Picard group of the product of a local normal scheme with $\mathbb{A}^2-\{0\}$ Let $X$ be a noetherian local normal scheme, we may even assume that $X$ is complete if necessary.
Consider $X\times (\mathbb{A}^2-\{0\})$, is it true that the Picard group of this scheme vanishes?
 A: My doubt in the first comment was incorrect.  There are, indeed, many formal invertible sheaves without trivializations over $\text{Spf} \ k[[t]].$  Thus, for instance, over $R=\text{Spec}\ k[t]/\langle t^n\rangle$ there are invertible sheaves on $\mathbb{A}^2_R\setminus\{0\}$ that are not isomorphic to the structure sheaf.  Nonetheless, for every locally Noetherian scheme $S$ that is normal, every invertible sheaf is the pullback of a unique invertible sheaf from $S$. 
A bit more generally, let $T$ be a locally Noetherian scheme, let $\mathbb{A}_{T}$ be a finite type, separated $T$-scheme, let $\mathbb{A}^{*}_{T}\subset \mathbb{A}_{T}$ be an open subset with closed complement $C,$ and let $\sigma:T\to \mathbb{A}_{T}$ be a section.  
Hypothesis 0. The $T$-scheme $\mathbb{A}_T$ is smooth, and for every morphism from a field, $t:\text{Spec}\ \kappa \to T,$ the base change scheme $\mathbb{A}_\kappa$ is integral, $C_\kappa$ has codimension $\geq 2$, every invertible sheaf on $\mathbb{A}_\kappa$ is isomorphic to the structure sheaf, and the following pullback homomorphism is an isomorphism,  $$\kappa^\times \to \mathcal{O}_{\mathbb{A}_\kappa}(\mathbb{A}_\kappa)^\times.$$ 
For every $T$-scheme $S$, denote by $\mathbb{A}_S$ the fiber product $\mathbb{A}_T\times_T S$ with its base change section $\sigma_S:S\to \mathbb{A}_S.$  Denote by $\pi_S:\mathbb{A}_S \to S$ the projection morphism.
Definition 1. For every invertible sheaf $\mathcal{L}$ on $\mathbb{A}^{*}_S$, a normalized trivialization is an isomorphism of $\mathcal{L}$ with $\pi_S^*\sigma_S^*\mathcal{L}$ whose pullback by $\sigma_S$ equals the tautological trivialization. The normalized invertible sheaf $\mathcal{N}$ is the invertible sheaf $$\mathcal{L}\otimes_{\mathcal{O}}\pi_S^*\sigma_S^*\mathcal{L}^\vee.$$  By construction, there is a canonical trivializing global section $n:\mathcal{O}_S\to \sigma_S^*\mathcal{N}.$
Lemma 2.  There exists a normalized trivialization of $\mathcal{L}$ if and only there exists a normalized trivialization of $\mathcal{N}.$
Proof This is just a matter of unwinding definitions. QED 
Lemma 3. If $S$ is Noetherian, then there exists a coherent sheaf $\overline{\mathcal{N}}$ on $\mathbb{A}_S$ whose restriction to $\mathbb{A}^{*}_S$ is isomorphic to $\mathcal{N}.$  If $S$ is also locally integral, then there exists such a coherent sheaf that is isomorphic to $\textit{Hom}_{\mathcal{O}}(\mathcal{M},\mathcal{O}_{\mathbb{A}^2_S})$ for a coherent, reflexive sheaf $\mathcal{M}.$
Proof.  By an Exercise II.5.15 of  Hartshorne's Algebraic geometry, there exists a coherent sheaf $\mathcal{N}^{\text{pre}}$ on $\mathbb{A}_S$ whose restriction to $\mathbb{A}^{*}_S$ equals $\mathcal{N}.$  Then both of the following sheaves are also coherent, $$ \mathcal{M} := (\mathcal{N}^{\text{pre}})^\vee=\textit{Hom}_{\mathcal{O}}(\mathcal{N}^\text{pre},\mathcal{O}_{\mathbb{A}_S}),\ \ \
\overline{\mathcal{N}}=\textit{Hom}_{\mathcal{O}}(\mathcal{M},\mathcal{O}_{\mathbb{A}_S}).
$$  QED
Lemma 4.  Let $X$ be a scheme that is locally Noetherian and normal.  Let $\phi:\mathcal{E}\to \mathcal{F}$ be an injective homomorphism of reflexive, coherent $\mathcal{O}_X$-modules whose cokernel has support of codimension $\geq 2$.  Then $\phi$ is an isomorphism.
Proof.  Denote by $U$ the open complement of the support $B$ of $\text{Coker}(\phi)$.  The restriction $\phi|_U$ is an isomorphism of $\mathcal{O}_U$-modules.  Denote by $\psi_U:\mathcal{F}|_U\to \mathcal{E}|_U$ the inverse of $\phi|_U$.  Since $X$ is normal, by Serre's criterion, $X$ is $S2$.  By hypothesis, the closed subset $B$ has codimension $\geq 2$ everywhere. 
 Thus, by Exercise III.3.5, p. 282, of Hartshorne's Algebraic geometry, there is a unique homomorphism of coherent $\mathcal{O}_X$-modules, $$\psi:\mathcal{F}\to \mathcal{E},$$ whose restriction to $U$ equals $\psi_U.$  By construction, both $\psi\circ \phi$ and $\phi\circ \psi$ restrict on $U$ to equal the respective identity homomorphisms.  Thus, by the uniqueness from Exercise III.3.5, these compositions equal the respective identity homomorphisms on all of $X$. QED
Proposition 5.     For every Noetherian, normal $T$-scheme $S,$ every invertible sheaf on $\mathbb{A}^{*}_S$ has a unique normalized trivialization.
Proof.  Algebraic geometry</I>. ->   Assume that $S=\text{Spec}\ R$ is affine and connected: the uniqueness guarantees the glueing hypothesis relative to an open affine cover of $S.$  By hypothesis, $R$ is an integral domain.  Denote by $K$ the fraction field of $R.$  Since Hom for finitely presented modules commutes with localization, the base change $\overline{\mathcal{N}}_K$ is the reflexive extension of $\mathcal{N}_K.$  By hypothesis, there is a unique normalized trivialization of $\mathcal{N}_K.$  Clearing denominators, there exists a unique normalized trivialization $s_r$ after base change to $\text{Spec}\ R[r^{-1}]$ for some nonzero $r\in R.$
By Krull's Hauptidealsatz, for every minimal prime $\mathfrak{p}$ over the principal ideal $\langle r \rangle$, the height of $\mathfrak{p}$ equals one.  Since $R$ is normal, it is $R1$.  Thus, the local ring $R_{\mathfrak{p}}$ is a DVR, and the scheme $$\mathbb{A}_{\mathfrak{p}}:=\mathbb{A}_{T}\times_T \text{Spec}\ R_{\mathfrak{p}}$$ is a smooth scheme over the regular ring $R_{\mathfrak{p}}$.  So $\mathbb{A}_{\mathfrak{p}}$ is a regular scheme.  Hence it is locally factorial by the Auslander-Buchsbaum-Nagata theorem.  Thus, the base change of the reflexive, rank one sheaf $\overline{\mathcal{N}}$ is an invertible sheaf, cf. Theorem II.6.11 of Hartshorne's Algebraic geometry.  In particular, the unique normalized trivialization over the residue field $\mathbb{A}_{\kappa(\mathfrak{p})}$ lifts to a section over $\mathbb{A}_{\mathfrak{p}}$.  
Since this section is surjective after base change to $\kappa(\mathfrak{p}),$ the section is surjective on $\mathbb{A}_{\kappa(\mathfrak{p})}$ by Nakayama's Lemma.  By Lemma 4, the section is an isomorphism, i.e., it is a trivialization.  After adjusting the section by the multiplicative inverse of the pullback by $\sigma,$ the section is a normalized trivialization.  Since the normalized trivialization is unique, this agrees with $s_r$.  
Therefore $s_r$ extends over every minimal prime over $\langle r \rangle$.  By the usual limit arguments, the maximal open subset on which $s_r$ extends to an $\mathcal{O}_{\mathbb{A}_R}$-module homomorphism that is surjective is an open subscheme of $\mathbb{A}_R$ that contains $\mathbb{A}_{R[r^{-1}]}$ as well as the generic point of every fiber $\mathbb{A}_{\mathfrak{p}}.$  Thus, the complement in $\mathbb{A}_{R[r^{-1}]}$ of this open is a closed subset $B$ of codimension $\geq 2.$  By Lemma 4 once more, $s_r$ extends to a normalized trivialization over all of $\mathbb{A}_R.$  Since the normalized trivialization over $K$ is unique, this normalized trivialization over $R$ is unique. QED
