For \(f: X \to Y\) and \(\mcf\) a sheaf of abelian groups on \(X\), \[ E_2^{p,q} = H^p\qty{Y, R^q f_* \mcf} \Longrightarrow H^{p+q}(X, \mcf) . \]
It is the derived-functor form of \(\globsec{X;\wait} = \globsec{Y;\wait} \circ f_*\), and the only thing usually needed from it is what happens when it degenerates.
If \(R^{q>0} f_* \mcf = 0\), the sequence collapses to the edge and \(H^i(X,\mcf) \cong H^i(Y, f_*\mcf)\) for all \(i\); that is the affine-morphism case above. In low degrees, with no vanishing at all, one always gets the exact sequence \[ 0 \to H^1(Y, f_* \mcf) \to H^1(X, \mcf) \to H^0(Y, R^1 f_* \mcf) \to H^2(Y, f_*\mcf) , \] which is the form most often actually used: a class on \(X\) is either pulled back from the base or detected fibrewise.
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D-COHRIFHigher direct imagesuses