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*}

Example 25.9.3. Let $k$ be a field. An example of a scheme which is not affine is given by the open subspace $U = \mathop{\mathrm{Spec}}(k[x, y]) \setminus \{ (x, y)\} $ of the affine scheme $X =\mathop{\mathrm{Spec}}(k[x, y])$. It is covered by two affines, namely $D(x) = \mathop{\mathrm{Spec}}(k[x, y, 1/x])$ and $D(y) = \mathop{\mathrm{Spec}}(k[x, y, 1/y])$ whose intersection is $D(xy) = \mathop{\mathrm{Spec}}(k[x, y, 1/xy])$. By the sheaf property for $\mathcal{O}_ U$ there is an exact sequence

\[ 0 \to \Gamma (U, \mathcal{O}_ U) \to k[x, y, 1/x] \times k[x, y, 1/y] \to k[x, y, 1/xy] \]

We conclude that the map $k[x, y] \to \Gamma (U, \mathcal{O}_ U)$ (coming from the morphism $U \to X$) is an isomorphism. Therefore $U$ cannot be affine since if it was then by Lemma 25.6.5 we would have $U \cong X$.


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