Divisor class group of a projective-space bundle with trivial vector bundle ($\mathrm{Cl}(X\times {\mathbb{P}}^n)$)

For a Noetherian integral scheme $X$ that is regular in codimension one,where $H=X\times {\mathbb{P}}^{n-1}$ is a hyperplane divisor. Restriction to $X\times {\mathbb{A}}^n$ gives the first summand, while restriction to the generic projective-space fiber detects the coefficient of $H$.