Contemporary Mathematics Volume 00, 2003

Spectral sequences associated to morphisms of locally free sheaves

Sándor J. Kovács

November 3, 2004

Let

0 ⟶ ℱ′ ⟶ ℱ ⟶ ℱ′′ ⟶ 0

be a short exact sequence of locally free sheaves. Then for any r ≥ 0 there is a finite filtration of ⋀^{r} ℱ,

⋀^{r} ℱ = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{r} ⊃ F^{r+1} = 0,

with quotients

F^{p}/F^{p+1} ≃ ⋀^{p} ℱ′ ⊗ ⋀^{r−p} ℱ′′

for each p.

This filtration in turn leads to the spectral sequence,

E_{1}^{p,q} = H^{p+q}(X, ⋀^{p} ℱ′ ⊗ ⋀^{r−p} ℱ′′) ⇒ H^{p+q}(X, ⋀^{r} ℱ).

This is a very useful tool, but requires ℱ′, ℱ, ℱ′′ be locally free. The aim of the present article is to give a construction that yields a similar spectral sequence in a more general case. More precisely, the goal is to define natural objects that will play the role of ⋀^{r−p} ℱ′′ if the cokernel of the morphism ℱ′ → ℱ is not locally free. These objects will not be single sheaves anymore, but objects of the derived category of O_{X}-modules. Nevertheless their functorial and cohomological properties will be strong enough to make applications possible.

This work originated in [Kovács96]. The principal goal was to make a cohomological argument work in a more general setting and the path to that goal led through the construction of certain objects whose properties resemble those of the components of the relative De Rham complex of a smooth morphism. In particular the analogue of the above spectral sequence was constructed in a special case (cf. [ibid.]).

\begin{matrix}c \\ ◯ \end{matrix}2003 American Mathematical Society

1991 Mathematics Subject Classification. Primary 14F05; Secondary 14F17.

Supported in part by NSF Grant DMS-0092165 and a Sloan Research Fellowship.

Subsequently I improved the construction and later I realized that similar arguments can be used elsewhere. The constructions needed were very similar, yet somewhat different cf. [Kovács97b, Kovács02]. It seems appropriate to give a common generalization, so one would only have to go through the construction once rather then to include it with every application, not to mention the hope that there will be new applications. The present article attempts to provide a unified treatment.

§1 contains some technical material that will be essential later. §§1.1 ought to be known, but I do not know a convenient reference, so some of the statements are proved here. §§1.2 is a simple generalization of the notion of a filtration to the derived categorical setting. (1.2.2) is the central result of this section, it is the analogue of its well-known relative about the spectral sequence associated to a filtration.

In §2 the complexes \underline{𝔔}_{θ_{X}}^{p} are constructed for a morphism θ_{X} between locally free sheaves. If θ_{X} is the first morphism of the above short exact sequence and its cokernel is locally free, then \underline{𝔔}_{θ_{X}}^{p} ≃ ⋀^{p} ℱ′′. The guiding principles of the construction are functoriality and the desired “filtration”. The main result is (2.7).

The results in this article were first developed for their use in [Kovács97b] in a more specific form. Later the similar, but still different needs of another project prompted this more general presentation. Hence the applications form an important part.

The first application is a vanishing theorem, proved in §3. This is a generalization of both [Kovács96, 2.13] and [Kovács97a, 1.4]. The functorial property of \underline{𝔔}_{θ_{X}}^{p} is proved in §4.

§5 gives a brief account of the more explicit applications of the theory. These statements are not proved here, the reader is referred to the cited articles for more details.

Definitions and Notation. Let X be a scheme over T. Then C(X) is the category of complexes of O_{X}-modules and for u ∈ Mor(C(X)), M(u) ∈ Ob(C(X)) denotes the mapping cone of u. K(X) is the category of homotopy equivalence classes of objects of C(X). A diagram in C(X) will be called a predistinguished triangle, if its image in K(X) is a distinguished triangle. D(X) denotes the derived category of complexes of O_{X}-modules. The superscripts +, −, b carry the usual meaning (bounded below, bounded above and bounded). Regarding these notions the basic reference will be [Hartshorne66]. S_{k} denotes the symmetric group of degree k.

Acknowledgements. The more complicated commutative diagrams were drawn with the help of the XY-pic package.

§1. Tools

§§1.1 Wedge products. Let Φ_{X} and Ψ_{X} be locally free sheaves on X of rank k and n respectively, and let θ_{X} : Φ_{X} → Ψ_{X} be a morphism.

1.1.1 Definition. Let η be a section of ⋀^{p} Ψ_{X} over an open set and ξ_{1}, …, ξ_{k} sections of Φ_{X} over the same set. Then η ⊗ (ξ_{1} ∧ ⋯ ∧ ξ_{k}) is a section of ⋀^{p} Ψ_{X} ⊗

SPECTRAL SEQUENCES ASSOCIATED TO MORPHISMS OF LOCALLY FREE SHEAVES 3

det Φ_{X}. For any σ ∈ S_{k} let

ξ_{σ,q} = θ_{X}(ξ_{σ(1)}) ∧ ⋯ ∧ θ_{X}(ξ_{σ(q)}),

and

ξ^{σ,q} = ξ_{σ(q+1)} ∧ ⋯ ∧ ξ_{σ(k)}.

Further let

S_{k,q} = {σ ∈ S_{k}|σ(1) < ⋯ < σ(q) and σ(q + 1) < ⋯ < σ(k)},

and

I_{σ} = {σ(1), …, σ(q)}.

It is easy to see that every σ ∈ S_{k,q} is determined by I_{σ}. Now define

⋏_{q}^{θ}(η ⊗ (ξ_{1} ∧ ⋯ ∧ ξ_{k})) ∈ ⋀^{p+q} Ψ_{X} ⊗ ⋀^{k−q} Φ_{X}

by the formula

⋏_{q}^{θ}(η ⊗ (ξ_{1} ∧ ⋯ ∧ ξ_{k})) = ∑_{σ∈S_{k,q}}(−1)^{sgn σ}(ξ_{σ,q} ∧ η) ⊗ ξ^{σ,q},

and extend it linearly.

To see that

⋏_{q}^{θ} : ⋀^{p} Ψ_{X} ⊗ det Φ_{X} → ⋀^{p+q} Ψ_{X} ⊗ ⋀^{k−q} Φ_{X}

is a well-defined morphism of sheaves, it is enough to verify the multi-linear and alternating properties. This is left to the reader.

1.1.2 Lemma. Let id denote id_{Φ_{X}} : Φ_{X} → Φ_{X}. Then

⋀^{p} Ψ_{X} ⊗ det Φ_{X} ⊗ det Φ_{X}  ⟶^{⋏_{q}^{θ}}  ⋀^{p+q} Ψ_{X} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X}
⋏_{q+r}^{θ} ↓  ↓^{⋏_{r}^{θ}}
⋀^{p+q+r} Ψ_{X} ⊗ ⋀^{k−q−r} Φ_{X} ⊗ det Φ_{X} ⟶_{⋏_{q}^{id}} ⋀^{p+q+r} Ψ_{X} ⊗ ⋀^{k−r} Φ_{X} ⊗ ⋀^{k−q} Φ_{X}

is a commutative diagram, i.e., ⋏_{r}^{θ} ∘ ⋏_{q}^{θ} = ⋏_{q}^{id} ∘ ⋏_{q+r}^{θ}.

Proof. Use the same notation as in (1.1.1). Then

⋏_{r}^{θ} ∘ ⋏_{q}^{θ} (η ⊗ (ξ_{1} ∧ ⋯ ∧ ξ_{k}) ⊗ (ξ_{1} ∧ ⋯ ∧ ξ_{k})) =

= ∑_{τ∈S_{k,r}} ∑_{σ∈S_{k,q}}(−1)^{sgn τ+sgn σ}(ξ_{τ,r} ∧ ξ_{σ,q} ∧ η) ⊗ ξ^{σ,q} ⊗ ξ^{τ,r}.

Let σ ∈ S_{k,q}, τ ∈ S_{k,r}. If

I_{τ} ∩ I_{σ} = {τ(1), …, τ(r)} ∩ {σ(1), …, σ(q)} ≠ ∅,

then ξ_{τ,r} ∧ ξ_{σ,q} = 0. Otherwise let μ = μ(σ, τ) ∈ S_{k,q+r} be defined by I_{μ} = I_{τ} ∪ I_{σ} and let ν = ν(σ, τ) = σ ∈ S_{k,q}.

⋏_{q}^{id} ∘ ⋏_{q+r}^{θ} (η ⊗ (ξ_{1} ∧ ⋯ ∧ ξ_{k}) ⊗ (ξ_{1} ∧ ⋯ ∧ ξ_{k})) =

∑_{ν∈S_{k,q}} ∑_{μ∈S_{k,q+r}}(−1)^{sgn μ+sgn ν}(ξ_{μ,q+r} ∧ η) ⊗ ξ^{ν,q} ⊗ (ξ_{ν,q} ∧ ξ^{μ,q+r})

and for ν ∈ S_{k,q}, μ ∈ S_{k,q+r}, ξ_{ν,q} ∧ξ^{μ,q+r} ≠ 0 let σ = σ(μ, ν) = ν and τ = τ(μ, ν) ∈ S_{k,r} be defined by I_{τ} = I_{μ} \ I_{ν}.

This gives a one-to-one correspondence between the pairs (σ, τ) and the pairs (μ, ν).

Observe that

(−1)^{sgn τ} (\underbrace{ξ_{τ,r} ∧ ξ_{σ,q}}_{±ξ_{μ,q+r}}) ⊗ ξ^{τ,r} = (−1)^{sgn μ}ξ_{μ,q+r} ⊗ (\underbrace{ξ_{ν,q} ∧ ξ^{μ,q+r}}_{±ξ^{τ,r}}),

so

(−1)^{sgn τ+sgn σ}(ξ_{τ,r} ∧ ξ_{σ,q} ∧ η) ⊗ ξ^{σ,q} ⊗ ξ^{τ,r} =
(−1)^{sgn μ+sgn ν}(ξ_{μ,q+r} ∧ η) ⊗ ξ^{ν,q} ⊗ (ξ_{ν,q} ∧ ξ^{μ,q+r}).

□

§§1.2 Hyperfiltrations and spectral sequences. Let 𝔄 be an abelian category and D(𝔄) its derived category. Let Φ : 𝔄 → 𝔄𝔟 be a left exact additive functor from 𝔄 to the category of abelian groups and assume that RΦ : D(𝔄) → D(𝔄𝔟), the right derived functor of Φ exists.

1.2.1 Definition. Let K ∈ Ob(D^{b}(𝔄)) be a bounded complex. A bounded hyperfiltration 𝔽of K consists of a set of objects 𝔽^{j}K ∈ Ob(D^{b}(𝔄)) for j = l, …, k+ 1, where l, k ∈ ℤ and morphisms

φ_{j} ∈ Hom_{D^{b}(𝔄)}(𝔽^{j+1}K, 𝔽^{j}K) for j = l, …, k,

where 𝔽^{l}K ≃ K and 𝔽^{k+1}K ≃ 0. 𝔽^{j}K will be denoted by 𝔽^{j} when no confusion is likely. For convenience let 𝔽^{i}K = K for i < l and 𝔽^{i}K = 0 for i > k.

The p-th associated graded complex of a hyperfiltration 𝔽 is

𝔾^{p}= 𝔾r_{𝔽}^{p}K = M(φ_{p})

the mapping cone^{1} of the morphism φ_{p}.

Then one has the following standard result:

1Strictly speaking this is the class of the mapping cone of a morphism in C(X) whose class in D(X) is φ_{p}. The point is that 𝔽^{p+1}⟶ 𝔽^{p}⟶ 𝔾^{p} ⟶^{+1} forms a distinguished triangle.

SPECTRAL SEQUENCES ASSOCIATED TO MORPHISMS OF LOCALLY FREE SHEAVES 5

1.2.2 Theorem. There exists a spectral sequence E_{r} with E_{1}^{p,q} = R^{p+q}Φ(𝔾^{p}) abutting to R^{p+q}Φ(K).

The proof of this theorem is relatively straightforward, but cumbersome and is not used elsewhere in this article. The key observation to make in order to prove this result is that one needs to start at the E_{1}-level since there is no E_{0}-level, so the proof has to be adjusted accordingly. An important point to notice is the following: once one passes to the E_{1}-level of this spectral sequence, one leaves the realm of derived categories behind and starts working with long exact sequences of abelian groups (or more generally of objects of an abelian category). This allows one to mimic the usual proof of the similar well-known result for filtrations. For the interested reader the proof is included in an appendix at the end of the article.

1.2.2.1 Remark. All of the above makes sense and remains true if one replaces 𝔄𝔟 by an arbitrary abelian category.

§2. Filtration diagrams

Let Φ_{X} and Ψ_{X} be locally free sheaves on X of rank k and n respectively, and let θ_{X} : Φ_{X} → Ψ_{X} be a morphism.

Let p, i ∈ ℕ. We are going to define an object, 𝔉_{i}^{p} = 𝔉_{i}^{p}(θ_{X}) ∈ Ob(C(X))

and a (p, i)-filtration diagram of θ_{X} diagram, 𝔉_{i}^{p} = 𝔉_{i}^{p}(θ_{X}) . This will be done

recursively, starting with i = 0 and then increasing i.

2.1 Definition. The (p, 0)-filtration diagram of θ_{X} is

𝔉_{0}^{p} = 𝔉_{0}^{p} = ⋀^{n−p} Ψ_{X}.

A 0-filtration morphism for some p, q, consists of locally free sheaves ℰ, ℱ and a morphism between ⋀^{n−p} Ψ_{X} ⊗ ℰ and ⋀^{n−q} Ψ_{X} ⊗ ℱ.

For instance,

⋏_{p}^{θ} : ⋀^{n−p} Ψ_{X} ⊗ det Φ_{X} ⟶ ⋀^{n} Ψ_{X} ⊗ ⋀^{k−p} Φ_{X}

is a 0-filtration morphism. Let

𝔉_{1}^{p} = M(⋏_{p}^{θ})[−1].

2.2 Definition. The (p, 1)-filtration diagram of θ_{X} consists of the predistinguished triangle,

𝔉_{1}^{p} ⟶ ⋀^{n−p} Ψ_{X} ⊗ det Φ_{X} ⟶ ⋀^{n} Ψ_{X} ⊗ ⋀^{k−p} Φ_{X} ⟶^{+1}

It is denoted by 𝔉_{1}^{p} . A 1-filtration morphism for some p, r, consists of locally

free sheaves ℰ, ℱ and morphisms between the corresponding terms of 𝔉_{1}^{p} ⊗ ℰ and

6 SÁNDOR J. KOVÁCS

𝔉_{1}^{r} ⊗ ℱ such that the resulting diagram,

𝔉_{1}^{p} ⊗ ℰ  ⟶  𝔉_{1}^{r} ⊗ ℱ
↓  ↓
⋀^{n−p} Ψ_{X} ⊗ det Φ_{X} ⊗ ℰ ⟶ ⋀^{n−r} Ψ_{X} ⊗ det Φ_{X} ⊗ ℱ
⋏_{p}^{θ}↓ ↓^{⋏_{r}^{θ}}
⋀^{n} Ψ_{X} ⊗ ⋀^{k−p} Φ_{X} ⊗ ℰ ⟶ ⋀^{n} Ψ_{X} ⊗ ⋀^{k−r} Φ_{X} ⊗ ℱ
↓^{+1}  ↓^{+1}

is commutative.

Consider the following commutative diagram (cf. (1.1.2)).

𝔉_{1}^{p} ⊗ det Φ_{X}  𝔉_{1}^{p−q} ⊗ ⋀^{k−q} Φ_{X}
↓ ↓
⋀^{n−p} Ψ_{X} ⊗ det Φ_{X} ⊗ det Φ_{X} ⟶^{⋏_{q}^{θ}} ⋀^{n−p+q} Ψ_{X} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X}
⋏_{p}^{θ}↓ ↓^{⋏_{p−q}^{θ}}
⋀^{n} Ψ_{X} ⊗ ⋀^{k−p} Φ_{X} ⊗ det Φ_{X} ⟶_{⋏_{q}^{id}} ⋀^{n} Ψ_{X} ⊗ ⋀^{k−p+q} Φ_{X} ⊗ ⋀^{k−q} Φ_{X}
↓^{+1}  ↓^{+1}

(2.2.1)

There exists a morphism,

α : 𝔉_{1}^{p} ⊗ det Φ_{X} → 𝔉_{1}^{p−q} ⊗ ⋀^{k−q} Φ_{X},

that makes the above diagram commutative. The diagram (2.2.1), combined with α gives a 1-filtration morphism

𝔉_{1}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{1}^{p−q} ⊗ ⋀^{k−q} Φ_{X},

with r = p − q, ℰ = det Φ_{X}, ℱ = ⋀^{k−q} Φ_{X}.

Let

𝔉_{2}^{p} = M(𝔉_{1}^{p} ⊗ det Φ_{X} → 𝔉_{1}^{1} ⊗ ⋀^{k−p+1} Φ_{X})[−1].

SPECTRAL SEQUENCES ASSOCIATED TO MORPHISMS OF LOCALLY FREE SHEAVES 7

Then there exists a distinguished triangle,

𝔉_{2}^{p} ⟶ 𝔉_{1}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{1}^{1} ⊗ ⋀^{k−p+1} Φ_{X} ⟶^{+1}

2.3 Definition. The (p, 2)-filtration diagram of θ_{X} consists of the diagram,

𝔉_{2}^{p} ⟶ 𝔉_{1}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{1}^{1} ⊗ ⋀^{k−p+1} Φ_{X}.

It is denoted by 𝔉_{2}^{p} . A 2-filtration morphism for some p, r, consists of locally free sheaves ℰ, ℱ and morphisms between the corresponding terms of 𝔉_{2}^{p} ⊗ ℰ and 𝔉_{2}^{r} ⊗ ℱ such that the resulting diagram is commutative.

More explicitly, the (p, 2)-filtration diagram of θ_{X} is:

𝔉_{2}^{p} ⟶  𝔉_{1}^{p} ⊗ det Φ_{X}  ⟶  𝔉_{1}^{1} ⊗ ⋀^{k−p+1} Φ_{X}
↓  ↓
⋀^{n−p} Ψ_{X} ⊗ det Φ_{X} ⊗ det Φ_{X} ⟶^{⋏_{p−1}^{θ}} ⋀^{n−1} Ψ_{X} ⊗ det Φ_{X} ⊗ ⋀^{k−p+1} Φ_{X}
⋏_{p}^{θ}↓ ↓^{⋏_{1}^{θ}}
⋀^{n} Ψ_{X} ⊗ ⋀^{k−p} Φ_{X} ⊗ det Φ_{X} ⟶_{⋏_{p−1}^{id}} ⋀^{n} Ψ_{X} ⊗ ⋀^{k−1} Φ_{X} ⊗ ⋀^{k−p+1} Φ_{X}
↓^{+1}  ↓^{+1}

Similarly, a 2-filtration morphism is:

[figure: 𝔉_{2}^{p} ^{0,0}⊗ ℰ; 𝔉_{2}^{p} ^{0,1}⊗ ℰ; 𝔉_{2}^{p} ^{0,2}⊗ ℰ; 𝔉_{2}^{r} ^{0,0}⊗ ℱ; 𝔉_{2}^{r} ^{0,1}⊗ ℱ; 𝔉_{2}^{r} ^{0,2}⊗ ℱ; 𝔉_{2}^{p} ^{1,1}⊗ ℰ; 𝔉_{2}^{p} ^{1,2}⊗ ℰ; 𝔉_{2}^{r} ^{1,1}⊗ ℱ; 𝔉_{2}^{r} ^{1,2}⊗ ℱ; 𝔉_{2}^{p} ^{2,1}⊗ ℰ; 𝔉_{2}^{p} ^{2,2}⊗ ℰ; 𝔉_{2}^{r} ^{2,1}⊗ ℱ; 𝔉_{2}^{r} ^{2,2}⊗ ℱ,]

where the (p, 2)-filtration diagram,

[figure: 𝔉_{2}^{p}; 𝔉_{1}^{p} ⊗ det Φ_{X}; 𝔉_{1}^{1} ⊗ ⋀^{k−p+1} Φ_{X}; ⋀^{n−p} Ψ_{X} ⊗ det Φ_{X} ⊗ det Φ_{X}; ⋀^{n−1} Ψ_{X} ⊗ det Φ_{X} ⊗ ⋀^{k−p+1} Φ_{X}; ⋀^{n} Ψ_{X} ⊗ ⋀^{k−p} Φ_{X} ⊗ det Φ_{X}; +1; ⋀^{n} Ψ_{X} ⊗ ⋀^{k−1} Φ_{X} ⊗ ⋀^{k−p+1} Φ_{X}; +1]

is represented by the simplified diagram,

\xymatrix{𝔉_{2}^{p} ^{0,0} \ar[r] & 𝔉_{2}^{p} ^{0,1} \ar[r] \ar[d] & 𝔉_{2}^{p} ^{0,2} \ar[d] \\  & _{𝔉_{2}^{p} _{1,1}} \ar[r] \ar[d] & _{𝔉_{2}^{p} _{1,2}} \ar[d] \\  & _{𝔉_{2}^{p} _{2,1}} \ar[r] & 𝔉_{2}^{p} ^{2,2}.}

To define the (p, i)-filtration diagram of θ_{X} and the i-filtration morphisms we will iterate this construction.

2.4 Inductive Hypotheses For a given i assume that the following holds for all p, q, r ∈ ℕ.

(2.4.1) The (p, i)-filtration diagram of θ_{X} is defined and denoted by 𝔉_{i}^{p} .

(2.4.2) An i-filtration morphism, by definition, consists of locally free sheaves ℰ, ℱ

and a morphism between the corresponding terms of 𝔉_{i}^{p} ⊗ ℰ and 𝔉_{i}^{r} ⊗ ℱ

such that the resulting diagram is commutative.

(2.4.3) 𝔉_{i}^{p} has a unique object, 𝔉_{i}^{p}, with only one adjacent arrow pointing out.

(2.4.4) 𝔉_{i}^{p} = 0 for p < i.

(2.4.5) There exists an i-filtration morphism,

⋏_{q}^{θ,i} : 𝔉_{i}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{i}^{p−q} ⊗ ⋀^{k−q} Φ_{X}.

(2.4.6) The diagram,

𝔉_{i}^{p} ⊗ det Φ_{X} ⊗ det Φ_{X}  ⟶^{⋏_{q}^{θ,i}}  𝔉_{i}^{p−q} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X}
⋏_{q+r}^{θ,i} ↓  ↓^{⋏_{r}^{θ,i}}
𝔉_{i}^{p−q−r} ⊗ ⋀^{k−q−r} Φ_{X} ⊗ det Φ_{X} ⟶_{⋏_{q}^{id}} 𝔉_{i}^{p−q−r} ⊗ ⋀^{k−r} Φ_{X} ⊗ ⋀^{k−q} Φ_{X}

is commutative.

2.5 Lemma-Definition. If (2.4) holds for i = 0, …, j, then 𝔉_{j+1}^{p} can be

defined so that (2.4) holds for i = j + 1.

Proof. If j > p, let 𝔉_{j+1}^{p} = 𝔉_{j+1}^{p} = 0. If j ≤ p then by (2.4.5) there exists a

j-filtration morphism,

⋏_{p−j}^{θ,j} : 𝔉_{j}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{j}^{j} ⊗ ⋀^{k−p+j} Φ_{X}.

Let ⋏_{p−j,res}^{θ,j} : 𝔉_{j}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{j}^{j} ⊗ ⋀^{k−p+j} Φ_{X} be the restriction of ⋏_{p−j}^{θ,j} to 𝔉_{j}^{p} ⊗ det Φ_{X} and let

𝔉_{j+1}^{p} =M(⋏_{p−j,res}^{θ,j})[−1].

Now 𝔉_{j+1}^{p} maps to 𝔉_{j}^{p} and they form the predistinguished triangle

𝔉_{j+1}^{p} ⟶ 𝔉_{j}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{j}^{j} ⊗ ⋀^{k−p+j} Φ_{X} ⟶^{+1}  (2.5.7)

such that this predistinguished triangle together with ⋏_{p−j}^{θ,j} form a diagram that is commutative. 2.5.8 The (p, j + 1)-filtration diagram of θ_{X} consists of the diagram

𝔉_{j+1}^{p} ⟶ 𝔉_{j}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{j}^{j} ⊗ ⋀^{k−p+j} Φ_{X}.

It is denoted by 𝔉_{j+1}^{p} . A j +1-filtration morphism for some p, r, consists of locally

free sheaves ℰ, ℱ and a morphism between the corresponding terms of 𝔉_{j+1}^{p} ⊗ ℰ

and 𝔉_{j+1}^{r} ⊗ ℱ such that the resulting diagram is commutative.

(2.4.1) and (2.4.2) follow from this definition, and (2.4.3) and (2.4.4) are also clear from the construction above.

SPECTRAL SEQUENCES ASSOCIATED TO MORPHISMS OF LOCALLY FREE SHEAVES 11

Next consider the following commutative diagram (cf. (2.4.6)).

𝔉_{j+1}^{p} ⊗ det Φ_{X}  𝔉_{j+1}^{p−q} ⊗ ⋀^{k−q} Φ_{X}
↓  ↓
𝔉_{j}^{p} ⊗ det Φ_{X} ⊗ det Φ_{X} ⟶^{⋏_{q}^{θ,j}}  𝔉_{j}^{p−q} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X}
⋏_{p−j}^{θ,j} ↓  ↓^{⋏_{p−q−j}^{θ,j}}
𝔉_{j}^{j} ⊗ ⋀^{k−p+j} Φ_{X} ⊗ det Φ_{X} ⟶_{⋏_{q}^{id}} 𝔉_{j}^{j} ⊗ ⋀^{k−p+q+j} Φ_{X} ⊗ ⋀^{k−q} Φ_{X}

Then the construction of 𝔉_{j+1}^{p} implies that there is a morphism,

𝔉_{j+1}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{j+1}^{p−q} ⊗ ⋀^{k−q} Φ_{X}

making the diagram commutative and so providing a j + 1-filtration morphism,

⋏_{q}^{θ,j+1} : 𝔉_{j+1}^{p} ⊗ det Φ_{X} ⟶ 𝔉_{j+1}^{p−q} ⊗ ⋀^{k−q} Φ_{X}.  (2.5.9)

This proves (2.4.5). Next consider (2.4.6) for i = j + 1.

𝔉_{j+1}^{p} ⊗ det Φ_{X} ⊗ det Φ_{X}  ⟶^{⋏_{q}^{θ,j+1}}  𝔉_{j+1}^{p−q} ⊗ ⋀^{k−q} Φ_{X} ⊗ det Φ_{X}
⋏_{q+r}^{θ,j+1}↓  ↓^{⋏_{r}^{θ,j+1}}
𝔉_{j+1}^{p−q−r} ⊗ det Φ_{X} ⊗ ⋀^{k−q−r} Φ_{X} ⟶_{⋏_{q}^{id}} 𝔉_{j+1}^{p−q−r} ⊗ ⋀^{k−q} Φ_{X} ⊗ ⋀^{k−r} Φ_{X}

(2.5.10)

This is shown in more detail in the following diagram. Due to the size of the diagram, it is broken up into two pieces.

[figure: 𝔉_{j+1}^{p} ⊗ (det Φ_{X})^{⊗2}; 𝔉_{j+1}^{p−q−r} ⊗ det Φ_{X} ⊗ ⋀^{k−q−r} Φ_{X}; 𝔉_{j}^{p} ⊗ (det Φ_{X})^{⊗3}; ⋏_{q}^{θ,j}; ⋏_{q+r}^{θ,j}; ⋏_{p−j}^{θ,j}; 𝔉_{j}^{p−q−r} ⊗ (det Φ_{X})^{⊗2} ⋀^{k−q−r} Φ_{X}; ⋏_{q}^{id}; ⋏_{p−q−r−j}^{θ,j}; 𝔉_{j}^{j} ⊗ ⋀^{k−p+j} Φ_{X} ⊗ (det Φ_{X})^{⊗2}; ⋏_{q+r}^{id}; ⋏_{q}^{id}; 𝔉_{j}^{j} ⊗ ⋀^{k−p+q+r+j} Φ_{X} ⊗ det Φ_{X} ⊗ ⋀^{k−q−r} Φ_{X}; ⋏_{q}^{id}]

[figure: 𝔉_{j+1}^{p−q} ⊗ ⋀^{k−q} Φ_{X} ⊗ det Φ_{X}; 𝔉_{j+1}^{p−q−r} ⊗ ⋀^{k−q} Φ_{X} ⊗ ⋀^{k−r} Φ_{X}; ⋏_{q}^{θ,j}; 𝔉_{j}^{p−q} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ det Φ_{X}; ⋏_{p−q−j}^{θ,j}; ⋏_{r}^{θ,j}; ⋏_{q}^{id}; 𝔉_{p−q−r}^{j} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ ⋀^{k−r} Φ_{X}; ⋏_{p−q−r−j}^{θ,j}; ⋏_{q}^{id}; 𝔉_{j}^{j} ⊗ ⋀^{k−p+q+j} Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ det Φ_{X}; ⋏_{r}^{id}; ⋏_{q}^{id}; 𝔉_{j}^{j} ⊗⋀^{k−p+q+r+j} Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ ⋀^{k−r} Φ_{X}]

The commutativity of vertical faces is either obvious or follows by the commutativity of (2.5.9).

𝔉_{j}^{p} ⊗ (det Φ_{X})^{⊗3}  ⟶^{⋏_{q}^{θ,j}}  𝔉_{j}^{p−q} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ det Φ_{X}

⋏_{q+r}^{θ,j} ↓  ↓^{⋏_{r}^{θ,j}}

𝔉_{j}^{p−q−r} ⊗ (det Φ_{X})^{⊗2} ⊗ ⋀^{k−q−r} Φ_{X} ⟶_{⋏_{q}^{id}} 𝔉_{j}^{p−q−r} ⊗ det Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ ⋀^{k−r} Φ_{X}

is commutative by (2.4.6), and

𝔉_{j}^{j} ⊗ ⋀^{k−p+j} Φ_{X} ⊗ (det Φ_{X})^{⊗2}  ⟶^{⋏_{q}^{id}}  𝔉_{j}^{j} ⊗ ⋀^{k−p+q+j} Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ det Φ_{X}

⋏_{q+r}^{id} ↓  ↓^{⋏_{r}^{id}}

𝔉_{j}^{j} ⊗ ⋀^{k−p+q+r+j} Φ_{X} ⊗ det Φ_{X} ⊗ ⋀^{k−q−r} Φ_{X} ⟶_{⋏_{q}^{id}} 𝔉_{j}^{j} ⊗ ⋀^{k−p+q+r+j} Φ_{X} ⊗ ⋀^{k−q} Φ_{X} ⊗ ⋀^{k−r} Φ_{X}

is commutative by (1.1.2). Now it is easy to see that (2.5.10) is commutative. □

2.6. We are ready to define \underline{𝔔}_{θ_{X}}^{p} ∈ Ob(D(X)) for p ∈ ℤ, p ≥ −k. Let \underline{𝔔}_{θ_{X}}^{p} be the class of

𝔉_{n−k−p}^{n−k−p} ⊗ (det Φ_{X})^{−(n−k−p+1)}

in Ob(D(X)) for −k ≤ p ≤ n − k, and let \underline{𝔔}_{θ_{X}}^{p} = 0 for p > n − k. It is easy to see that

\underline{𝔔}_{θ_{X}}^{n−k} = det Ψ_{X} ⊗ (det Φ_{X})^{−1}

and that there is a distinguished triangle:

\underline{𝔔}_{θ_{X}}^{n−k−1} ⊗ det Φ_{X} ⟶ ⋀^{n−1} Ψ_{X} ⟶ \underline{𝔔}_{θ_{X}}^{n−k} ⊗ ⋀^{k−1} Φ_{X} ⟶^{+1}

In general for j ≥ p − n + k let 𝔽^{j}⋀^{p} Ψ_{X} be the class of 𝔉_{n−k−p+j}^{n−p} ⊗ (det Φ_{X})^{−(n−k−p+j)}

in Ob(D(X)). The predistinguished triangle (2.5.7),

𝔉_{n−k−p+j+1}^{n−p} ⟶ 𝔉_{n−k−p+j}^{n−p} ⊗ det Φ_{X} ⟶ 𝔉_{n−k−p+j}^{n−k−p+j} ⊗ ⋀^{j} Φ_{X} ⟶^{+1}

gives the distinguished triangle,

𝔽^{j+1}⋀^{p} Ψ_{X} ⟶ 𝔽^{j}⋀^{p} Ψ_{X} ⟶ \underline{𝔔}_{θ_{X}}^{p−j} ⊗ ⋀^{j} Φ_{X} ⟶^{+1} .

Now 𝔽^{k+1}⋀^{p} Ψ_{X} = 0 by (2.4.4) and by definition 𝔽^{p−n+k}⋀^{p} Ψ_{X} = ⋀^{p} Ψ_{X}.

Observe that if p − n + k < 0, then

𝔽^{0}⋀^{p} Ψ_{X} ≃ 𝔽^{−1}⋀^{p} Ψ_{X} ≃ ⋯ ≃ 𝔽^{p−n+k}⋀^{p} Ψ_{X} = ⋀^{p} Ψ_{X},

since ⋀^{j} Φ_{X} = 0 for j < 0. If p − n + k ≥ 0, define 𝔽^{j}⋀^{p} Ψ_{X} = ⋀^{p} Ψ_{X} for j = 0, …, p − n + k.

Therefore we proved:

2.7 Theorem. Let Φ_{X} and Ψ_{X} be locally free sheaves on X of rank k and n respectively, and let θ_{X} : Φ_{X} → Ψ_{X} be a morphism. Then there exists a \underline{𝔔}_{θ_{X}}^{r} ∈ Ob(D(X)) for all r ∈ ℤ, r ≥ −k with the following property. For any p ∈ ℕ there exists a hyperfiltration 𝔽^{j}⋀^{p} Ψ_{X} of ⋀^{p} Ψ_{X} with j = 0, …, k + 1, such that

𝔽^{0}⋀^{p} Ψ_{X} ≃ ⋀^{p} Ψ_{X},

𝔽^{k+1}⋀^{p} Ψ_{X} ≃ 0

and

𝔾^{j}⋀^{p} Ψ_{X} ≃ \underline{𝔔}_{θ_{X}}^{p−j} ⊗ ⋀^{j} Φ_{X}.

Furthermore, for r > n − k,

\underline{𝔔}_{θ_{X}}^{r} ≃ 0.

Now using (1.2.2) one obtains the following corollary:

2.8 Corollary. Let Φ_{X} and Ψ_{X} be locally free sheaves on X of rank k and n respectively, and let θ_{X} : Φ_{X} → Ψ_{X} be a morphism. Then for any locally free sheaf ℰ and for any p ∈ ℕ there exists a spectral sequence,

E_{1}^{r,s} = ℍ^{r+s}(X, \underline{𝔔}_{θ_{X}}^{p−r}⊗ ⋀^{r} Φ_{X} ⊗ ℰ) ⇒ H^{r+s}(X, ⋀^{p} Ψ_{X} ⊗ ℰ).

2.9 Proposition. If the cokernel sheaf, Ξ_{X}, of θ_{X} is locally free, then \underline{𝔔}_{θ_{X}}^{p} is simply the p-th exterior power of Ξ_{X}, and the spectral sequence above is the well-known spectral sequence associated to the filtration,

⋀^{p} Ψ_{X} = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{p} ⊃ F^{p+1} = 0,

with quotients

F^{j}/F^{j+1} ≃ ⋀^{p−j} Ξ_{X} ⊗ ⋀^{j} Φ_{X}

for each j.

Proof. By definition one has that

\underline{𝔔}_{θ_{X}}^{p} ≃ det Ψ_{X} ⊗ (det Φ_{X})^{−1} ≃ det Ξ_{X}.

Then the statement follows using descending induction, the filtration associated to the short exact sequence of locally free sheaves and the distinguished triangle,

𝔽^{j+1}⋀^{p} Ψ_{X} ⟶ 𝔽^{j}⋀^{p} Ψ_{X} ⟶ \underline{𝔔}_{θ_{X}}^{p−j} ⊗ ⋀^{j} Φ_{X} ⟶^{+1} .

□

2.10 Example. Let k = 1, i.e., assume that Φ_{X} is a line bundle. Then the hyperfiltration in (2.7) is simply a distinguished triangle,

\underline{𝔔}_{θ_{X}}^{p−1} ⊗ Φ_{X} ⟶ ⋀^{p} Ψ_{X} ⟶ \underline{𝔔}_{θ_{X}}^{p} ⟶^{+1} .

and then the spectral sequence in (2.8) reduces to a long exact hypercohomology sequence:

… ⟶ ℍ^{s}(X, \underline{𝔔}_{θ_{X}}^{p−1}⊗Φ_{X} ⊗ ℰ) ⟶ H^{s}(X, ⋀^{p} Ψ_{X} ⊗ ℰ) ⟶

⟶ ℍ^{s}(X, \underline{𝔔}_{θ_{X}}^{p} ⊗Φ_{X} ⊗ ℰ) ⟶ ℍ^{s+1}(X, \underline{𝔔}_{θ_{X}}^{p−1}⊗Φ_{X} ⊗ ℰ) ⟶ …

§3. A vanishing theorem

Throughout this section X is an algebraic veriety over a field k. The following well-known fact is included for ease of reference.

3.1 Fact. Let ℰ be a locally free sheaf of rank r. Assume that there is a filtration

ℰ = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{r} = 0

of ℰ such that

F^{i−1}/F^{i} = ℒ_{i}

is a line bundle for all i = 1, …, r. Then for every 1 ≤ t ≤ r there is a filtration

⋀^{t} ℰ = F_{t}^{0} ⊃ F_{t}^{1} ⊃ ⋯ ⊃ F \binom{r}{t}_{t} = 0

of ⋀^{t} ℰ such that

F_{t}^{i−1}/F_{t}^{i} = ℒ_{i_{1}} ⊗ ℒ_{i_{2}} ⊗ ⋯ ⊗ ℒ_{i_{t}}

for all i = 1, …, \binom{r}{t} and a suitable set of indicies 1 ≤ i_{1} < i_{2} < ⋯ < i_{t} ≤ r. □

3.2 Definition. A locally free sheaf E on X is called semi-positive or nef if for every smooth proper curve C and every map γ : C ⟶ X, deg Q|_{C} ≥ 0 for any quotient bundle Q of γ^{∗}ℰ.

3.3 Definition. Let ℰ be a locally free sheaf of rank r. ℰ will be called semi-negative of splitting type if ℰ has a filtration

ℰ = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{r} = 0

such that

F^{i−1}/F^{i} = ℒ_{i}

is a semi-negative line bundle, i.e., ℒ_{i}^{−1} is nef.

3.4 Definition. A class, ℭ, of line bundles on X will be called nef-invariant, if for all ℒ ∈ ℭ and 𝒩 nef line bundle on X, ℒ ⊗ 𝒩 ∈ ℭ.

3.5 Example. If X is projective, then the class of ample line bundles on X is nef-invariant.

3.6 Lemma. Let Φ_{X} and Ψ_{X} be locally free sheaves on X of rank k and n respectively, and let θ_{X} : Φ_{X} → Ψ_{X} be a morphism. Let ℭ be a nef-invariant class of line bundles on X. Let c(n) ∈ ℕ be a constant such that H^{p}(X, ⋀^{q} Ψ_{X} ⊗ ℒ) = 0 for every line bundle ℒ ∈ ℭ and p + q > c(n). Let i, j ∈ ℕ be natural numbers such that i + j + k ≥ c(n). Assume that Φ_{X} is semi-negative of splitting type and that for every natural number l > j and line bundle ℒ ∈ ℭ on X,

ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ det Φ_{X} ⊗ ℒ) = 0.

Then

ℍ^{i+1}(X, \underline{𝔔}_{θ_{X}}^{j} ⊗ det Φ_{X} ⊗ ℒ) = 0

for every line bundle ℒ ∈ ℭ.

Proof. By (2.8) there exists a spectral sequence

E_{1}^{r,s} = ℍ^{r+s}(X, \underline{𝔔}_{θ_{X}}^{j+k−r} ⊗ ⋀^{r} Φ_{X} ⊗ ℒ) ⇒ H^{r+s}(X, ⋀^{j+k} Ψ_{X} ⊗ ℒ).

Since i+j+k ≥ c(n), H^{i+1}(X, ⋀^{j+k} Ψ_{X} ⊗ℒ) = 0 by assumption. Hence E_{∞}^{r,i+1−r} = 0 for all r. In particular E_{∞}^{k,i+1−k} = 0. Suppose now that

E_{1}^{k,i+1−k} = ℍ^{i+1}(X, \underline{𝔔}_{θ_{X}}^{j} ⊗ det Φ_{X} ⊗ ℒ) ≠ 0.

Observe that E_{w}^{u,v} = 0 for every u > k and arbitrary v, w, so in order to have E_{∞}^{k,i+1−k} = 0, there must be a t ≥ 1 such that E_{t}^{k−t,i−k+t} ≠ 0. Then E_{1}^{k−t,i−k+t} ≠ 0 for the same t, so l = j + t > j is such that

E_{1}^{k−t,i−k+t} = ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ ⋀^{k−t} Φ_{X} ⊗ ℒ) ≠ 0.

Now by assumption Φ_{X} has a filtration

Φ_{X} = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{k} = 0

such that F^{i−1}/F^{i} = ℒ_{i} and ℒ_{i}^{−1} is a nef line bundle. Then by (3.1) there exist i_{1}, …, i_{k−t} such that

ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ ℒ_{i_{1}} ⊗ ⋯ ⊗ ℒ_{i_{k−t}} ⊗ ℒ) ≠ 0.

Therefore

ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ det Φ_{X} ⊗ ℒ ⊗ ℒ_{i_{k−t+1}}^{−1} ⊗ ⋯ ⊗ ℒ_{i_{k}}^{−1}) ≠ 0,

for {i_{k−t+1}, …, i_{k}} = {1, …, k} \ {i_{1}, …, i_{k−t}}.

Since ℒ ⊗ ℒ_{i_{k−t+1}}^{−1} ⊗ ⋯ ⊗ ℒ_{i_{k}}^{−1} ∈ ℭ, this non-vanishing violates the assumption. Hence the statement follows. □

3.7 Theorem. Let Φ_{X} and Ψ_{X} be locally free sheaves on X of rank k and n respectively, and let θ_{X} : Φ_{X} → Ψ_{X} be a morphism. Let ℭ be a nef-invariant class of line bundles on X. Assume that H^{p}(X, ⋀^{q} Ψ_{X} ⊗ ℒ) = 0 for every line bundle ℒ ∈ ℭ and p + q > n. Assume further that Φ_{X} is semi-negative of splitting type. Then for every line bundle ℒ ∈ ℭ,

ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ det Φ_{X} ⊗ ℒ) = 0 for i + l > n − k.

Proof. By (2.7) \underline{𝔔}_{θ_{X}}^{l} = 0 for l > n − k, so
ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ det Φ_{X} ⊗ ℒ) = 0

for l > n − k and i ≥ 0. Then

ℍ^{i+1}(X, \underline{𝔔}_{θ_{X}}^{n−k} ⊗ det Φ_{X} ⊗ ℒ) = 0

for i ≥ 0 by (3.6). Hence

ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ det Φ_{X} ⊗ ℒ) = 0

for l > n − k − 1 and i ≥ 1. Then again

ℍ^{i+1}(X, \underline{𝔔}_{θ_{X}}^{n−k−1} ⊗ det Φ_{X} ⊗ ℒ) = 0

for i ≥ 1 by (3.6). Hence again

ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ det Φ_{X} ⊗ ℒ) = 0

for l > n − k − 2 and i ≥ 2.

Iterating this process one sees that

ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{l} ⊗ det Φ_{X} ⊗ ℒ) = 0

for i + l > n − k. □

3.8 Example. If X is a smooth projective variety over ℂ and ℭ is the class of ample line bundles on X, then Ψ_{X} = Ω_{X} satisfies the assumption by the Kodaira-Akizuki-Nakano vanishing theorem.

3.9 Corollary. [Kovács97a, 1.4] Let f : X ⟶ S be a smooth morphism of smooth projective varieties over ℂ of dimension n and k respectively. Let ℒ be an ample line bundle and assume that f^{∗}Ω_{S} is semi-negative of splitting type. Then

H^{i}(X, Ω_{X/S}^{l} ⊗ f^{∗}ω_{S} ⊗ ℒ) = 0 for i + l > n − k.
Proof. (3.8), (3.7), (2.9.1). □

§4. Functoriality

Let S be a scheme and 𝔖𝔠𝔥_{S} the category of S-schemes. Let 𝔏𝔬𝔠𝔉𝔯𝔢𝔢_{S} be the category of pairs (X, ℰ) where X ∈ Ob(𝔖𝔠𝔥_{S}) and ℰ is a locally free sheaf on X. A morphism in Hom_{𝔏𝔬𝔠𝔉𝔯𝔢𝔢_{S}}((Y, ℱ), (X, ℰ)) consists of pairs (ϕ, φ) where ϕ ∈ Hom_{𝔖𝔠𝔥_{S}}(Y, X) and φ ∈ Hom_{X}(ℰ, ϕ_{∗}ℱ) ≃ Hom_{Y} (ϕ^{∗}ℰ, ℱ).

A functor Φ = (Φ_{1}, Φ_{2}) : 𝔖𝔠𝔥_{S} → 𝔏𝔬𝔠𝔉𝔯𝔢𝔢_{S} will be called honest if Φ_{1} is the identity. In this case Φ_{X} will denote Φ_{2}(X), i.e., Φ_{X} is a locally free sheaf on X.

4.1 Theorem. Let Φ and Ψ be honest functors from 𝔖𝔠𝔥_{S} to 𝔏𝔬𝔠𝔉𝔯𝔢𝔢_{S}, and θ : Φ → Ψ a natural transformation. Then the definition of 𝔽^{q}⋀^{p} Ψ_{X} and \underline{𝔔}_{θ_{X}}^{p} is functorial in the following sense, if ϕ : Y → X is a morphism of S-schemes, such that rk Φ_{X} = rk Φ_{Y} = k, then there are natural maps in D(X) forming the commutative diagram:

𝔽^{q+1}⋀^{p} Ψ_{X} ⟶ 𝔽^{q}⋀^{p} Ψ_{X} ⟶ 𝔾^{q}⋀^{p} Ψ_{X}  ⟶^{+1}
↓  ↓  ↓
Rϕ_{∗}𝔽^{q+1}⋀^{p} Ψ_{Y} ⟶ Rϕ_{∗}𝔽^{q}⋀^{p} Ψ_{Y} ⟶ Rϕ_{∗}𝔾^{q}⋀^{p} Ψ_{Y} ⟶^{+1} .

In particular, there are natural maps in D(X) forming the commutative diagram:

⋀^{p} Ψ_{X} ⟶ \underline{𝔔}_{θ_{X}}^{p}
↓  ↓
Rϕ_{∗} ⋀^{p} Ψ_{Y} ⟶ Rϕ_{∗}\underline{𝔔}_{θ_{Y}}^{p} .

Proof. Notice that θ induces a natural transformation, θ^{p} : ⋀^{p} Φ → ⋀^{p} Ψ, between the functors ⋀^{p} Φ : X ⟼ (X, ⋀^{p} Φ_{X}) and ⋀^{p} Ψ : X ⟼ (X, ⋀^{p} Ψ_{X}). Then add the following to (2.4.5):

(2.4.5′) ⋏_{q}^{θ,i} is natural, i.e., there exists a commutative diagram:

𝔉_{i}^{p}(θ_{X}) ⊗ det Φ_{X} ⟶^{⋏_{q}^{θ,i}} 𝔉_{i}^{p−q}(θ_{X}) ⊗ ⋀^{k−q} Φ_{X}
↓^{θ^{k}(ϕ)}  ↓^{θ^{k−q}(ϕ)}
𝔉_{i}^{p}(θ_{Y}) ⊗ det Φ_{Y} ⟶^{⋏_{q}^{θ,i}}  𝔉_{i}^{p−q}(θ_{Y}) ⊗ ⋀^{k−q} Φ_{Y} .

Now repeating the proof of (2.5) for X and Y simultaneously it is easy to see that the first statement is a simple consequence of (2.4.5′) and the construction of 𝔽^{q} ⋀^{p} Ψ_{X}. Then the second statement follows directly from (2.7). □

4.1.1 Remark. In the above statements “natural” should be understood the following way: In the definition of \underline{𝔔}_{θ_{X}}^{p} one needs to make non-canonical choices when defining objects as the third vertex of a distinguished triangle. Here “natural” means that once those choices are made, the construction of the above maps is natural.

§5. Applications

The common theme of the applications is that certain arguments that involve ample line bundles and vanishing theorems on smooth varieties can be extended to certain singular varieties. These results about singular varieties in turn play an important role toward proving other results about smooth varieties and the key line bundles being only nef and big instead of ample.

5.1 Definition. A line bundle K on Y is called big if Y is proper and K^{⊗m} gives a birational map for some m > 0.

5.2 Example. Let f : Y ⟶ X be a birational morphism between projective varieties and let L be an ample line bundle on X. Then f^{∗}L is nef and big on Y.

For the rest of this section we assume that everything is defined over ℂ.

§§5.3. Let S be a smooth variety. Let Ψ be the functor of Kähler differentials, i.e., for any S-scheme X, smooth over ℂ, Ψ_{X} = Ω_{X}, and let Φ be the functor on 𝔖𝔠𝔥_{S} that is the pullback of Ψ from S, i.e., for f : X → S, Φ_{X} = f^{∗}Ω_{S}. Then f^{∗} induces a natural transformation θ : Φ → Ψ.

Let f : X ⟶ S be a morphism of algebraic varieties of dimension n and k respectively, such that S is smooth. Then one obtains the definition of the p-th relative De Rham complexes^{2}, \underline{Ω}_{X/S}^{p}, for non-smooth morphisms. The germ of this construction first appeared in [Kovács96], but there the base was required to be a smooth curve. The initial motivation to define the relative De Rham complexes was to prove the following statement:

5.3.1 Theorem. [Kovács96] Let g : Y ⟶ C be a smooth family of projective varieties of general type with a nef canonical bundle and C a smooth projective curve of genus at most one. Then the fibers of g are birational.

5.3.1.1 Remark. The case of dim Y = 2 is well-known and the case of dim Y = 3 was first proved by [Migliorini95].

The construction presented here allows the generalization of the p-th relative De Rham complexes to the case when the base of the family is higher dimensional (but still smooth). This is discussed and the following generalization of (5.3.1) is proved in [Kovács97b].

5.3.2 Theorem. [Kovács97b] Let g : Y ⟶ S be a smooth family of projective varieties of general type with a nef canonical bundle and S a smooth projective variety. Assume that Ω_{S} is semi-negative. Then the fibres of g are birational.

Further generalizations were obtained in [Kovács02], where logarithmic analogues of the above were constructed. For details the reader is referred to the original article.

§6. The ’nef & big on smooth’ ≈ ’ample on singular’ principal

It may not be entirely clear at this point how the results of the present article are applied to prove the above theorems, or why they are necessary at all. The statements are about smooth varieties whereas the results in this paper are only interesting for singular ones.

The main point is that without these results the arguments that are used in the applications would only prove the statements for ample line bundles. By working with singular varieties we are able to extend the argument to include nef and big line bundles as explained below.

The essence of the applications may be summarized as the

“Nef & big line bundle on smooth ≈ ample line bundle on singular” principle.

In fact, it is a new approach to this principle:

The usual approach sets its goal to obtain a statement for a possibly singular X and an ample line bundle L on X. This is achieved by working with a desingularization, f : Y → X, and a nef and big line bundle, f^{∗}L, on Y.

2Note that the word complex here refers to an object in a derived category. These are analogues of the components of the usual De Rham complex and not the total complex. For more details please refer to [Kovács97b].

Our “upside down” approach aims for a statement for a smooth Y and a nef and big line bundle K on Y. If K^{⊗m} is generated by global sections, then we may work with the morphism, f : Y → X, induced by K^{⊗m}, so K^{⊗m} = f^{∗}L for some L ample on X. The main idea is that working on X we may assume that our line bundle is ample. The price to pay is that we no longer can assume that the variety is smooth, but fortunately this is not a fatal problem.

The detailed plan is the following:

6.1 Plan (6.1.2) First prove the statement for Y smooth, K ample. (6.1.3) Check if K^{⊗m} is generated by global sections. This gives f : Y → X

induced by K^{⊗m}, so K^{⊗m} = f^{∗}L for some L ample on X. (6.1.4) Extend the argument (from (6.1.2)) for X singular, L ample. (6.1.5) “Pull back” the result.

Next we will sketch how this plan is executed in the applications mentioned in the previous section. Let us remark that the line bundle K is chosen to be the canonical bundle, ω_{Y}.

6.2. The key ingredient for (6.1.2) is the spectral sequence from the introduction. For more details please refer to the actual applications.

6.3. (6.1.3) follows easily from the Basepoint-free Theorem [CKM88, 9.3]. It provides the needed global generation for a power of the canonical bundle.

6.4. To complete (6.1.4) the necessary ingredients are the argument from (6.1.2), (2.7) and (2.8). In addition we use that if g : Y → S is as in (5.3.2), then \underline{𝔔}_{θ_{Y}}^{j} ≃ Ω_{Y/S}^{j} by (2.9).

6.5. Functoriality becomes important for (6.1.5):

For the morphism f : Y → X induced by ω_{Y}^{⊗m} and for all j there exists a natural morphism,

ϕ^{∗} : \underline{𝔔}_{θ_{X}}^{j} → Rϕ_{∗}\underline{𝔔}_{θ_{Y}}^{j} .

One can also prove that there exists a natural morphism ρ : O_{X} → \underline{𝔔}_{θ_{X}}^{0} cf. [Kovács97b].

Next we need the following:

6.6 Definition. [Kovács97b] f : X → S is an SP-morphism if

ρ : O_{X} → \underline{𝔔}_{θ_{X}}^{0}

has a left inverse in D(X), i.e., ∃ a morphism ρ′ : \underline{𝔔}_{θ_{X}}^{0} → O_{X} such that ρ′ ∘ ρ : O_{X} → O_{X} is a quasi-isomorphism.

6.7 Observation. If f : X → S is an SP-morphism, then

H^{i}(X, L) = ℍ^{i}(X, O_{X} ⊗ L) ⊆ ℍ^{i}(X, \underline{𝔔}_{θ_{X}}^{0} ⊗ L)

for all i and for all locally free sheaves L.

SPECTRAL SEQUENCES ASSOCIATED TO MORPHISMS OF LOCALLY FREE SHEAVES 23

6.8 Example. If f is smooth it is SP. The importance of this notion is shown by the next simple lemma. 6.9 Lemma.

Y ⟶^{ϕ} X
g↓ ↓^{f}
S  S

Let S and g be smooth, ϕ birational and assume that X has rational singularities. Then f is an SP-morphism

Proof.

O_{X}  ⟶^{ρ}  \underline{𝔔}_{θ_{X}}^{0}
↓^{≃}  ↓^{ϕ^{∗}}
Rϕ_{∗}O_{Y} ⟶_{Rϕ_{∗}ρ}^{≃} Rϕ_{∗}Ω_{Y/S}^{0} (≃ Rϕ_{∗}\underline{𝔔}_{θ_{Y}}^{0} )

□ Now (3.7) implies the following: 6.10 Theorem. Let f : X ⟶ S be a morphism of projective algebraic varieties

of dimension n and k respectively, S an Abelian variety and L an ample line bundle on X. Then

ℍ^{p}(X, \underline{𝔔}_{θ_{X}}^{q} ⊗ L) = 0 p + q > n − k.

6.11 Corollary. If in addition f is an SP-morphism, then

H^{p}(X, L) = 0 p > n − k.

This in turn implies that ω_{X} is not ample and that can be used to show that the fibres of f : X → S are isomorphic. Hence the fibres of g are birational.

§A. Appendix: The proof of (1.2.2)

The definition of 𝔾^{p} implies that there exists an exact sequence,

... ⟶ R^{p+q}Φ(𝔽^{p+1}) ⟶^{α_{p,q}} R^{p+q}Φ(𝔽^{p}) ⟶^{β_{p,q}} R^{p+q}Φ(𝔾^{p}) ⟶^{γ_{p,q}} R^{p+q+1}Φ(𝔽^{p+1}) ⟶ ....

Therefore one has a morphism,

β_{p+1,q} ∘ γ_{p,q} : R^{p+q}Φ(𝔾^{p}) → R^{p+q+1}Φ(𝔾^{p+1}),

giving

d_{1}^{p,q} : E_{1}^{p,q} → E_{1}^{p+1,q},

the first differential of E_{r}. Here γ ∘ β = 0 shows that indeed d^{2} = 0.

Next let

x ∈ ker d_{1}^{p,q} ⊆ E_{1}^{p,q} = R^{p+q}Φ(𝔾^{p}).

24 SÁNDOR J. KOVÁCS

Since β_{p+1,q}(γ_{p,q}(x)) = 0, there exists a

y_{p,q}^{1}(x) ∈ R^{p+q+1}Φ(𝔽^{p+2})

such that

α_{p+1,q}(y_{p,q}^{1}(x)) = γ_{p,q}(x).

If

z_{p,q}^{1}(x) ∈ R^{p+q+1}Φ(𝔽^{p+2})

is another element such that

α_{p+1,q}(z_{p,q}^{1}(x)) = γ_{p,q}(x),

then there exists a

w ∈ R^{p+q}Φ(𝔾^{p+1}) = E_{1}^{p+1,q−1}

such that

z_{p,q}^{1}(x) = y_{p,q}^{1}(x) + γ_{p+1,q−1}(w).

Thus

β_{p+2,q−1}(z_{p,q}^{1}(x)) = β_{p+2,q−1}(y_{p,q}^{1}(x)) + \underbrace{β_{p+2,q−1} ∘ γ_{p+1,q−1}(w)}_{d_{1}^{p+1,q−1}(w)} .

Therefore

β_{p+2,q−1}(y_{p,q}^{1}(x)) + im d_{1}^{p+1,q−1}

does not depend on the choice of y_{p,q}^{1}(x). Fix such a y_{p,q}^{1}(x) for each x. Then one has a well-defined natural morphism,

d_{2}^{p,q} : E_{2}^{p,q} = ker d_{1}^{p,q}/im d_{1}^{p−1,q} → E_{2}^{p+2,q−1} = ker d_{1}^{p+2,q−1}/im d_{1}^{p+1,q−1}. Now let

K_{2}^{p,q} = {u ∈ ker d_{1}^{p,q} | u + im d_{1}^{p−1,q} ∈ ker d_{2}^{p,q}}

and let x ∈ K_{2}^{p,q}. Then there exists a

y_{p,q}^{1}(x) ∈ R^{p+q+1}Φ(𝔽^{p+2})

such that

α_{p+1,q}(y_{p,q}^{1}(x)) = γ_{p,q}(x)  (A.1)
β_{p+2,q−1}(y_{p,q}^{1}(x)) = β_{p+2,q−1} ∘ γ_{p+1,q−1}(x′)

for some x′ ∈ R^{p+q}Φ(𝔾^{p+1}), and so

y_{p,q}^{1}(x) − γ_{p+1,q−1}(x′) ∈ ker β_{p+2,q−1} = im α_{p+2,q−1}.

Then there exists a

y_{p,q}^{2}(x) ∈ R^{p+q+1}Φ(𝔽^{p+3})

such that

α_{p+2,q−1}(y_{p,q}^{2}(x)) = y_{p,q}^{1}(x) − γ_{p+1,q−1}(x′).  (A.2)

Now

β_{p+3,q−2}(y_{p,q}^{2}(x)) ∈ ker d_{1}^{p+3,q−2}

and the class, β_{p+3,q−2}(y_{p,q}^{2}(x)) + im d_{1}^{p+2,q−2} ∈ ker d_{1}^{p+3,q−2}/im d_{1}^{p+2,q−2} = E_{2}^{p+3,q−2} does not depend on the choice of y_{p,q}^{2}(x) as long as it satisfies (A.2). Furthermore, since γ ∘ β = 0,

β_{p+3,q−2}(y_{p,q}^{2}(x)) + im d_{1}^{p+2,q−2} ∈ ker d_{2}^{p+3,q−2}.

Thus one can define

d_{3}^{p,q} : E_{3}^{p,q} = ker d_{2}^{p,q}/im d_{2}^{p−2,q+1} → E_{3}^{p+3,q−2} = ker d_{2}^{p+3,q−2}/im d_{2}^{p+1,q−1}

by the formula

d_{3}^{p,q} (x + im d_{1}^{p−1,q} + im d_{2}^{p−2,q+1}) = β_{p+3,q−2}(y_{p,q}^{2}(x))+im d_{1}^{p+2,q−2}+im d_{2}^{p+1,q−1}.

Claim. d_{3}^{p,q} is well-defined.

Proof. One needs to prove that d_{3}^{p,q} does not depend on the choice of y_{p,q}^{1}(x) and x′. Let z_{p,q}^{1}(x) ∈ R^{p+q+1}Φ(𝔽^{p+2}) and z_{p,q}^{2}(x) ∈ R^{p+q+1}Φ(𝔽^{p+3}) be such that

α_{p+1,q}(z_{p,q}^{1}(x)) = γ_{p,q}(x)  (A.3)
β_{p+2,q−1}(z_{p,q}^{1}(x)) = β_{p+2,q−1} ∘ γ_{p+1,q−1}(x′′)

for some x′′ ∈ R^{p+q}Φ(𝔾^{p+1}) and

α_{p+2,q−1}(z_{p,q}^{2}(x)) = z_{p,q}^{1}(x) − γ_{p+1,q−1}(x′′).
By (A.1) and (A.3) there exists an x′′′ ∈ R^{p+q}Φ(𝔾^{p+1}) such that
z_{p,q}^{1}(x) − γ_{p+1,q−1}(x′′) = y_{p,q}^{1}(x) − γ_{p+1,q−1}(x′) + γ_{p+1,q−1}(x′′′).

Then

γ_{p+1,q−1}(x′′′) = α_{p+2,q−1}(z_{p,q}^{2}(x) − y_{p,q}^{2}(x)),

hence by definition,

d_{2}^{p+1,q−1}(x′′′ + im d_{1}^{p,q−1}) = β_{p+3,q−2}(z_{p,q}^{2}(x) − y_{p,q}^{2}(x)) + im d_{1}^{p+2,q−2}.

In other words

β_{p+3,q−2}(z_{p,q}^{2}(x) − y_{p,q}^{2}(x)) + im d_{1}^{p+2,q−2} ∈ im d_{2}^{p+1,q−1}

which means that d_{3}^{p,q} is indeed well-defined. □

In the general case one has to work in a similar way. Assume that for a fixed r ∈ ℕ and for all p, q and 1 ≤ i ≤ r,

d_{i}^{p,q} : E_{i}^{p,q} → E_{i}^{p+i,q−i+1}

is defined and

E_{i}^{p,q} ≃ ker d_{i−1}^{p,q}/im d_{i−1}^{p−i+1,q+i−2}.

Let K_{0}^{p,q} = E_{1}^{p,q} and

K_{i}^{p,q} = {x ∈ E_{1}^{p,q} | x ∈ ker d_{1}, x + im d_{1} ∈ ker d_{2}, x + im d_{1} + im d_{2} ∈ ker d_{3}, … …, x + im d_{1} + im d_{2} + ⋯ + im d_{i−1} ∈ ker d_{i}}.

26 SÁNDOR J. KOVÁCS

Then K_{i−1}^{p,q} maps surjectively onto E_{i}^{p,q}. Let the kernel of this surjection be I_{i−1}^{p,q}, i.e.,

E_{i}^{p,q} ≃ K_{i−1}^{p,q}/I_{i−1}^{p,q}.

Then I_{i−1}^{p,q} is the preimage of im d_{i−1}^{p−i+1,q+i−2} in K_{i−1}^{p,q}. Let

α_{p,q}^{i} = α_{p+1,q} ∘ α_{p+2,q−1} ∘ ⋯ ∘ α_{p+i,q−i+1}.

Further assume that there exist (set-theoretic) functions,

y_{p,q}^{i−1} : K_{i−1}^{p,q} → R^{p+q+1}Φ(𝔽^{p+i})

such that

α_{p,q}^{i−1}(y_{p,q}^{i−1}(x)) = γ_{p,q}(x) ∈ R^{p+q+1}Φ(𝔽^{p+1}).  (A.4)

Further notice, that

d_{i}^{p,q} : E_{i}^{p,q} → E_{i}^{p+i,q−i+1}

is given by

d_{i}^{p,q} (x + I_{i−1}^{p,q}) = β_{p+i,q−i+1}(y_{p,q}^{i−1}(x)) + I_{i−1}^{p+i,q−i+1}  (A.5)

and d_{i}^{p,q} does not depend on the choice of y_{p,q}^{j}(x) as long as it satisfies (A.4).

Note also that y_{p,q}^{0}(x) = γ_{p,q}(x). Let I_{r}^{p,q} be the preimage of im d_{r}^{p−r,q+r−1} in K_{r}^{p,q} and let

E_{r+1}^{p,q} ≃ K_{r}^{p,q}/I_{r}^{p,q}.
Suppose now that x ∈ K_{r}^{p,q}, i.e., x + I_{r−1}^{p,q} ∈ ker d_{r}^{p,q} or equivalently
β_{p+r,q−r+1}(y_{p,q}^{r−1}(x)) ∈ I_{r−1}^{p+r,q−r+1}.

By (A.5) there exists an x_{j} ∈ K_{j}^{p+r−j−1,q−r+j+1} for j = 0, …, r − 2 such that

β_{p+r,q−r+1}(y_{p,q}^{r−1}(x) − ∑_{j=0}^{r−2} y_{p+r−j−1,q−r+j+1}^{j}(x_{j})) = 0.

Therefore there exists a

y_{p,q}^{r}(x) ∈ R^{p+q+1}Φ(𝔽^{p+r+1})

such that

α_{p+r,q−r+1}(y_{p,q}^{r}(x)) = y_{p,q}^{r−1}(x) − ∑_{j=0}^{r−2} y_{p+r−j−1,q−r+j+1}^{j}(x_{j}).

By (A.4), and since α ∘ γ = 0,

\begin{aligned}α_{p,q}^{r}(y_{p,q}^{r}(x)) & = α_{p,q}^{r−1} ∘ α_{p+r,q−r+1}(y_{p,q}^{r}(x)) \\ & = α_{p,q}^{r−1}(y_{p,q}^{r−1}(x) − ∑_{j=0}^{r−2} y_{p+r−j−1,q−r+j+1}^{j}(x_{j})) \\ & = α_{p,q}^{r−1}(y_{p,q}^{r−1}(x)) = γ_{p,q}(x). \end{aligned}  (A.6)

Now one can define d_{r+1}^{p,q} : E_{r+1}^{p,q} = K_{r}^{p,q}/I_{r}^{p,q} → E_{r+1}^{p+r+1,q−r} = K_{r}^{p+r+1,q−r}/I_{r}^{p+r+1,q−r} by

d_{r+1}^{p,q} (x + I_{r}^{p,q}) = β_{p+r+1,q−r}(y_{p,q}^{r}(x)) + I_{r}^{p+r+1,q−r}.

Claim. d_{r+1}^{p,q} is well-defined.

Proof. One needs to prove that any set of functions z_{p,q}^{i} for p, q and 0 ≤ i ≤ r satisfying (A.4), (A.5) and (A.6) gives the same d_{r+1}^{p,q}. This is true for d_{i}^{p,q}, 1 ≤ i ≤ r, by induction.

Since

α_{p+1,q} ∘ α_{p+1,q−1}^{i−1}(z_{p,q}^{i}(x)) = α_{p,q}^{i}(z_{p,q}^{i}(x)) = γ_{p,q}(x),

one finds that

α_{p+1,q−1}^{i}(z_{p,q}^{i}(x)) = z_{p,q}^{1}(x) + γ_{p+1,q−1}(x_{i})

for some x_{i} ∈ E_{1}^{p+1,q−1} for all i ≥ 2. Now replace z_{p,q}^{i}(x) by z_{p,q}^{i}(x) − z_{p+1,q−1}^{i−1}(x_{i}) for i ≥ 2. Then α_{p,q}^{i}(z_{p,q}^{i}(x)) = γ_{p,q}(x) remains true and β_{p+i+1,q−i}(z_{p,q}^{i}(x)) is only changed by an element of I_{i}^{p+i+1,q−i}. Hence one may assume that

α_{p+1,q−1}^{i}(z_{p,q}^{i}(x)) = z_{p,q}^{1}(x).

Repeating this argument for j = 2, …, r−1, using α_{p+j,q−j}^{i−j} in place of α_{p+1,q−1}^{i−1}, one may assume that

α_{p+j,q−j}^{i−j}(z_{p,q}^{i}(x)) = z_{p,q}^{j}(x) for all i > j.

One may also assume the same for the y’s, i.e., for all p, q and 0 ≤ j ≤ r − 1,

α_{p+j,q−j}^{i−j}(y_{p,q}^{i}(x)) = y_{p,q}^{j}(x) for all i > j.  (A.7)

Henceforth it will be assumed that any set of y’s satisfies (A.7). Observe that

α_{p+1,q}(z_{p,q}^{1}(x)) = γ_{p,q}(x) = α_{p+1,q}(y_{p,q}^{1}(x)),

so there exists an x′ ∈ E_{1}^{p+1,q−1} such that

z_{p,q}^{1}(x) = y_{p,q}^{1}(x) + γ_{p+1,q−1}(x′).

Then replace y_{p,q}^{i}(x) by y_{p,q}^{i}(x) + y_{p+1,q−1}^{i−1}(x′) for i ≥ 1. This way

α_{p,q}^{i}(y_{p,q}^{i}(x)) = γ_{p,q}(x)

and (A.7) remain true and β_{p+i+1,q−i}(y_{p,q}^{i}(x)) is only changed by an element of I_{i}^{p+i+1,q−i}. Hence one may assume that

z_{p,q}^{1}(x) = y_{p,q}^{1}(x).

Next observe that

α_{p+2,q−1}(z_{p,q}^{2}(x)) = z_{p,q}^{1}(x) = y_{p,q}^{1}(x) = α_{p+2,q−1}(y_{p,q}^{2}(x)),

hence there exists an x′′ ∈ E_{1}^{p+2,q−2} such that

z_{p,q}^{2}(x) = y_{p,q}^{2}(x) + γ_{p+2,q−2}(x′′).
Now replace y_{p,q}^{i}(x) by y_{p,q}^{i}(x) + y_{p+2,q−2}^{i−2}(x′′) for i ≥ 2. Again
α_{p,q}^{i}(y_{p,q}^{i}(x)) = γ_{p,q}(x)

and (A.7) remain true and β_{p+i+1,q−i}(y_{p,q}^{i}(x)) is only changed by an element of I_{i}^{p+i+1,q−i}. Hence one can assume that

z_{p,q}^{2}(x) = y_{p,q}^{2}(x)

also.

Iterating this procedure one finds that d_{r+1}^{p,q} is the same defined either via the y’s or the z’s. □

Now let z ∈ R^{p+q}Φ(𝔽^{p}). Then β_{p,q}(z) ∈ K_{i}^{p,q} for all i, since γ ∘ β = 0. On the other hand if x ∈ K_{k−p}^{p,q}, then y_{p,q}^{k−p}(x) = 0, since 𝔽^{k+1}= 0. Then

γ_{p,q}(x) = α_{p,q}^{k−p}(y_{p,q}^{k−p}(x)) = 0,

so there exists a z ∈ R^{p+q}Φ(𝔽^{p}) such that x = β_{p,q}(z).

Therefore β_{p,q} induces a surjective morphism

β̄_{p,q} : R^{p+q}Φ(𝔽^{p}) ↠ E_{∞}^{p,q}.

Let

F^{p}R^{p+q} = α_{l−1,p+q−l}^{p−l}(R^{p+q}Φ(𝔽^{p})) ⊆ R^{p+q}Φ(𝔽^{l}).

From the construction of the spectral sequence it is clear that β̄_{p,q}(x) = 0 for any x ∈ R^{p+q}Φ(𝔽^{p}) such that α_{l−1,p+q−l}^{p−l}(x) = 0, hence β̄_{p,q} factors through F^{p}R^{p+q} inducing a surjective morphism

β̂_{p,q} : F^{p}R^{p+q} ↠ E_{∞}^{p,q}.

Now ker β_{p,q} = im α_{p,q} implies that ker β̂_{p,q} = F^{p+1}R^{p+q}. Hence for any m

R^{m}Φ(𝔽^{l}) = F^{l}R^{m} ⊇ F^{l+1}R^{m} ⊇ ⋯ ⊇ F^{k}R^{m} ⊇ F^{k+1}R^{m} = 0

is a filtration of R^{m}Φ(K) = R^{m}Φ(𝔽^{l}) such that

E_{∞}^{p,m−p} ≃ ^{F^{p}R^{m}/}F^{p+1}R^{m}.

This completes the proof of (1.2.2). □

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University of Washington, Department of Mathematics, Seattle, WA 98195

    E-mail address: kovacs@math.washington.edu