Proof. Let _X^*K' be the Godement resolution of sheaves of abelian groups on X. The direct image K of K' is a differential graded module for the sheaf _S^* of diferential forms on S. The irrelevant ideal _S^>1 induces an ideal theoretic filtration F on K. Since the morphism X S is proper and smooth we see that

Furthermore, the d₁ differential of this sequence can be identified with the morphism arising out of the Gauss-Manin connection (see [5]). Now by applying the lemma (3.7) we have the required spectral sequence.

Proof. Let f : X S be a proper submersion of C-manifolds. Let A_X (resp. A_S) be the sheaf of differential forms on X (resp. S). These are sheaves of differential graded algebras and f_*(A_X) is in addition a sheaf of differential graded modules for A_S. The irrelevant ideal A_S^>1 thus gives rise to a filtration F on f_*(A_X). The E₁ terms for the spectral sequence (2.2) for this complex give us a complex

Now as above we can show that

And the d₁ differential can be identified with the morphism arising out of the Gauss-Manin connection on the vector bundle associated with the local system R^qf_*(_X). But then the above complex becomes an exact sequence

Now the sheaves A_S^p are fine and hence are acyclic for the functor of global sections. Thus if G denotes the trivial filtration on f_*(A_X) we have a level-2 (S,^.)-acyclic resolution (f_*(A_X),G) (f_*(A_X),F). We can apply the theorem (1.1) to conclude that the two E₂ spectral sequences coincide by naturality.

Proof. Let f_*(A_X) be the complex with the natural filtrations as in the proof of the previous corollary. By the Poincaré lemma we have a quasi-isomorphism _X A_X of complex on X. Thus we have a quasi-isomorphism i : Rf_*(_X) f_*(A_X). The former is a sheaf of differential graded algebras which is a differential graded module for _S. Thus we have a filtration on Rf_*(_X) induced by the irrelevant ideal _S^>1. This makes the above morphism i a morphism of filtered complexes on S. This gives a morphism of spectral sequences constructed using Lemma (3.7):

By the regularity of the Gauss-Manin system this is an isomorphism. Now we combine this with the previous corollary to obtain the result.

Proof. We will use the exactness of the Gersten complex as proved by Bloch and Ogus [1]. Let Y denote a fine site associated with a variety X. Let K' be a complex of injective sheaves which computes the cohomology of Y in a suitable Poincaré duality theory. Let K be the resulting complex of sheaves on X obtained by taking direct image. Then for any Zariski open set U in X the global sections K(U) give a complex that computes the cohomology on the fine site associated with U. We have a natural filtration F on K by the codimension of support.

Let Z be a subset of U which is of pure codimension p and W Z be a subset which is pure of codimension p + 1 in U. Then we have the complexes K_Z(U) = ker(K(U) K(U - Z) and K_W(U) = ker(K(U) K(U - W) which compute the cohomology of U with supports in Z and W respectively. We see that the quotient complex K_Z(U)/K_W(U) is naturally isomorphic to K_Z-W(U - W) = ker(K(U - W) K(U - Z)). Now as we take direct limits over pairs (Z,W) we obtain F^pK(U) =  K_Z(U) and F^p+1K =  K_W(U). Furthermore, we see that the cohomology of the complex gr_F^pK at the q-th place is the term _xX^p(i_x)_*H^q(k(x)) which is a flasque sheaf and hence in particular is (X,^.)-acyclic. Thus if G denotes the trivial filtration on K we have a level-2 acyclic resolution K,G) (K,F) by applying the result of Bloch and Ogus (loc. cit. ; Theorem 4.2). Thus we see that the conditions of the theorem (1.1) are satisfied and the two spectral sequences coincide.