The Stacks project

Lemma 40.15.1. Let $X$ be an ind-quasi-affine scheme. Let $E \subset X$ be an intersection of a nonempty family of quasi-compact opens of $X$. Set $A = \Gamma (E, \mathcal{O}_ X|_ E)$ and $Y = \mathop{\mathrm{Spec}}(A)$. Then the canonical morphism

\[ j : (E, \mathcal{O}_ X|_ E) \longrightarrow (Y, \mathcal{O}_ Y) \]

of Schemes, Lemma 26.6.4 determines an isomorphism $(E, \mathcal{O}_ X|_ E) \to (E', \mathcal{O}_ Y|_{E'})$ where $E' \subset Y$ is an intersection of quasi-compact opens. If $W \subset E$ is open in $X$, then $j(W)$ is open in $Y$.

Proof. Note that $(E, \mathcal{O}_ X|_ E)$ is a locally ringed space so that Schemes, Lemma 26.6.4 applies to $A \to \Gamma (E, \mathcal{O}_ X|_ E)$. Write $E = \bigcap _{i \in I} U_ i$ with $I \not= \emptyset $ and $U_ i \subset X$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $U_ k \subset U_ i \cap U_ j$. Set $A_ i = \Gamma (U_ i, \mathcal{O}_{U_ i})$. We obtain commutative diagrams

\[ \xymatrix{ (E, \mathcal{O}_ X|_ E) \ar[r] \ar[d] & (\mathop{\mathrm{Spec}}(A), \mathcal{O}_{\mathop{\mathrm{Spec}}(A)}) \ar[d] \\ (U_ i, \mathcal{O}_{U_ i}) \ar[r] & (\mathop{\mathrm{Spec}}(A_ i), \mathcal{O}_{\mathop{\mathrm{Spec}}(A_ i)}) } \]

Since $U_ i$ is quasi-affine, we see that $U_ i \to \mathop{\mathrm{Spec}}(A_ i)$ is a quasi-compact open immersion. On the other hand $A = \mathop{\mathrm{colim}}\nolimits A_ i$. Hence $\mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Spec}}(A_ i)$ as topological spaces (Limits, Lemma 32.4.6). Since $E = \mathop{\mathrm{lim}}\nolimits U_ i$ (by Topology, Lemma 5.24.7) we see that $E \to \mathop{\mathrm{Spec}}(A)$ is a homeomorphism onto its image $E'$ and that $E'$ is the intersection of the inverse images of the opens $U_ i \subset \mathop{\mathrm{Spec}}(A_ i)$ in $\mathop{\mathrm{Spec}}(A)$. For any $e \in E$ the local ring $\mathcal{O}_{X, e}$ is the value of $\mathcal{O}_{U_ i, e}$ which is the same as the value on $\mathop{\mathrm{Spec}}(A)$.

To prove the final assertion of the lemma we argue as follows. Pick $i, j \in I$ with $U_ i \subset U_ j$. Consider the following commutative diagrams

\[ \xymatrix{ U_ i \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A_ i) \ar[d] \\ U_ i \ar[r] & \mathop{\mathrm{Spec}}(A_ j) } \quad \quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A_ i) \ar[d] \\ W \ar[r] & \mathop{\mathrm{Spec}}(A_ j) } \quad \quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A) \ar[d] \\ W \ar[r] & \mathop{\mathrm{Spec}}(A_ j) } \]

By Properties, Lemma 28.18.5 the first diagram is cartesian. Hence the second is cartesian as well. Passing to the limit we find that the third diagram is cartesian, so the top horizontal arrow of this diagram is an open immersion. $\square$

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