Lemma 31.28.4. Let R be a UFD. The Picard groups of the following are trivial.
\mathop{\mathrm{Spec}}(R) and any open subscheme of it.
\mathbf{A}^ n_ R = \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]) and any open subscheme of it.
In particular, the Picard group of any open subscheme of affine n-space \mathbf{A}^ n_ k over a field k is trivial.
Proof.
Any polynomial ring over R is a UFD by Algebra, Lemma 10.120.10. Thus the question is to show that \mathop{\mathrm{Pic}}\nolimits (U) is trivial if U \subset \mathop{\mathrm{Spec}}(R) is a (nonempty) open.
In the Noetherian case, we may use Lemma 31.28.3 to reduce to showing \mathop{\mathrm{Pic}}\nolimits (\mathop{\mathrm{Spec}}(R)) is trivial. This in turn is proved in More on Algebra, Lemma 15.117.3. We urge the reader to skip the rest of the proof.
Let U \subset \mathop{\mathrm{Spec}}(R) be a nonempty open. Choose a nonzero f \in R with D(f) \subset U. Since f is a product of prime elements of R we conclude that there are finitely many prime elements a_1, \ldots , a_ n \in R such that (a_ i) \not\in U (or equivalentaly U \cap V(a_ i) = \emptyset ). After replacing R by R_{a_1 \ldots a_ n} we may assume that U contains all principal prime ideals. (Note that a localization of a UFD is a UFD, for example by Algebra, Lemma 10.120.7.)
Assume U contains all principal prime ideals. Write U = \mathop{\mathrm{Spec}}(R) \setminus V(I) for some ideal I \subset R. Pick f \in I nonzero and write f = a_1 \ldots a_ n as a product of prime elements. Since (a_ i) \not\in V(I) we may choose g \in I with g \not\in (a_ i) for all i (prime avoidance). Picture
U' = D(f) \cup D(g) \subset U \subset \mathop{\mathrm{Spec}}(R)
By More on Algebra, Lemma 15.117.3 the Picard groups of D(f) and D(g) are trivial and by an elementary argument about UFDs the map R_ f^* \times R_ g^* \to R_{fg}^*, (u, v) \mapsto uv^{-1} is surjective. We conclude that \mathop{\mathrm{Pic}}\nolimits (U') is trivial.
Thus it suffices to show that an invertible module \mathcal{L} on U whose pullback by j : U' \to U' is trivial, is trivial. To do this it suffices to show that the map \mathcal{L} \to j_*j^*\mathcal{L} is an isomorphism. For this in turn, let u \in U \setminus U'. Then f, g \in \mathfrak m_ u \subset \mathcal{O}_{U, u} = A are elements of a local UFD which do not have a prime factor in common. The reader easily shows that this means that
0 \to A \to A_ f \times A_ g \to A_{fg}
is exact by looking at prime factors. Since A = \mathcal{L}_ u because \mathcal{L} is invertible and since (j_*j^*\mathcal{L})_ u is the kernel of A_ f \times A_ g \to A_{fg} we conclude.
\square
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