◆ DE RHAM SINGULAR COMPARISON

HODGE-DE RHAM SPECTRAL SEQUENCE // COMPARISON ISOMORPHISM // DEGENERATION

I. De Rham Theorem

For a smooth manifold M, the de Rham theorem establishes a canonical isomorphism between de Rham and singular cohomology:

H^n_dR(M) = H^n(Ω^•(M), d) ≅ H^n_sing(M, R)

The map is given by integration of differential forms over singular chains. For algebraic varieties over C, this extends to the algebraic de Rham cohomology.

II. Hodge-de Rham Spectral Sequence

The stupid filtration on the de Rham complex gives the Hodge-to-de Rham spectral sequence:

E_1^{p,q} = H^q(X, Ω^p_X) ⇒ H^{p+q}_dR(X)

For smooth proper varieties over a field of characteristic 0, this spectral sequence degenerates at E_1 (Deligne-Illusie). This gives the Hodge decomposition:

H^n_dR(X/C) = ⊕_{p+q=n} H^q(X, Ω^p_X)

III. Comparison Isomorphisms

For a smooth proper variety X/C, we have three cohomology theories connected by comparison isomorphisms:

H^n_B(X, Z) ⊗ C ≅ H^n_dR(X/C) ≅ H^n_et(X, Q_l) ⊗ C

The Betti-de Rham comparison preserves the Hodge filtration. The Betti-etale comparison is Artin's comparison theorem.

IV. Algebraic de Rham (GAGA)

Serre's GAGA theorem ensures that for proper X/C, algebraic and analytic de Rham cohomology coincide. The algebraic de Rham complex computes the same groups as the analytic one:

H^n(X^{an}, C) ≅ H^n(X, Ω^•_{X/C}) (GAGA comparison)
C446 // DE RHAM COMPARISON // BBOBOP VOID MATHEMATICS