# Picard group of $\mathbb{A}^1-\{0\}$. [duplicate]

I know that $$\text{Pic}(\mathbb{A}^n)=0$$, but it turns out that $$\text{Pic}(\mathbb{A}^1-\{0\})=0$$ also. Is there a simple explanation for this?

$$\mathbb{A}^1_k - (0) = Spec(k[x]_x)$$, which is the spec of a Noetherian UFD, see here .
Hartshorne proposition II.6.2 : Let $$A$$ be a Noetherian domain. Then, $$A$$ is a unique factorization domain if and only if $$X = Spec(A)$$ is normal and $$Cl(X) = 0$$.
Then just use that $$Cl(X) = Pic(X)$$ for schemes that are regular at all points and satisfy $$(*)$$ (Noetherian, integral, separated, regular in codimension 1). This is proposition II.6.11 in Hartshorne.