Let \(X\) be a noetherian integral separated scheme that is regular in codimension one, and let \(n\ge1\). Show that \(\PP_X^n=X\times_{\Spec\ZZ}\PP_\ZZ^n\) has the same properties and \[ \Cl(\PP_X^n)\cong\Cl(X)\oplus\ZZ. \]
(Dimension zero)
For \(n=0\), \(\PP_X^0\cong X\) and the class group is \(\Cl(X)\), without an additional summand. For example, \(X=\Spec k\) for a field \(k\) has no prime divisors, so \(\Cl(\PP_X^0)=0\), not \(\ZZ\). Thus the extra hyperplane summand in the source formula requires \(n\ge1\) [@Har10a, Exercise II.6.1].
Solution
Put \(P=\PP_X^n\), let \(p:P\to X\) be the projection, and let \(H\subseteq P\) be the relative hyperplane defined by \(x_0=0\). Its complement is \(U=D_+(x_0)\cong\AA_X^n\). The class groups are Weil-divisor class groups.
1. The scheme \(P\) is noetherian, integral, separated, and regular in codimension one.
Projective space over a noetherian scheme is noetherian, and \(P\to X\) is separated; composing with the separated structure morphism of \(X\) makes \(P\) separated. Its standard open cover consists of \(n+1\) copies of \(\AA_X^n\). Iterating [@Har10a, Proposition II.6.6] shows that each such chart is integral and regular in codimension one. Their pairwise intersections are nonempty opens in integral schemes, since they contain the corresponding nonempty opens of the generic fiber over \(X\). Thus their union is irreducible and reduced, hence integral. Regularity in codimension one is local on open subsets and therefore holds on \(P\).
2. Pullback of prime divisors along \(p\) gives a homomorphism \(p^*:\Cl(X)\to\Cl(P)\), and its composite with restriction to \(U\) is an isomorphism \(\Cl(X)\to\Cl(U)\).
For a prime divisor \(D\subseteq X\), the closed subscheme \(p^{-1}(D)=\PP_D^n\) is integral and has codimension one in \(P\). Define \(p^*D=\PP_D^n\) and extend linearly to divisors. At its generic point, the local ring is a localization of \[ \OO_{X,\eta_D}[t_1,\ldots,t_n] \] at the prime generated by the uniformizer of the DVR \(\OO_{X,\eta_D}\). That uniformizer still generates the maximal ideal, so its valuation on \(K(X)^\times\) is the original valuation \(v_D\). Prime divisors dominating \(X\) have zero valuation on \(K(X)^\times\), since all nonzero elements of \(K(X)\) are units on the generic fiber. The prime-divisor classification in the proof of [@Har10a, Proposition II.6.6], iterated on the affine charts, shows that these and the vertical divisors \(\PP_D^n\) exhaust the prime divisors of \(P\). Consequently, for \(g\in K(X)^\times\), \[ \operatorname{div}_P(p^*g)=p^*(\operatorname{div}_X g). \] Thus the construction descends to class groups. Its restriction to \(U=\AA_X^n\) is precisely the iterated pullback isomorphism of Proposition II.6.6, proving the last assertion.
3. Every class in \(\Cl(P)\) is of the form \(p^*\alpha+m[H]\), with \(\alpha\in\Cl(X)\) and \(m\in\ZZ\).
The hyperplane \(H\cong\PP_X^{n-1}\) is an irreducible codimension-one closed subset. The divisor restriction sequence is therefore exact: \[ \ZZ\xrightarrow{m\mapsto m[H]}\Cl(P)\xrightarrow{j^*}\Cl(U)\longrightarrow0, \] where \(j:U\hookrightarrow P\) is the open inclusion [@Har10a, Proposition II.6.5(c)]. Given \(\beta\in\Cl(P)\), step 2 gives an \(\alpha\in\Cl(X)\) with \(j^*p^*\alpha=j^*\beta\). The difference \(\beta-p^*\alpha\) is in the displayed kernel and is therefore an integer multiple of \([H]\).
4. If \(p^*\alpha+m[H]=0\), then \(\alpha=0\) and \(m=0\).
Restrict to \(U\). The class \([H]\) restricts to zero, and step 2 makes \(j^*p^*\) injective, so \(\alpha=0\). Thus \(mH\) is a principal divisor on \(P\).
Let \(K=K(X)\). Restriction to the generic fiber \(P_K=\PP_K^n\) discards vertical prime divisors and retains the divisors dominating \(X\). It takes principal divisors to principal divisors: on an affine chart, passing to the generic fiber localizes by the nonzero base elements, so the local rings and valuations at the surviving codimension-one primes are unchanged. It takes \(H\) to the ordinary hyperplane \(H_K\). Thus \(mH_K\) is principal on \(\PP_K^n\).
A rational function on \(\PP_K^n\) is the quotient of two homogeneous polynomials of the same degree. Its divisor has degree zero, whereas \(\deg(mH_K)=m\) [@Har10a, Proposition II.6.4]. It follows that \(m=0\).
5. Q.E.D.
Step 1 proves the scheme-theoretic assertions. Steps 3 and 4 show that \[ \boxed{\Cl(X)\oplus\ZZ\xrightarrow{\cong}\Cl(\PP_X^n),\qquad (\alpha,m)\longmapsto p^*\alpha+m[H]} \] is a bijective group homomorphism.