3 Extensions to [3]

We now extend the definition ([3]; 1.3.6) slightly. Let us first introduce the notation E_r,p,q(K,F) for the complex of E_r terms which contains the term E_r^p,q(K,F) at the p + q-th position. When the integers p and q are irrelevant we will abbreviate this to E_r(K,F). Note that we have the equality of complexes E_r,p,q = E_r,p+r,q-r+1.

A level-1 filtered quasi-isomorphism is what was earlier (2.5) called a filtered quasi-isomorphism.

A similar definition can be given with the property injective replaced by the property D-acyclic in the context of a left-exact functor C -->

C' as before. A level-1 filtered injective resolution is what was earlier (2.6) called a filtered injective resolution.

Example 3.3. Let K be any complex on objects in C. We put the trivial filtration F on K by defining F⁰K = K and F¹K = 0. Then we note as in ([3]; 1.4.6) that

{ 0 if n > 1- p Dec*(F )pKn = d(K -p) if n = 1- p n K if n < 1- p

Thus, gr_{Dec ^*(F)}^pK is the complex concentrated in degrees -p and 1 - p.

K -p/d(K -p-1)-- > d(K- p)

There is a natural morphism to this from the single term complex H^-p(K) concentrated in degree -p which is clearly a quasi-isomorphism.

Fact 3.4. Let L denote the total complex of a Cartan-Eilenberg resolution [2] I of K. Let G^p(L) be the total complex of the subcomplex I^>p. Let F be the trivial filtration on K. Then the natural morphism (K,F) --> (L,G) is a level-2 filtered injective resolution.

Proof. As noted above we have

E02,n(K, F) = E-1 n,2n(K,Dec*(F)) = Hn(K)

and the remaining E₂ terms are 0. We have the identity E₀^p,n(J,G) = I^p,n and so we deduce E₁^p,n(J,G) = Hⁿ(I^p). Since I is a Cartan-Eilenberg resolution these E₁ terms give an injective resolution of Hⁿ(K). Thus we have the result.

Proof. Let (K,(Dec^*)^r-1(F)) --> (L,G) be a (level-1) filtered injective resolution (which exists by (2.7)). Consider the composite morphism

r-1 * r- 1 r-1 (K, F)-- > (K,Bac (Dec ) (F )) --> (L,Bac (G))

By (3) we see that the first morphism is a level-r quasi-isomorphism. Also by (1) we see that the second morphism is a level-r quasi-isomorphism. Hence the composite is also a level-r quasi-isomorphism. Now by (1) and (2.3) we have for r' < r

Er'(L,Bacr-1(G)) = Er'-r+1(L,G) = E0(L,G)

By assumption the latter terms are injective.

Proof. Note that E_r-1(f) : E_r-1(K,F) --> E_r-1(L,G) is a quasi-isomorphism of complexes of injectives. Hence there is a morphism g_r-1 : E_r-1(L,G) --> E_r-1(K,F) such that E_r-1(f)og_r-1 and g_r-1oE_r-1(f) are homotopic to identity. By induction we assume that we are given the morphism g_r' : E_r'(L,G) --> E_r'(K,F). We wish to find a morphism g_r'-1 : E_r'-1(L,G) --> E_r'-1(K,F) such that it induces g_r' on the cohomology of the E_r'-1 terms (which is E_r'). This is possible since the E_r'-1,p,q’s are bounded below complexes of injectives. Thus we obtain g_0,p : gr_G^pL --> gr_F^pK. Again we have that K and L are bounded below complexes of injectives and so we obtain the required morphism g which satisfies gr^p(g) = g_0,p.

Lemma 3.7. Let D : C --> C' be a left-exact functor and assume that C has enough injectives. Let ⁱ denote the associated hypercohomology functors and let (K,F) be a good filtered complex in C. Then for any r > 1 we have a natural spectral sequence

p,q p+q p+q Er = D (Er-1,p,q) ===> D (K)

Proof. Let (K,F) --> (L,G) be a level-r filtered D-acyclic resolution (for example we can take a level-r filtered injective resolution by lemma (3.5)). Consider the good filtered complex (D(L),D(G)) in C'. The associated spectral sequence is

p,q p p+q p+q E0 = grD(G) D(L) ===> H (D(L))

Now, by (2.8) the latter term is ^p+q(K). Since gr_G^p(Lⁿ) are D-acyclic for all integers p and n and the filtrations are finite, we see that gr_D(G)^p(D(L)) = D(gr_G^p(L)).

Now by definition E_l^p,q = H^p+q(E_l-1,p,q). Thus we obtain

Ep,q = Hp+q(D(grp(L))) = Dp+q(grp(L)) 1 G G

since gr_G^p(Lⁿ) are D-acyclic. We now claim by induction that

p,q Er' = Dp+q(Er'- 1,p,q(L, G));for r'< r

Assume this for r'- 1. Now since r'- 1 < r we have E_r'-1(L,G) consists of D-acyclics. Thus we see by (2.9) that

p,q p+q ' p+q ' p,q Er'-1 = D (Er- 2,p,q(L,G)) = D(H (Er- 2,p,q(L,G))) = D(Er'- 1)

But then by definition of ⁱ’s we have

p,q p+q p+q p+q E r' = H (Er'- 1,p,q) = H (D(Er'-1,p,q(L,G))) = D (Er'-1,p,q)

This proves the claim by induction.

Now we have E_r-1,p,q(K,F) --> E_r-1,p,q(L,G) is an D-acyclic resolution. Thus

Dp+q(Er- 1,p,q(K,F)) = Dp+q(Er-1,p,q(L, G))

Hence we obtain the required spectral sequence. The naturality of this spectral sequence easily follows from the lemma (3.6) by the usual techniques.

Now we note that for r = 1 this spectral sequence is exactly the one constructed by Deligne in ([3];1.4.5). For r = 2 we see that this is the Leray spectral sequence for hypercohomology by applying the level-2 injective resolution given by Cartan-Eilenberg (3.4). From the above proof we see that we obtain our “new” spectral sequence. On the other hand the E₂ spectral sequence associated with the Cartan-Eilenberg resolution is precisely what is called the Leray spectral sequence for hypercohomology. This completes the proof of the main theorem (1.1).

3. Extensions to [3]