= Divisor class group of a projective-space bundle with trivial vector bundle
{title2=$\operatorname{Cl}(X\times\mathbb P^n)$}
For a <Noetherian> integral <scheme> $X$ regular in codimension one and an <integer> $n\ge1$,
$$
\operatorname{Cl}(X\times\mathbb P^n)
\cong\operatorname{Cl}(X)\oplus\mathbb Z[H],
\qquad H=X\times\mathbb P^{n-1}.
$$
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 $\mathbb P^n_{K(X)}$ 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 $\operatorname{Cl}(X\times\mathbb P^0)=\operatorname{Cl}(X)$, with no hyperplane summand.
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