Is the divisor class group $Cl(X \times \mathbb{A}^n)$ isomorphic to $Cl(X)$?

Question

Let $X$ be a noetherian integral separated scheme which is regular in codimension one. According to Hartshorne's Algebraic Geometry, Prop. II.6.6, The divisor class group $Cl(X)$ is isomorphic to $Cl(X \times \mathbb{A}^1)$. Is the statement true when $\mathbb{A}^1$ is replaced with $\mathbb{A}^n$?

Answer 1

Sure because

$Cl(X\times \mathbb{A}^n)=Cl((X\times \mathbb{A}^{n-1})\times \mathbb{A})\cong$

$\cong Cl(X\times \mathbb{A}^{n-1})\cong \dots \cong Cl(X)$