Locally free sheaves are vector bundles (Tag T-MODVB)
\(\mcf\) is free if \(\mcf \cong \bigoplus_{i \in I} \OO_X\) for some index set \(I\), whose cardinality is its rank; it is free of rank \(n\) if \(\mcf \cong \OO_X\sumpower{n}\). \(\mcf\) is locally free if \(X\) has an open cover \(\ts{U_j}\) with each \(\ro{\mcf}{U_j}\) free, and locally free of rank \(n\) if each \(\ro{\mcf}{U_j}\) is free of rank \(n\). The rank of a locally free sheaf is constant on each connected component of \(X\). An invertible sheaf is a locally free sheaf of rank \(1\).
(Bundles and sheaves)
Taking a rank-\(n\) vector bundle \(\pi: E \to X\) to its sheaf of sections \(U \mapsto \ts{s: U \to E \mid \pi s = \id}\) is an equivalence between rank-\(n\) vector bundles on \(X\) and locally free \(\OO_X\)-modules of rank \(n\).
Both objects are glued from trivialisations on a cover and transition functions in \(\GL_n(\OO_X(U_{ij}))\) satisfying the cocycle condition; the equivalence identifies these two constructions. The sheaf side is the one to compute with, because it lives in an abelian category where kernels, cokernels and cohomology exist; the bundle side is the one that makes geometric statements about fibres legible.
A locally free sheaf is quasicoherent, and on a locally Noetherian scheme it is coherent when the rank is finite; local freeness additionally requires the transition maps to lie in \(\GL_n\) on a trivializing cover. The fibre \(\mcf \tensor \kappa(x)\) recovers the bundle’s fibre, and a coherent sheaf on a reduced locally Noetherian scheme is locally free exactly when the fibre dimension \(x \mapsto \dim_{\kappa(x)} \mcf \tensor \kappa(x)\) is locally constant.
The correspondence identifies locally free sheaves with vector bundles through sheaves of sections and transition functions. Locally free sheaves do not form an abelian category: the cokernel of \(\OO_{\AA^1}\xrightarrow{x}\OO_{\AA^1}\) is the skyscraper sheaf \(\OO_{\AA^1}/(x)\), which is not locally free, so kernels and cokernels are taken among all \(\OO_X\)-modules.
Rank one
The Picard group (Tag D-MODPIC)
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].
On a noetherian integral separated locally factorial scheme, \(\Pic(X)\cong\Cl(X)\) (divisors). Global sections \(s_0,\ldots,s_n\) generating an invertible sheaf \(\mcl\) give a morphism \(\varphi\colon X\to\PP^n\) with \(\varphi^*\OO(1)\cong\mcl\) and \(\varphi^*x_i=s_i\); \(\mcl\) is very ample over \(k\) when some such \(\varphi\) is an immersion.
\(\Pic(\PP^n_k)\cong\ZZ\), generated by \(\OO(1)\): quasicoherence and twisting.
Vector bundles on the projective line
Vector bundles on the projective line split (Tag T-BIRKGROTH)
Let \(k\) be a field. Every locally free sheaf of finite rank on \(\PP^1_k\) is isomorphic to \(\OO(a_1) \oplus \cdots \oplus \OO(a_r)\) for integers \(a_1 \geq \cdots \geq a_r\), and these integers are uniquely determined. [@Har10a, Exercise V.2.6]
The tangent bundle of \(\PP^2\) restricted to a line \(\ell\) is \(\OO_\ell(2) \oplus \OO_\ell(1)\): the tangent bundle of the line is the first summand and the normal bundle \(\OO_\ell(1)\) the second.