Appendix B. Filtered derived categories

We re-prove the main results using the language of derived categories ([6]).

Let C be an abelian category. Let FK denote the category whose objects are pairs (K,F) where K is a cochain complex with terms in C and F is a filtration on K such that K is bounded below and the filtration F is finite on each term of K. Moreover, we assume that the differential on K is compatible with the filtration. The group Hom_FK((K,F),(L,G)) is the group of morphisms of complexes compatible with the filtration modulo the subgroup of homotopically trival morphisms. We call FK the Homotopy category of filtered complexes in C. One checks that this is a triangulated category.

A morphism f : (K,F) (L,G) in FK is called a filtered quasi-isomorphism if it induces quasi-isomorphisms gr_F^p(K) gr_G^p(L) for every p. It is well known that filtered quasi-isomorphisms form a saturated multiplicatively closed set and hence we can form the localised category DF with a functor FKDF which is universal for the property that all filtered quasi-isomorphisms become isomorphisms under this functor.

In section (3) we defined level-r filtered quasi-isomorphisms and the Dec shift operation on filtrations. The result of ([3]; section 1.3.4) can then be restated as follows:

Let us apply this to the composite functor

By the lemma we see that the set of level-(l + 1) filtered quasi-isomorphims is precisely the set of morphisms that become isomorphisms under the composite. Since Dec is clearly a triangulated functor we can apply the results of [6] to conclude that level-(l + 1) filtered quasi-isomorphisms form a saturated multiplicatively closed set. Hence we can form the quotient category of FK by inverting such morphisms. We denote this category by DF_l+1. Note that DF and DF₁ are identical.

The operation Bac on filtrations also gives a functor Bac : FKFK which carries level-r filtered quasi-isomorphmisms to level-r + 1 filtered quasi-isomorphisms. It thus induces a functor Bac : DF_r DF_r+1.

Let D denote the derived category of bounded below co-chain complexes with terms in C. We have natural forgetful functors DF_r D by forgetting the filtrations (note that a level-r filtered quasi-isomorphism is in particular a quasi-isomorphism of the underlying complexes). We also have for each integer p a functor gr^p : DFD. More generally, for each integer r > 1 and each p we have E_r-1,p,0 : DF_r D.

Let T : CA be a left-exact functor with values in an abelian category A. Moreover, let us assume that C has sufficiently many injectives. Then we have the hypercohomology functors RTⁱ : DA. The spectral sequence for the hypercohomology of a filtered complex is then

which is naturally associated with any element (K,F) of DF. We can also write the E₁ terms as E₁^p,n-p = RTⁿ(E_0,p,n-p(K,F)).

We then construct a level-r spectral sequence for the hypercohomology of a filtered complex

One way to construct such a sequence is as follows. We have a natural spectral sequence

But we use (2.4) to re-write this as an E_r spectral sequence by setting

is an isomorphism. This gives us the required spectral-sequence which is natural for elements of DF_r.