(Picard group)
Let \((X,\OO_X)\) be a ringed space with commutative structure sheaf. Its Picard group \(\Pic(X)\) is the group of isomorphism classes of invertible \(\OO_X\)-modules under tensor product, with identity \([\OO_X]\) and inverse \([\dualof{\mcl}]\), where \(\dualof{\mcl}=\sheafhom_{\OO_X}(\mcl,\OO_X)\) [@Har10a, Chapter II, §6].
For \(\mcl\) invertible, the evaluation map \(\mcl\otimes_{\OO_X}\dualof{\mcl}\to\OO_X\) is an isomorphism: on every trivializing open it is the multiplication isomorphism \(\OO_X\otimes\OO_X\cong\OO_X\). There is a natural group isomorphism \(\Pic(X)\cong H^1(X,\OO_X^\times)\), obtained by sending local frames to their transition-unit cocycle, as proved in P-AGH345PICH1.
(Rank and tensor inverses)
On a nonempty scheme, a locally free sheaf of constant rank \(r\ge2\) has no tensor inverse. At any point, an inverse would give \(\kappa(x)^r\otimes_{\kappa(x)}V\cong\kappa(x)\) for a vector space \(V\). For \(V=0\) the left side is zero; otherwise it contains at least \(r\) independent vectors and cannot be one-dimensional.
(Affine and projective examples)
For a PID or a local ring \(A\), one has \(\Pic(\Spec A)=0\). Under the affine sheaf-module correspondence, invertible sheaves correspond to finite projective modules of rank one; these are free over either such ring [@Har10a, Chapter II, §§5 and 6]. For a field \(k\) and \(n\ge1\), one has \(\Pic(\PP_k^n)\cong\ZZ\), with generator \([\OO(1)]\) [@Har10a, Proposition II.6.4 and Corollary II.6.16]. For \(n=0\), projective space is \(\Spec k\) and its Picard group is zero.
(Cartier--Weil comparison)
For an integral noetherian separated locally factorial scheme \(X\), the correspondence \(D\mapsto\OO_X(D)\) identifies the Weil divisor class group with \(\Pic(X)\) [@Har10a, Corollary II.6.16]. In particular it applies to a smooth integral variety over a field, since its regular local rings are factorial [@Har10a, Remark II.6.11.1A].
Dependencies
T-MODVBLocally free sheaves are vector bundlesuses
Backlinks
T-CRVJACFUNThe Jacobian represents $\Pic^0(X/-)$, and is a smooth proper group scheme of dimension $g$uses