2 Recapitulation some facts from [3]

2. Recapitulation some facts from [3]

We recall some facts proved by Deligne in ([3]; 1.3 and 1.4) with slightly different notations. The details can also be found in the appendix A. Let C be an abelian category. All objects, morphisms, etc. will be with reference to this category.

Definition 2.1. We say (K,F) is good filtered complex in C if K is a co-chain complex in C which is bounded below and F is a filtration on it which is compatible with the differential and is finite on Kⁿ for each integer n.

We will always work with good filtered complexes in this paper. For any co-chain complex K let Hⁿ(K) denote its n-th cohomology.

Fact 2.2 ([3]; 1.3.1). For any good filtered complex (K,F) there is a spectral sequence

Ep,n- p= Ep,n-p(K, F) = grp Kn ===> Hn(K) 0 0 F

such that the filtration induced by this spectral sequence on Hⁿ(K) is the same as that induced by F.

Fact 2.3. The definitions of E_r^p,q(K,F) given in (loc. cit. ) make sense for all integers r (not only positive integers r). The equalities E_r^p,q = E₀^p,q for r < 0 hold and the differentials d_r are 0 for r < 0 due the compatibility of the filtration F with the differential on K.

Next Deligne defines various shifted filtrations associated with the given one. First of all we define ([3]; proof of 1.3.4)

Bac(F)pKn := Fp-nKn

Deligne shows that (loc. cit. ),

p,n- p p-n,2n- p Er (K, Bac(F)) = Er-1 (K, F)

for all integers r. By induction on l we obtain

Epr,n-p(K,Bacl(F)) = Epr--lln,(l+1)n-p(K,F ); for all integers r

(1)

Moreover, Deligne notes the following fact about renumbering spectral sequences (actually he only notes it for s-r=1).

Fact 2.4. Let E_a^p,q be the terms of a spectral sequence which starts at a = r, then we can obtain another spectral sequence E'_b^p,q starting at the term b = s by setting

E'pb,q = Epb--((s-s- rr))(p+q),q+(s-r)(p+q)

Next we consider the dual shifted filtration ([3]; section 1.3.5),

* p n p+n-1 n-1 n p+n n p+n-1,1- p Dec (F )K := im(F K --> K )+ F K = B1 (K, F)

Deligne shows (loc. cit. ) that

Epr,n-p(K,Dec*(F)) = Ep+r+n1,-p(K,F )

for all r > 1. By induction on l we obtain

Ep,n-p(K,(Dec*)l(F)) = Ep+ln,(1-l)n-p(K,F ); for all integers r > 1. r r+l

(2)

Combining the results for Bac and Dec we see that

p,n-p l * l p,n-p Er (K, Bac ((Dec ) (F ))) = Er (K,F ); for all r > (l+ 1).

(3)

Definition 2.5 ([3]; 1.3.6). A morphism f : (K,F) --> (L,G) of good filtered complexes is said to be a filtered quasi-isomorphism if the morphisms gr^p(f) : gr_F^p(K) --> gr_G^p(L) is a quasi-isomorphism, i. e. E₁^p,q(f) are isomorpisms for all integers p and q.

Definition 2.6 ([3]; 1.4.5). A filtered injective resolution of a good filtered complex (K,F) is a filtered quasi-isomorphism (K,F) --> (L,G) such that the terms gr_L^pGⁿ are injective for all integers p and n.

A similar definition can be given with the property injective replaced by the property D-acyclic in the context of a left-exact functor D : C --> C' as in ([3]; 1.4.1).

The following well-known fact is used in ([3]; 1.4.5)

Fact 2.7. If C has sufficiently many injectives then any good filtered complex (K,F) has a filtered injective resolution.

If D : C --> C' is a left-exact functor and C has enough injectives, then we have the hypercohomologies ⁱ(K) in C' associated with any bounded below cochain complex K

Fact 2.8 ([3]; 1.4.4). The ⁱ(K) are computed as the cohomologies of the complex D(L) for any quasi-isomorphism K --> L where the terms of L are D-acyclic.

We will also need the following well-known fact.

Fact 2.9. If K is a bounded below cochain complex such that H^p(K) are all D-acyclic then ^p(K) = D(H^p(K)).

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