The Stacks project

Lemma 30.29.3. Let $X$ be an integral locally Noetherian normal scheme. Let $\mathcal{F}$ be a rank 1 coherent reflexive $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{F})$. Let

\[ U = \{ x \in X \mid s : \mathcal{O}_{X, x} \to \mathcal{F}_ x \text{ is an isomorphism}\} \]

Then $j : U \to X$ is an open subscheme of $X$ and

\[ j_*\mathcal{O}_ U = \mathop{\mathrm{colim}}\nolimits (\mathcal{O}_ X \xrightarrow {s} \mathcal{F} \xrightarrow {s} \mathcal{F}^{[2]} \xrightarrow {s} \mathcal{F}^{[3]} \xrightarrow {s} \ldots ) \]

where $\mathcal{F}^{[1]} = \mathcal{F}$ and inductively $\mathcal{F}^{[n + 1]} = (\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{F}^{[n]})^{**}$.

Proof. The set $U$ is open by Modules, Lemmas 17.9.4 and 17.12.6. Observe that $j$ is quasi-compact by Properties, Lemma 27.5.3. To prove the final statement it suffices to show for every quasi-compact open $W \subset X$ there is an isomorphism

\[ \mathop{\mathrm{colim}}\nolimits \Gamma (W, \mathcal{F}^{[n]}) \longrightarrow \Gamma (U \cap W, \mathcal{O}_ U) \]

of $\mathcal{O}_ X(W)$-modules compatible with restriction maps. We will omit the verification of compatibilities. After replacing $X$ by $W$ and rewriting the above in terms of homs, we see that it suffices to construct an isomorphism

\[ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{O}_ X, \mathcal{F}^{[n]}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_ U, \mathcal{O}_ U) \]

Choose an open $V \subset X$ such that every irreducible component of $X \setminus V$ has codimension $\geq 2$ in $X$ and such that $\mathcal{F}|_ V$ is invertible, see Lemma 30.12.13. Then restriction defines an equivalence of categories between rank $1$ coherent reflexive modules on $X$ and $V$ and between rank $1$ coherent reflexive modules on $U$ and $V \cap U$. See Lemma 30.12.12 and Serre's criterion Properties, Lemma 27.12.5. Thus it suffices to construct an isomorphism

\[ \mathop{\mathrm{colim}}\nolimits \Gamma (V, (\mathcal{F}|_ V)^{\otimes n}) \longrightarrow \Gamma (V \cap U, \mathcal{O}_ U) \]

Since $\mathcal{F}|_ V$ is invertible and since $U \cap V$ is equal to the set of points where $s|_ V$ generates this invertible module, this is a special case of Properties, Lemma 27.17.2 (there is an explicit formula for the map as well). $\square$


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