The proof by \textit{P. Deligne} and \textit{L. Illusie} [Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)] of the degeneration of the \(E_ 1\)-term of the Hodge spectral sequence of a smooth variety X, after its reduction to the characteristic \(p\) case, is embellished by using explicit quasi- isomorphisms instead of working locally in the derived category of complexes of \({\mathcal O}_ X\)-modules.