Lemma 52.27.3. Let (A, \mathfrak m) be a Noetherian local ring and f \in \mathfrak m. Assume
the conditions of Lemma 52.27.1 hold, and
for every maximal ideal \mathfrak p \subset A_ f the punctured spectrum of (A_ f)_\mathfrak p has trivial Picard group.
Let U, resp. U_0 be the punctured spectrum of A, resp. A/fA. Then
is surjective.
Proof. Let \mathcal{L}_0 \in \mathop{\mathrm{Pic}}\nolimits (U_0). By Lemma 52.27.1 there exists an open U_0 \subset U' \subset U and \mathcal{L}' \in \mathop{\mathrm{Pic}}\nolimits (U') whose restriction to U_0 is \mathcal{L}_0. Since U' \supset U_0 we see that U \setminus U' consists of points corresponding to prime ideals \mathfrak p_1, \ldots , \mathfrak p_ n as in (2). By assumption we can find invertible modules \mathcal{L}'_ i on \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i}) agreeing with \mathcal{L}' over the punctured spectrum U' \times _ U \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i}) since trivial invertible modules always extend. By Limits, Lemma 32.20.2 applied n times we see that \mathcal{L}' extends to an invertible module on U. \square
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