Degeneration of the Hodge de-Rham spectral sequence Ÿ0 ...

Degeneration of the Hodgede-Rham spectral sequence
Ÿ0 Introduction
Let S be a locally noetherian scheme, and let f : X → S be a smooth proper morphism. If
Ω X/S is the coherent sheaf of relative Kähler dierentials, we dene the de Rham complex to
be its exterior algebra with dierential the exterior derivative, and the de Rham cohomology
is the cohomology of the de Rham complex (relative to S):
Ω • X/S = ∧ •
O X
Ω X/S and H ∗ dR(X/S) = R ∗ f ∗ Ω • X/S.
By virtue of the latter being the cohomology of a complex, i.e. hypercohomology, there
is a spectral sequence having the form
E p,q
1 = R q f ∗ Ω p X/S
=⇒ Hp+q (X/S),
dR
which we call the Hodgede-Rham spectral sequence (HdRSS). Using the algebra structure
on Ω • X/S
one obtains an algebra structure the de Rham cohomology, and the HdRSS is a
spectral sequence of graded algebras.
The basic question of this talk is: When is the HdRSS degenerate at E 1 ?
Ÿ1 Statement of the main result
To save writing in the sequel, we dene the Hodge cohomology of X/S to be the graded
algebra H ∗ Hodge
(X/S) with
H
n Hodge(X/S) = ⊕
p+q=n
R q f ∗ Ω p X/S .
The spectral sequence machine endows H n dR
(X/S) with a canonical ltration, called the
Hodge lrration, whose associated graded satises Gr p H n (X/S) ∼ dR = E∞
p,n−p . Thus, if the
HdRSS is degenerate at E 1 , i.e. E 1 = E ∞ , then Gr ∗ H n (X/S) ∼ dR = H n Hodge
(X/S) for all n. In
fact, a converse holds under our niteness hypotheses; we will explain it next, but only in
the special case that matters to us.
Let us take S = Spec k with k a eld. As one passes from E n to E n+1 , each En
p,q is
replaced by one of its subquotients. Hence E p,q
1 and E∞
p,q are equal precisely when they have
the same dimension as k-vector spaces, and so degeneracy at E 1 amounts to having the a
priori inequality dim k E p,q
1 ≥ dim k E∞
p,q be an equality. In fact, one easily sees that it is the
same to show that, for each n ≥ 0, one has
∑
p+q=n
dim k E p,q
1 = ∑
p+q=n
dim k E p,q
∞ .
The left hand side of this identity is clearly dim k H n Hodge
(X/k). On the other hand, the HdRSS
abuts to H ∗ (X/k), meaning that there some a ltration on dR H∗ dR
(X/k) whose associated
graded is E ∞ . Since taking gradeds preserves dimensions, we nd that the right hand side of
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the above identity is dim k H ∗ dR
(X/k). To summarize, the degeneracy of HdRSS for a scheme
X over a eld k amounts to the simple claim that
dim k H n Hodge(X/k) = dim k H n dR(X/k)
for all n. The converse alluded to above now follows: if there exists any ltration on
H n (X/k) whose associated graded is isomorphic to dR Hn Hodge
(X/k), then the above identity
of dimensions holds.
The following is the main result of this talk.
Theorem 1. Let k be a eld of characterstic 0, and X/k a smooth proper variety. Then
HdRSS degenerates, i.e. dim k H ∗ Hodge (X/k) = dim k H ∗ dR (X/k).
There are three known proofs. For the rst, one makes use of the Hodge decomposition
of projective varieties over the complex numbers C. For the second, one makes use of the
HodgeTate decomposition of projective varieties over p-adic local elds K, upon their base
change to a completed algebraic closure C K . (When K = Q p , the latter is often written C p .)
In both proofs, one extends to the smooth proper case by Chow's lemma and Hironaka's
resolution of singularities. Then, one deduces the result for arbitrary k using the Lefschetz
principle. The third proof, due to DeligneIllusie, gives a criterion for the degeneration of
the HdRSS in characteristic p, and then uses a variant of the Lefschetz principle to reduce
to this case.
We give in this talk an exposition of DeligneIllusie's proof, following their paper very
closely. The structure of the proof consists of several reduction steps. Each reduction leads
us to an interesting new thing to see or prove.
Ÿ2 Reduction to characteristic p>0
We assume from the outset that X is connected, because if the theorem holds for the
connected components of X, then it clearly holds for X. In particular, X has constant
dimension d.
A standard variant of the Lefschetz principle guarantees the existence of the following
very noncanonical data: a nitely generated Z-algebra A, a ring homomorphism A → k,
a smooth proper A-scheme X, and an isomorphism X k
∼ = X. After possibly localizing A,
we can assume that the coherent A-modules H n (X/A) and dR Hn Hodge
(X/A) are locally free.
Furthermore, invert in A all rational primes p with p ≤ d.
Because our cohomology groups are locally free, at base change implies that their formation
commutes with base arbitrary change: for any g : T → Spec A, one has g ∗ H⋆ n (X/A) ∼ =
H⋆ n (X T /T ), with ⋆ = dR or Hodge. In particular, for any any ring map A → l to a eld l
one has
dim l H⋆ n (X l /l) = rank A H⋆ n (X/A) = dim k H⋆ n (X/k),
and the question of degeneracy of the HdRSS is equivalent for all the X l /l. Let us now
choose a particular member of this last family.
Pick a closed point t ∈ Spec(A ⊗ Q), and let T = {t} ⊆ Spec A be its scheme-theoretic
closure. Then T → Spec Z is quasi-nite at. Choose a closed point s ∈ T at which
T → Spec Z is étale. By our setup, the local ring O T,s has maximal ideal is generated by a
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ational prime p, and its residue eld l is nite. Noting that l is perfect, this implies that
O T,s /(p 2 ) ∼ = W 2 (l), the ring of p-typical Witt vectors of length 2 over l.
Supposing S ↩→ ˜S is a closed immersion dened by a square-zero ideal, and X → S is a
at morphism, by a lift of X to ˜S we mean a at map ˜X → ˜S equipped with an isomorphism
˜X × eS S ∼ = X. If X/S is smooth then ˜X/ ˜S automatically is too. Thus, in the preceding
setup, X OT,s /(p 2 ) is clearly a smooth lift of X l to W 2 (l), and the degeneracy of HdRSS for our
original X/k will follow from the following theorem.
Theorem 2. Let k be a perfect eld of characteristic p > 0, and let X/k be any smooth
proper scheme of dimension d < p. If X admits a smooth lift to W 2 (k), then the HdRSS of
X/k is degenerate at E 1 .
The degeneracy of HdRSS fails quite often in characteristic p, when the requirement that
d < p is relaxed; see DeligneIllusie for some references. (Somehow, widespread degeneracy
of the HdRSS is a hallmark of the characteristic 0 world. In fact, in crystalline deformation
theory, one obtains a deformation from char p to char 0 in exchange for supplying a Hodge
ltration on the putative de Rham cohomology, i.e. evidence of a HdRSS degeneration!)
In spite of this, DeligneIllusie work henceforth entirely in characteristic p geometryeven
working over a more general base F p -scheme S, although we stick to S = Spec k to keep our
statements simple.
Ÿ3 Reduction to decomposition of the de Rham complex
We begin this section with some preliminaries.
Recall that, for S any F p -scheme, there is a morphism of schemes F S : S → S called the
absolute Frobenius morphism, acting as the identity on |S| and as the pth power map on O S .
If f : X → S is any map, then we set X ′ = X × S S with S → S via F S , and call X ′ the
Frobenius twist of X (relative to S). In down-to-earth terms, X ′ is obtained by taking the
dening equations of X and raising their coecients to the pth power.
Moreover, one has F S ◦ f = f ◦ F X , and this commutativity leads to a map F X/S , called
the relative Frobenius morphism of X (considered as an S-scheme), characterized as the
unique dotted arrow making the following diagram commutative.
X F X
X ′ GX/S X
f
f ′ f
S F S
S
We will denote F X/S more briey by F when no confusion my arise. In down-to-earth terms,
given a solution to the equations dening X, the map F raises its coordinates to the pth
power, after which they satisfy the equations dening X ′ . The map F is an S-map and is
nite at of degree p, so that F ∗ is exact and fully faithful, and H ∗ (X ′ , F ∗ M) ∼ = H ∗ (X, M)
for any quasicoherent O X -module M.
Finally, we recall here an important construction due to Cartier. For a complex M • of
sheaves, use H ∗ M • to denote the cohomology sheaves of M • with respect to the dierentials
(as opposed to sheaf cohomology).
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Theorem 3. Let S be any F p -scheme and let X/S be smooth. Then there exists a unique
homomorphism of graded O X ′-algebras
C −1 : ⊕ i≥0
Ω i X ′ /S −→ ⊕ i≥0
H i (F ∗ O X/S )
with the property that, for every local section s of O X with image G ∗ X/S s = x ⊗ 1 in O X the ′
map C −1 sends d(x ⊗ 1) to the class of x p−1 dx. Moreover, C −1 is an isomorphism of graded
O X ′-algebras.
The map C −1 above is called the (inverse of the) Cartier isomorphism.
Let us specialize to S = Spec k and X/k smooth proper of dimension d < p, as in
Theorem 2. Again, we can assume X is connected. Our present step is to implement the
following wishful thinking. The two sides related by the Cartier isomorphism are naturally
the H i of complexes on X ′ , respectively
⊕
i
Ω i X ′ /k[−i] and F ∗ Ω • X/k,
the rst of these being the complex with Ω i X ′ /k
in each degree i and with dierentials d = 0.
Experience with derived categories tells us that natural operations on cohomology tend to
come from operations on the underlying complexes. So we may wish that C −1 be the H ∗
of a map C0 −1 between these complexes in D(X ′ ) := D b (qCoh(X ′ )), the bounded derived
category of quasicoherent O X ′-modules. (In particular, C −1 being an isomorphism, C0
−1
would be a quasi-isomorphism, i.e. an isomorphism in D(X ′ ).)
Such maps C0 −1 do not always exist, essentially because the HdRSS does not always
degenerate when d > p, and, we shall now see, the HdRSS degenerates when a C0 −1 exists.
Suppose given a C0 −1 , and that X/k is proper. Then
HdR(X/k) n = H n (X, Ω • X/k) = H n (X ′ , F ∗ Ω • X/k) C−1 0 ⊕
∼ = H n−i (X ′ , Ω i X/k) = HHodge(X n ′ /k),
whence dim k H n (X/k) = dim dR k H n Hodge (X′ /k) for all n. On the other hand, the pth power
map k → ∼ k is an extension of elds, hence at, and one has H n Hodge (X′ /k) = H n (X/k)⊗ Hodge kk.
This gives dim k H n Hodge (X′ /k) = dim k H n Hodge
(X/k) for all n, and the desired identity of
dimensions follows.
We now see that we are reduced to proving the following theorem.
Theorem 4. Let k be a prefect eld of characteristic p, and X/k a smooth (not necessarily
proper) variety. One can associate to each lift ˜X of X to W2 (k) a canonical morphism
i
ϕ = ϕ eX : ⊕ i

As mentioned previously, the property that H ∗ ϕ = C −1 implies that ϕ is an isomorphism
in D(X ′ ). We say that ϕ realizes a decomposition of the (image of the) de Rham complex
(under F ∗ ) by equating it in D(X ′ ) to a complex with dierential d = 0. DeligneIllusie
actually prove a more general theorem, in two ways. First, they give conditions that generalize
the base k to an F p -scheme S. Second, they give a result that is valid for schemes
of dimension d > p, by producing a morphism ϕ relating only the truncations of the two
complexes to their components of degree < p. This implies that E i,j
1 = E ∞ i, j for i + j < p
in the HdRSS.
Ÿ4 Construction of ϕ eX
We must dene, for i ≥ 0, maps ϕ i : Ω i X ′ /k → F ∗Ω • X/k
. For i = 0, the only option for ϕ0
is the composite
Ω 0 X /k[0] −→ C−1
H 0 (F ′ ∗ Ω • X/k) ⊆ F ∗ Ω • X/k.
And, supposing we have dened ϕ 1 , we dene ϕ i for i > 1 to be the composite
Ω i X ′ /k[−i] = ∧ i
O X ′
(
ΩX ′ /k[−1] ) → ( Ω X ′ /k[−1] ) ⊗i (ϕ 1 )
−−−→ ⊗i
(F ∗ Ω • X/k) ⊗i mult.
−−−→ F ∗ Ω • X/k,
the rst arrow being the usual section of the projection Ω ⊗i
X ′ /k ↠ ∧ i
O Ω X ′ X ′ /k given by
ω 1 ∧ · · · ∧ ω i ↦→ 1 i!
∑
sgn(σ)ω σ(1) ⊗ · · · ⊗ ω σ(i) .
σ∈S i
Note that, oddly enough, this is the only point in the proof where we use the hypothesis
d < p (so that the denominator i! is nonzero). Anyhow, because C −1 is a graded algebra
homomorphism, and our ϕ i are clearly multiplicative, we see that H 1 ϕ = C −1 implies H i ϕ =
C −1 for all i.
As another aside, we point out that we only use ˜X in order to construct ϕ 1 . In their
paper, DeligneIllusie prove (as an isomorphism of gerbes) that the datum ˜X is equivalent
to the datum of a decoposition of the truncation of F ∗ Ω • X/k
to degrees at most 1. The upshot
is that, given this decomposition, if it agrees with the Cartier isomorphism, then one gets a
decomposition in degrees below p for free.
We are left with the task of dening ϕ 1 : Ω 1 X ′ /k [−1] → F ∗Ω • so that X/k H1 ϕ = C −1 .
Zariski-locally on X, there exist lifts of F to ˜X, i.e. W 2 (k)-maps ˜F : ˜X → ˜X′ , where ˜X ′ =
˜X ⊗ W2 (k) W 2 (k) via W 2 (F k ): W 2 (k) → W 2 (k), such that ˜F | X = F. Let us begin by working
locally, assuming that such a lift is dened on all of X, and use it to write down ϕ 1 explicitly;
then we will use a sort of ƒech patching argument in the derived category to obtain ϕ 1
globally.
In the case where F lifts, we construct ϕ 1 following an idea due to Mazur. As in the
preceding section one has F ∗ (x ⊗ 1) = x p , which implies that F ∗ : Ω X ′ /S → F ∗ Ω X/S is the
zero map. In terms of our lift, ˜F ∗
takes values in p ˜F ∗ Ω . Now eX/W2 (k) Ω X/W2 e (k)
is locally free,
and therefore so is ˜F ∗ Ω . It follows that, via the isomorphism eX/W2 (k) pW 2(k) ← ∼ k given by
multiplication by p, one has p ˜F ∗ Ω ∼ eX/W2 (k) = F ∗ Ω X/k . Considering that the composition
Ω 1 X e [−1] F e ∗ [−1]
′ /W 2
−−−−→ p ˜F (k) ∗ Ω 1 p−1
X/W2 e [−1] ∼
(k) = F∗ Ω 1 X/k[−1] ↩→ F ∗ Ω • X/k
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is killed by p, it uniquely factors through Ω eX ′ /W 2 (k) [−1] ↠ Ω X/k[−1] to give our map ϕ 1 . We
must check that ϕ 1 is a chain map, which amounts to dϕ 1 = 0, and that H 1 ϕ 1 = C −1 . Since
˜F lifts F, one must have ˜F ∗ (x ⊗ 1) = x p + p u(x) for every local section x of O eX , with u(x)
some section of O eX . Given a local section x 0 of O X , we choose a lift x to O eX and unwind
the denition of ϕ 1 . We calculate that
ϕ 1 (dx 0 ⊗ 1) = p −1 d( ˜F ∗ (x ⊗ 1)) = p −1 d(x p + p u(x))
= p −1 (px p−1 dx + p du(x)) = x p−1
0 dx 0 + du(x).
One reads our remaining two claims immediately from this equation.
Let us describe our patching mechanism. Let U = {U i } i∈I be an open cover of X, and
M a quasicoherent O X -module. The ƒech resolution associated to M over U is dened
as follows. For each n ≥ 0 let ∆ n = {0, 1, . . . , n}, and for any map s: ∆ n → I write
U s = ⋂ i∈∆ n
U s(n) and j s : U s ↩→ X for the inclusion. Dene the sheaves
ˇ C n (U , M) =
∏
s: ∆ n→I
j s∗ j ∗ sM.
The collection of C ˇ n
is made into a complex in the standard simplicial manner. Notice from
the denition that there is a canonical map M → C ˇ0
(U , M). In fact, the map of complexes
M[0] → C ˇ•
(U , M) deduced from this is a quasi-isomorphism (i.e. the ƒech resolution really
is a resolution): PROOF. We leave to the reader to deduce as a corollary that, if the U i are
ane, then the old-fashioned ƒech cohomology groups compute quasicoherent sheaf cohomology.
Given a complex M • of quasicoherent sheaves on X, we dene the ƒech resolution
C ˇ • (U , M • ) of M • to be the total complex associated to the obvious double complex. The
maps M i → C ˇ0
(U , M i ) assemble to a map M • → C ˇ•
(U , M • ) that is a quasi-isomorphism,
hence an isomorphism in D(X).
Now return to the situation of our smooth X/k that lifts to W 2 (k), and relax our
requirement that F lift to ˜X. Recall that F lifts to ˜X locally on X, and choose an
open cover U = {U i } i∈I such that F lifts to each U i . Hitting the quasi-isomorphism
Ω • ∼
X/k
→ C ˇ (U , Ω • X/k ) with F ∗, we get a quasi-isomorphism F ∗ Ω • ∼
X/k
→ F ∗C ˇ • (U , Ω • X/k ) in
D(X ′ ). To dene our map ϕ 1 in D(X ′ ), it suces to construct a morphism from Ω 1 into
X ′ /k
F ∗ ˇ C 1 (U , Ω • X/k) = F ∗ ˇ C 1 (U , O X ) ⊕ F ∗ ˇ C 0 (U , Ω X/k ),
and postcompose it with the inverse of the preceding quasi-isomorphism.
On each U i , x a lift ˜F i of F | Ui to Ũi (where Ũi is the unique open subscheme of ˜X
whose preimage in X is U i ), and let ϕ 1 i = p −1 ∗ ˜F i be the morphism ϕ 1 as dened over each
U i where ˜F exists. We clearly want to take, for second summand, the map sending ω to
the tuple (ϕ 1 i ω| Ui ) i . The rst summand is chosen in the only reasonable way that ensures
that dϕ 1 = 0 (i.e., it is a chain map). To this end, for x a local section x of O eUi write
˜F i ∗ (x ⊗ 1) = x p + p u i (x) for u i (x) some section of O eUi . For x 0 a local section of O X , dene
h i,j (dx 0 ⊗ 1) = u j (x| Ui ∩U j
) − u i (x| Ui ∩U j
), using any lift x of x 0 to O eX whatsoever.
It is easy to check the following facts, in order: The map ω ↦→ ((h i,j (ω)) i,j , (ϕ 1 i (ω)) i ) is
well-dened and independent of choices of lifts up to coboundary, and gives a chain map
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as desired. The resulting morphism is independent (up to the obvious quasi-isomorphism)
under replacing U with a renement, and hence is completely independent of U (any two
covers have a comon renement). To see that H 1 ϕ 1 = C −1 , it suces to see that these two
morphisms agree locally on X ′ ; but locally F lifts to ˜X, and we have checked the identity in
this case, and, as just noted, our construction is independent of how X is covered by liftable
situations.
This completes the proof.
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