(Filtration of a coherent sheaf)
Let \(X\) be a Noetherian scheme and \(\mcf\) a coherent \(\OO_X\)-module. There is a filtration \(0 = \mcf_0 \subseteq \mcf_1 \subseteq \cdots \subseteq \mcf_m = \mcf\) by coherent subsheaves such that for each \(j\) there are an integral closed subscheme \(i_j \colon Z_j \hookrightarrow X\) and a nonzero coherent ideal sheaf \(\mathcal{I}_j \subseteq \OO_{Z_j}\) with \(\mcf_j / \mcf_{j-1} \cong i_{j*} \mathcal{I}_j\).
(Dévissage)
Dévissage is the reduction of a statement about all coherent sheaves on a Noetherian scheme to the sheaves \(i_* \mathcal{I}\) of the proposition. If a property \(P\) of coherent sheaves holds for an extension whenever it holds for the subsheaf and the quotient, and holds for \(i_* \mathcal{I}\) for every integral closed subscheme \(i \colon Z \hookrightarrow X\) and nonzero coherent ideal \(\mathcal{I} \subseteq \OO_Z\), then \(P\) holds for every coherent sheaf, by induction along the filtration.
Grothendieck’s proof that \(R^i f_* \mcf\) is coherent for \(f\) proper and \(\mcf\) coherent uses dévissage together with Noetherian induction on the support: it reduces to sheaves on an integral scheme, where Chow’s lemma reduces the proper case to the projective case.