Quasicoherence, coherence, twisting sheaves, and line bundles.
For \(X\) projective over a noetherian ring and \(\mcf\) coherent on \(X\), there is a surjection \(\bigoplus_{i=1}^r\OO_X(-d_i)\to\mcf\), so long exact sequences reduce statements about \(H^j(X,\mcf)\) to the cohomology of the sheaves \(\OO(d)\) on projective space.
Quasicoherence and twisting, the local test, and coherent sheaves as quotients of finite sums of twists.
Differentials, the relative and conormal sequences, the canonical sheaf, and adjunction.
Operations and functoriality, the module operations, the two image functors and their adjunction, and ideal sheaves.
Locally free sheaves and the Picard group, the vector bundle dictionary, and the group \(\Pic X\) of invertible sheaves under tensor product.
Sheaves on projective space, graded modules, global generation and ampleness, and Serre’s theorems.