
Example 25.14.3 (Affine space with zero doubled). Let $k$ be a field. Let $n \geq 1$. Let $X_1 = \mathop{\mathrm{Spec}}(k[x_1, \ldots , x_ n])$, let $X_2 = \mathop{\mathrm{Spec}}(k[y_1, \ldots , y_ n])$. 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 $X_1, X_2 \subset X$ as open subschemes. There is a morphism $f : X \to \mathop{\mathrm{Spec}}(k[t_1, \ldots , t_ n])$ which on $X_1$ (resp. $X_2$) 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.

Comment #2799 by Kat Christianson on

In the last sentence of the first paragraph ("There is a morphism $f$. . ."), there seems to be a very minor issue with indices: you write $X_i$ but then use the index $i$ to define the $k$-algebra maps later in the sentence. I think replacing $X_i$ with "$X_1$ (resp. $X_2$)" resolves the issue.

On an unrelated note, the "preview" button in this comment box seems not to be working for me. I'm not sure if this is the place to describe the issue, but in case it helps: I'm using Google Chrome (Version 61.0.3163.79) on Windows 10, and I tried disabling browser plugins with no success.

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