In mathematics, the Hodge–de Rham spectral sequence (named in honor of W. V. D. Hodge and Georges de Rham) is an alternative term sometimes used to describe the Frölicher spectral sequence (named after Alfred Frölicher, who actually discovered it).
On a compact Kähler manifold, the sequence degenerates, thereby leading to the Hodge decomposition of the de Rham cohomology.
-page of the spectral sequence, is the cohomology with values in the sheaf of holomorphic differential forms.
The existence of the spectral sequence as stated above follows from the Poincaré lemma, which gives a quasi-isomorphism of complexes of sheaves together with the usual spectral sequence resulting from a filtered object, in this case the Hodge filtration of
[3] For smooth proper varieties over a field of characteristic 0, the spectral sequence can also be written as where
In this guise, all terms in the spectral sequence are of purely algebraic (as opposed to analytic) nature.
In particular, the question of the degeneration of this spectral sequence makes sense for varieties over a field of characteristic p>0.
Deligne & Illusie (1987) showed that for a smooth proper scheme X over a perfect field k of positive characteristic p, the spectral sequence degenerates, provided that dim(X)
When applied to the category of perfect complexes on a smooth proper variety X, these invariants give back differential forms, respectively, de Rham cohomology of X. Kontsevich and Soibelman conjectured in 2009 that for any smooth and proper dg category C over a field of characteristic 0, the Hodge–de Rham spectral sequence starting with Hochschild homology and abutting to periodic cyclic homology, degenerates: This conjecture was proved by Kaledin (2008) and Kaledin (2016) by adapting the above idea of Deligne and Illusie to the generality of smooth and proper dg-categories.