Let \(X\) be a regular noetherian scheme, and \(\mce\) a locally free coherent sheaf of rank \(\geq 2\) on \(X\).
If \(X\) is nonempty and connected, show that \(\Pic \PP(\mce) \cong \Pic X \times \ZZ\). For an arbitrary regular noetherian \(X\), prove the componentwise formula \[ \Pic\PP(\mce)\cong\Pic X\times\ZZ^{\pi_0(X)}, \] where \(\pi_0(X)\) is the finite set of connected components.
If \(\mce'\) is another locally free coherent sheaf on \(X\), show that \(\PP(\mce) \cong \PP(\mce')\) over \(X\) if and only if there is an invertible sheaf \(\mcl\) on \(X\) with \(\mce' \cong \mce \tensor \mcl\).
(The connectedness requirement)
For \(X=\Spec k\sqcup\Spec k\) and \(\mce=\OO_X^{\oplus2}\), one has \(\Pic(X)=0\) and \(\PP(\mce)=\PP_k^1\sqcup\PP_k^1\), whose Picard group is \(\ZZ^2\). Thus the single-\(\ZZ\) formula in the original card cannot hold without a nonempty connected base; the general formula has one independent fibre degree per connected component.
Solution
Put \(P=\PP(\mce)=\operatorname{Proj}_X\operatorname{Sym}^{\bullet}\mce\), and let \(\pi:P\to X\) be the projection. Use the quotient convention, with tautological quotient \(\pi^*\mce\twoheadrightarrow\OO_P(1)\). For a locally constant function \(n:X\to\ZZ\), write \(\OO_P(n)\) for the invertible sheaf whose restriction over a component where \(n=a\) is \(\OO_P(a)\).
1. The scheme \(X\) is a finite disjoint union of integral open-and-closed components, and \(\pi_*\OO_P\cong\OO_X\), \(\pi_*\OO_P(1)\cong\mce\) canonically.
A regular local ring is a domain [@Har10a, Remark II.6.11.1A]. Thus distinct irreducible components of \(X\) cannot meet: such an intersection would give a local ring with more than one minimal prime. There are finitely many irreducible components because \(X\) is noetherian. Each is therefore open and closed, and each is reduced and irreducible, hence integral. These are exactly the connected components.
On a framed affine open \(V=\Spec A\) where \(\mce\) has rank \(r\ge2\), the projection is \(\PP_A^{r-1}\to\Spec A\). The identities \[ \Gamma(\PP_A^{r-1},\OO)=A,\qquad \Gamma(\PP_A^{r-1},\OO(1))=\bigoplus_{i=1}^r A T_i \] are the projective-space section calculation [@Har10a, Theorem III.5.1]. The first identification is the structure map, and the second sends a frame of \(\mce\) to its degree-one coordinate sections. They respect changes of frame and restrictions and give the two global direct-image identifications.
2. Every invertible sheaf \(\mathcal M\) on \(P\) has, on each sufficiently small affine open \(V\subseteq X\), a unique form\[ \mathcal M|_{P|_V}\cong\pi_V^*\mathcal N_V\otimes\OO_{P|_V}(n_V), \qquad [\mathcal N_V]\in\Pic(V),\quad n_V\in\ZZ. \]
Choose \(V\) inside a component of \(X\) and with a frame for \(\mce\); then \(V\) is regular, integral, affine and noetherian, and \(P|_V\cong\PP_V^{r-1}\) with \(r-1\ge1\). Every local ring of \(V\) is factorial because it is regular [@Har10a, Remark II.6.11.1A]. The local rings of \(\PP_V^{r-1}\) are also factorial: on a standard chart, the local ring at a prime over \(p\in V\) is a localization of a polynomial ring over the factorial ring \(\OO_{V,p}\). Both schemes are integral, noetherian and separated, so the Cartier–Weil comparison identifies their class groups with their Picard groups.
The class-group isomorphism proved in P-AGH261CLPROJBUN consequently gives \[ \Pic(\PP_V^{r-1})\cong\Pic(V)\oplus\ZZ, \] with the two summands mapped by pullback and by powers of \(\OO(1)\). This proves existence and uniqueness of the displayed local form. No global separatedness assumption on \(X\) is used.
3. The integers \(n_V\) define a locally constant function \(n:X\to\ZZ\), and \(\mathcal M\cong\pi^*\mathcal N\otimes\OO_P(n)\) for a unique invertible sheaf class \([\mathcal N]\in\Pic(X)\).
On the fibre over a point \(x\in V\), the local form in step 2 restricts to \(\OO_{\PP^{r-1}_{\kappa(x)}}(n_V)\), since a line bundle pulled back from \(V\) is trivial on that fibre. The Picard group of projective space over a field is freely generated by \(\OO(1)\) when its dimension is positive [@Har10a, Proposition II.6.4 and Corollary II.6.16]. Hence two integers computed on overlapping affine opens agree at every point of the overlap. They define a locally constant function \(n\).
Put \(\mathcal M_0=\mathcal M\otimes\OO_P(-n)\) and define \(\mathcal N=\pi_*\mathcal M_0\). On each open \(V\) of step 2, one has \(\mathcal M_0|_{P|_V}\cong\pi_V^*\mathcal N_V\). For an invertible sheaf \(\mathcal N_V\), the identity \(\pi_{V*}\pi_V^*\mathcal N_V\cong\mathcal N_V\) follows from \(\pi_{V*}\OO=\OO_V\) by trivializing \(\mathcal N_V\) on an open cover of \(V\). Thus \(\mathcal N|_V\cong\mathcal N_V\), and \(\mathcal N\) is invertible on all of \(X\).
The counit \(\pi^*\pi_*\mathcal M_0\to\mathcal M_0\) is an isomorphism on every such open, because there \(\mathcal M_0\) is a pulled-back invertible sheaf and \(\pi_*\OO=\OO_X\). It is therefore an isomorphism globally. This proves the asserted form of \(\mathcal M\). Its degree function is uniquely determined by its fibres, and then \(\mathcal N\) is recovered as \(\pi_*(\mathcal M\otimes\OO_P(-n))\), proving uniqueness.
4. The componentwise isomorphism in (a) is\[ \boxed{\Pic(X)\oplus\ZZ^{\pi_0(X)}\xrightarrow{\cong}\Pic(P), \qquad ([\mathcal N],n)\longmapsto[\pi^*\mathcal N\otimes\OO_P(n)].} \]
The map preserves the group operation because pullback commutes with tensor products and \(\OO_P(n)\otimes\OO_P(m)\cong\OO_P(n+m)\). Steps <1>2–<1>3 prove that it is bijective. By step 1, a locally constant integer-valued function is exactly a choice of one integer for each of the finitely many components. For nonempty connected \(X\) this gives one copy of \(\ZZ\). For empty \(X\), the component set is empty and both Picard groups and the degree summand are zero.
5. If \(\varphi:\PP(\mce)\xrightarrow{\cong}\PP(\mce')\) is an isomorphism over \(X\), then \(\mce'\cong\mce\otimes\mcl\) for an invertible sheaf \(\mcl\) on \(X\).
The isomorphic fibres have equal dimensions, so \(\mce'\) has the same rank as \(\mce\) at every point. Apply step 3 to \(\varphi^*\OO_{\PP(\mce')}(1)\), obtaining \[ \varphi^*\OO_{\PP(\mce')}(1)\cong\pi^*\mcl\otimes\OO_P(n) \] for an invertible \(\mcl\) and a locally constant integer \(n\). On each fibre, \(\varphi\) induces an automorphism between the Picard groups of two projective spaces of the same positive dimension. Thus it sends the generator to either its positive or negative, so the fibre value of \(n\) is \(1\) or \(-1\). The pulled-back \(\OO(1)\) is globally generated on that fibre, whereas \(\OO(-1)\) has no nonzero global section. Consequently \(n=1\) on every component.
Let \(\pi':\PP(\mce')\to X\) be the other projection. Pushing forward the isomorphism with \(n=1\) and using \(\pi=\pi'\circ\varphi\) gives \[ \mce'\cong\pi'_*\OO_{\PP(\mce')}(1) \cong\pi_*(\pi^*\mcl\otimes\OO_P(1)) \cong\mcl\otimes\pi_*\OO_P(1) \cong\mcl\otimes\mce. \] Here step 1 supplies the degree-one direct images; the tensor identity follows by trivializing the invertible sheaf \(\mcl\) locally on \(X\).
6. Conversely, an isomorphism \(\mce'\cong\mce\otimes\mcl\) induces an isomorphism of the projective bundles over \(X\).
On an open set where \(\mcl\) has a frame, it identifies \(\mce\otimes\mcl\) with \(\mce\) and therefore identifies their symmetric algebras and relative Proj schemes. If the frame is multiplied by a unit \(u\), the comparison on the degree-\(d\) piece is multiplied by \(u^d\). On every degree-zero homogeneous localization, numerator and denominator have the same total degree, so these powers cancel. Hence this frame change acts as the identity on relative Proj. The local projective-bundle isomorphisms therefore agree on overlaps and glue to \(\PP(\mce\otimes\mcl)\cong\PP(\mce)\) over \(X\). Composing with the isomorphism induced by \(\mce'\cong\mce\otimes\mcl\) proves the converse.
7. Q.E.D.
Step 4 proves (a), and steps <1>5–<1>6 prove both directions of (b).