= 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 $X$ that is regular in codimension one,
$$
\operatorname{Cl}(X\times\mathbb P^n)\cong\operatorname{Cl}(X)\oplus\mathbb Z[H],
$$
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 $H$.
Back to article page
