For any ringed space \((X, \mco_X)\), let \(\Pic X\) be the group of isomorphism classes of invertible sheaves (II, §6). Show that \(\Pic X \cong H^1(X, \mco_X^*)\), where \(\mco_X^*\) denotes the sheaf whose sections over an open set \(U\) are the units in the ring \(\Gamma(U, \mco_X)\), with multiplication as the group operation.
Hint: For any invertible sheaf \(\mcl\) on \(X\), cover \(X\) by open sets \(U_i\) on which \(\mcl\) is free, and fix isomorphisms \(\varphi_i: \mco_{U_i} \iso \ro{\mcl}{U_i}\). Then on \(U_i \intersect U_j\), we get an isomorphism \(\inverseof{\varphi_i} \circ \varphi_j\) of \(\mco_{U_i \intersect U_j}\) with itself. These isomorphisms give an element of \(\check{H}^1(\mathfrak{U}, \mco_X^*)\). Now use (Ex. 4.4).
Solution
1. For an open cover \(\mathfrak{U} = \{U_i\}\) of \(X\), isomorphism classes of invertible sheaves free on every \(U_i\) correspond bijectively to \(\check{H}^1(\mathfrak{U}, \mco_X^*)\), compatibly with tensor products.
For such \(\mcl\) fix \(\varphi_i \colon \mco_{U_i} \iso \mcl|_{U_i}\) as in the hint. On \(U_i \cap U_j\) the automorphism \(\varphi_i^{-1} \circ \varphi_j\) of \(\mco_{U_i \cap U_j}\) is multiplication by a unit \(g_{ij} \in \Gamma(U_i \cap U_j, \mco_X^*)\), and \(g_{ij} g_{jk} = g_{ik}\) on \(U_i \cap U_j \cap U_k\), so \((g_{ij})\) is a Čech \(1\)-cocycle. Replacing each \(\varphi_i\) by \(\varphi_i \circ h_i\) with \(h_i \in \Gamma(U_i, \mco_X^*)\), or \(\mcl\) by an isomorphic sheaf, changes \((g_{ij})\) by the coboundary of \((h_i)\); a cocycle that is a coboundary allows the \(\varphi_i\) to be chosen compatibly, so they glue to an isomorphism \(\mco_X \iso \mcl\). Every cocycle arises by gluing the sheaves \(\mco_{U_i}\) along multiplication by \(g_{ij}\), and the cocycle of \(\mcl \otimes \mcl'\) is the product of the cocycles.
2. \(\Pic X \cong \check{H}^1(X, \mco_X^*) = \varinjlim_{\mathfrak{U}} \check{H}^1(\mathfrak{U}, \mco_X^*)\).
Every invertible sheaf is free on the members of some open cover, and the bijections of step 1 commute with the refinement maps.
3. Q.E.D.
By Exercise III.4.4 the natural map \(\check{H}^1(X, \mcf) \to H^1(X, \mcf)\) is an isomorphism for every sheaf of abelian groups \(\mcf\); with step 2 this gives \(\Pic X \cong H^1(X, \mco_X^*)\). The comparison is needed because \(\mco_X^*\) is not a quasicoherent sheaf, so the equality of Čech and derived-functor cohomology for affine covers of Noetherian separated schemes does not apply to it.