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

For a Noetherian integral scheme $X$ regular in codimension one and an integer $n\ge 1$,The first summand is the pullback of base Weil divisors; the second is the hyperplane divisor in the projective fibre.

Remove $H$ to obtain $X\times {\mathbb{A}}^n$. The localization sequence for the divisor class group, together with invariance under adjoining affine-space coordinates, shows that the base classes and $H$ generate. Restriction to the generic fibre ${\mathbb{P}}_{K(X)}^n$ sends a base class to zero and $H$ to its degree-one generator, so no nonzero multiple of $H$ can be a base class. This detects the extra summand geometrically by fibre degree.

The positive-dimension hypothesis is essential: ${\mathbb{P}}^0$ is a point and $\mathrm{Cl}(X\times {\mathbb{P}}^0)=\mathrm{Cl}(X)$, with no hyperplane summand.