The Stacks project

Remark 31.29.5. Let $A$ be a Noetherian normal domain. Let $M$ be a rank $1$ finite reflexive $A$-module. Let $s \in M$ be nonzero. Let $\mathfrak p_1, \ldots , \mathfrak p_ r$ be the height $1$ primes of $A$ in the support of $M/As$. Then the open $U$ of Lemma 31.29.3 is

\[ U = \mathop{\mathrm{Spec}}(A) \setminus \left(V(\mathfrak p_1) \cup \ldots \cup \mathfrak p_ r)\right) \]

by Lemma 31.29.4. Moreover, if $M^{[n]}$ denotes the reflexive hull of $M \otimes _ A \ldots \otimes _ A M$ ($n$-factors), then

\[ \Gamma (U, \mathcal{O}_ U) = \mathop{\mathrm{colim}}\nolimits M^{[n]} \]

according to Lemma 31.29.3.


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