsets $T \subset X$ (take the closure of any non closed point in $X_1$ for example). But unless $T = \{0_1\}$, or $T = \{0_2\}$ we have $0_1 \in T \Leftrightarrow 0_2 \in T$. Proof omitted.

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 2513–2556 (see updates for more information).

    \begin{example}[Affine space with zero doubled]
    \label{example-affine-space-zero-doubled}
    Let $k$ be a field. Let $n \geq 1$.
    Let $X_1 = \Spec(k[x_1, \ldots, x_n])$,
    let $X_2 = \Spec(k[y_1, \ldots, y_n])$.
    Let $0_1 \in X_1$ be the point corresponding to the maximal ideal
    $(x_1, \ldots, x_n) \subset k[x_1, \ldots, x_n]$.
    Let $0_2 \in X_2$ be the point corresponding to the maximal ideal
    $(y_1, \ldots, y_n) \subset k[y_1, \ldots, y_n]$.
    Let $U_{12} = X_1 \setminus \{0_1\}$ and
    let $U_{21} = X_2 \setminus \{0_2\}$. Let
    $\varphi_{12} : U_{12} \to U_{21}$ be the isomorphism
    coming from the isomorphism of $k$-algebras
    $k[y_1, \ldots, y_n] \to k[x_1, \ldots, x_n]$
    mapping $y_i$ to $x_i$ (which induces $X_1 \cong X_2$ mapping
    $0_1$ to $0_2$).
    Let $X$ be the scheme obtained from the glueing data
    $(X_1, X_2, U_{12}, U_{21}, \varphi_{12},
    \varphi_{21} = \varphi_{12}^{-1})$. Via the slight abuse
    of notation introduced above the example we think of
    $X_i \subset X$ as open subschemes.
    There is a morphism $f : X \to \Spec(k[t_1, \ldots, t_n])$
    which on $X_i$ corresponds to $k$ algebra map
    $k[t_1, \ldots, t_n] \to k[x_1, \ldots, x_n]$
    (resp.\ $k[t_1, \ldots, t_n] \to k[y_1, \ldots, y_n]$)
    mapping $t_i$ to $x_i$ (resp.\  $t_i$ to $y_i$).
    It is easy to see that this morphism identifies
    $k[t_1, \ldots, t_n]$ with $\Gamma(X, \mathcal{O}_X)$. Since
    $f(0_1) = f(0_2)$ we see that $X$ is not affine.
    
    \medskip\noindent
    Note that $X_1$ and $X_2$ are affine opens of $X$.
    But, if $n = 2$, then $X_1 \cap X_2$ is the scheme
    described in Example \ref{example-not-affine} and hence not affine.
    Thus in general the intersection of affine opens of a scheme
    is not affine. (This fact holds more generally for any $n > 1$.)
    
    \medskip\noindent
    Another curious feature of this example is the following.
    If $n > 1$ there are many irreducible closed subsets $T \subset X$
    (take the closure of any non closed point in $X_1$ for example).
    But unless $T = \{0_1\}$, or $T = \{0_2\}$ we have
    $0_1 \in T \Leftrightarrow 0_2 \in T$. Proof omitted.
    \end{example}

    Comments (0)

    There are no comments yet for this tag.

    There are also 5 comments on Section 25.14: Schemes.

    Add a comment on tag 01JD

    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 lower-right corner).

    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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?