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?



Picard group of the product of a local normal scheme with $\mathbb{A}^2-\{0\}$
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**


