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Appendix A. Spectral sequences and filtered complexes

We reproduce some facts proved by Deligne in ([3]; 1.3 and 1.4) with slightly different notations. Let (K,F) be a filtered co-chain complex which is bounded below and such that the filtration is finite on each term of the complex. Moreover, we assume that the differential on K is compatible with the filtration. We define for each integer r,

Zpr,n- p(K, F) := ker(F pKn --> Kn+1/F p+rKn+1) Bp,n- p(K, F) := im(F p-r+1Kn -1-- > Kn)+ F p+1Kn rp,n- p p,n- p p,n-p p,n-p Er (K, F) := Zr /(Z r /~\ Br )
Note that Er = E0 for all r < 0 because the differential is compatible with the filtration. One easily shows that the Erp,q’s are the terms of a spectral sequence
Ep,n-p= Ep,n-p(K, F) ===> Hn(K) 0 0

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

Next we define various shifted filtrations associated with the given one. First of all we define

p n p-n n Bac(F) K := F K

One then computes that for all integers r,

p,n-p p-n n n+1 p+r-n-1 n+1 Z r (K,Bac(F )) = kerp(-Fn,2n-Kp --> K /F K ) = Zr-1 (K, F) Bpr,n-p(K,Bac(F )) = im(F p-r-n+2Kn -1-- > Kn) +F p+1-nKn p-n,2n-p = Br-1 (K,F ) Epr,n-p(K,Bac(F )) = Epr--n1,2n-p(K,F )
In particular, we see that Er(K,Bac(F)) = E1(K,Bac(F)) = E0(K,F) for all r < 1.

Next we consider the dual shifted filtration,

Dec*(F )pKn := im(Fp+n-1Kn -1 --> Kn) + Fp+nKn = Bp1+n-1,1- p(K, F)

One computes the following equations for all r > 0,

Zpr,n- p(K, Dec*(F )) = ker(Bp1+n- 1,1-p(K, F) --> Kn+1/Bp+1r+n,1-r-p(K, F)) = im(F p+n-1Kn -1-- > Kn) + Zpr++n1,- p(K, F) p,n- p * p+n-r-1,r-p n p+n,- p Br (K, Dec (F )) = im(B 1 (K,F )-- > K ) +B 1 (K, F) = Bpr++n1,- p(K, F)
Now for r > 1 we have Fp+n+rKn <Br+1p+n,-p. Hence one deduces that
p,n-p * p+n,-p Er (K,Dec (F)) = Er+1 (K,F )

for all r > 1.

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