Slogan
If you have some objects which are locally trivial but globally are possily not trivial which live over \(X\), then the isomorphism classes of such objects are classified by \(H^1(X, \underline{Aut})\) where \(\underline{Aut}\) is the sheaf of automorphisms of your objects.
Example
- Consider vector bundles \(\pi:E\to X\). Then locally \(E\) looks like \(U\times \mathbb A^n\), so vector bundles are classified by \(H^1(X, \GL(n, \mathcal O_X))\).
- Now consider locally free sheaves over \(X\). These locally look like \(\mathcal O^{\oplus n}_X\), whose automorphism group is also \(\GL(n, \mathcal O_X)\), so they are classified by \(H^1(X, \GL(n, \mathcal O_X))\).
- Finally, consider line bundles over \(X\). This is a special case of (2) above, so these are classified by \(H^1(X,\GL(1,\mathcal O_X)) \cong H^1(X, \mathcal O^*_X)\) as expected.