Let \(f:X\to Y\) be a morphism of schemes. Show that \(\mathcal L\mapsto f^*\mathcal L\) induces a homomorphism \(f^*:\Pic(Y)\to\Pic(X)\).
If \(f\) is a finite morphism of nonsingular curves, show that this homomorphism agrees with the divisor-class pullback \(f^*:\Cl(Y)\to\Cl(X)\) under the isomorphisms of Corollary II.6.16.
Let \(X\) be a locally factorial integral closed subscheme of \(\PP_k^n\), with inclusion \(f:X\to\PP_k^n\). Show that Picard pullback agrees, under the same isomorphisms, with the class-group homomorphism defined by hypersurface intersection in Exercise II.6.2.
Solution
The Cartier-divisor/Picard comparison sends a Cartier divisor class represented by \(D\) to \([\OO(D)]\). For the nonsingular curves in part (b), and for the locally factorial schemes in part (c), it identifies the Weil class group with the Picard group [@Har10a, Corollary II.6.16].
1. Pullback of invertible sheaves induces a group homomorphism \(f^*:\Pic(Y)\to\Pic(X)\) for every morphism of schemes.
By definition, \[ f^*\mathcal L=f^{-1}\mathcal L\otimes_{f^{-1}\OO_Y}\OO_X. \] On the inverse image of an open set where \(\mathcal L\) is free of rank one, this pullback is free of rank one as well. Thus it is invertible, and the pullback functor takes isomorphisms to isomorphisms, so the operation is well-defined on isomorphism classes.
Tensor associativity gives canonical isomorphisms \[ f^*(\mathcal L\otimes_{\OO_Y}\mathcal M) \cong f^*\mathcal L\otimes_{\OO_X}f^*\mathcal M, \qquad f^*\OO_Y\cong\OO_X. \] For example, the first isomorphism sends \((\ell\otimes m)\otimes a\) to \((\ell\otimes a)\otimes(m\otimes1)\), with inverse multiplying the two \(\OO_X\) coefficients. These local tensor maps are compatible with restrictions and therefore define the stated sheaf isomorphism. They show that pullback preserves the group operation and identity, proving part (a). No flatness assumption on \(f\) is needed.
2. For a finite morphism of nonsingular curves and every divisor \(D\) on \(Y\), there is a canonical isomorphism\[ f^*\OO_Y(D)\cong\OO_X(f^*D). \]
The finite morphism \(f\) is dominant: its image is closed, and its zero-dimensional fibers exclude the possibility that the whole curve maps to a point. It therefore gives an inclusion \(K(Y)\hookrightarrow K(X)\). For a closed point \(Q\in Y\), choose a uniformizer \(t_Q\) of the DVR \(\OO_{Y,Q}\). The divisor pullback is defined as follows [@Har10a, definition preceding Proposition II.6.9]: \[ f^*[Q]=\sum_{f(P)=Q}e(P/Q)[P],\qquad e(P/Q)=v_P(f^\#t_Q). \] Replacing \(t_Q\) by a unit multiple leaves these valuations unchanged.
Every divisor \(D\) on the nonsingular curve \(Y\) is Cartier. Choose rational local equations \(g_i\) for it on an open cover \((V_i)\). Then \(\OO_Y(D)|_{V_i}\) is generated by \(g_i^{-1}\), and the transition functions are ratios of these rational functions, which are regular units on overlaps. On \(f^{-1}(V_i)\), pullback is generated by \((f^\#g_i)^{-1}\) with the pulled-back transition functions. The functions \(f^\#g_i\) are nonzero because \(f\) is dominant.
At \(P\) above \(Q\), writing the local equation of \(D\) as a unit times \(t_Q^{n_Q}\) gives \[ v_P(f^\#g_i)=n_Qe(P/Q), \] since units pull back to units. Thus these pulled-back local equations define exactly the divisor \(f^*D\) specified above. Their inverse local generators and transition functions are those of \(\OO_X(f^*D)\), proving the canonical isomorphism. Passing to classes proves part (b), including negative divisor coefficients.
3. For the inclusion in part (c), Picard pullback agrees with hypersurface-intersection pullback on class groups.
If \(\dim X=0\), then \(X\) is the spectrum of a field: an integral zero-dimensional scheme has one point, and its local ring is a field. Hence \(\Cl(X)=\Pic(X)=0\), so both homomorphisms are zero. This also handles the case \(n=0\).
Suppose now that \(\dim X\ge1\). The group \(\Cl(\PP_k^n)\) is infinite cyclic, generated by a hyperplane class \([H]\) [@Har10a, Proposition II.6.4]. Choose \(H=V_+(\ell)\) not containing \(X\). Such a hyperplane exists: some projective coordinate is nonzero at the generic point of \(X\). Let \(D=X.H\) be its hypersurface-intersection divisor.
On a standard chart \(D_+(x_i)\), the hyperplane has local equation \(\ell/x_i\). Its restriction is nonzero on the integral scheme \(X\), so these restricted equations define an effective Cartier divisor on \(X\). At each codimension-one point, their valuations equal the intersection multiplicities, namely the lengths of the corresponding DVR quotients. Thus the associated Weil divisor is precisely \(D\), as in Exercise II.6.2.
The inverse equations \((\ell/x_i)^{-1}\) are local generators of \(\OO_{\PP^n}(H)\). After restriction to \(X\) they give the local generators and transition functions of \(\OO_X(D)\). Consequently \[ f^*\OO_{\PP^n}(H)\cong\OO_X(D). \] Under Corollary II.6.16, Picard pullback therefore sends \([H]\) to \([X.H]\), exactly as the homomorphism of Exercise II.6.2 does. Both are homomorphisms and \([H]\) generates the source, so they agree everywhere. The argument uses a representative not containing \(X\); it does not assume that every ambient Cartier divisor can be pulled back as a Cartier divisor along the inclusion.
4. Q.E.D.
Step 1 constructs Picard pullback for arbitrary morphisms, step 2 proves compatibility with finite-curve divisor pullback, and step 3 proves compatibility with hypersurface intersection for closed inclusions.