\(H^1(X,\OO_X^\times)\cong\Pic X\), and for \(\OO_X\)-modules \(\mcf,\mcg\) with \(\mcg\) locally free, \(\Ext^1(\mcg,\mcf)\cong H^1(X,\mcf\otimes\mcg^\vee)\) classifies extensions of \(\mcg\) by \(\mcf\). By Serre's criterion, a noetherian scheme \(X\) is affine if and only if \(H^1(X,\mci)=0\) for every coherent ideal sheaf \(\mci\subseteq\OO_X\).
Computing cohomology, the Čech comparison, the cohomology of \(\OO_{\PP^n}(d)\), and \(H^1\) as the obstruction to lifting sections and as \(\Pic X\).
Vanishing and duality, Serre’s affineness criterion, Serre duality, and Riemann–Roch for curves and surfaces.
Derived functors and vanishing, right derived functors of global sections, flasque resolutions, affine vanishing, and Grothendieck vanishing.
Cohomology of projective schemes, finiteness, Serre vanishing for large twists, and the Euler characteristic and Hilbert polynomial in flat families.
Cohomology in families, higher direct images, formal functions, and when cohomology commutes with base change.