Big De Rham-Witt Cohomology: Basic Results

Andre Chatzistamatiou^{1}

Received: March 21, 2013

Communicated by Lars Hesselholt

Abstract. Let X be a smooth projective R-scheme, where R is a smooth ℤ-algebra. As constructed by Hesselholt, we have the absolute big de Rham-Witt complex 𝕎Ω_{X}^{∗} of X at our disposal. There is also a relative version 𝕎Ω_{X/R}^{∗} with 𝕎(R)-linear differential. In this paper we study the hypercohomology of the relative (big) de Rham-Witt complex after truncation with finite truncation sets S. We show that it is a projective 𝕎_{S}(R)-module, provided that the de Rham cohomology is a flat R-module. In addition, we establish a Poincaré duality theorem. explicit description of the relative de Rham-Witt complex of a smooth λ-ring, which may be of independent interest.

2010 Mathematics Subject Classification: 14F40, 14F30

Introduction

Let X be a scheme over a perfect field k of characteristic p > 0. The de Rham-Witt complex W Ω_{X/k}^{∗} was defined by Illusie [Ill79] relying on ideas of Bloch, Deligne and Lubkin. It is a projective system of complexes of W (k)- modules on X, which is indexed by the positive integers. If X is smooth then the hypercohomology of W_{n}Ω_{X/k}^{∗} admits a natural comparison isomorphism to the crystalline cohomology of X with respect to W_{n}(k). Langer and Zink have extended Illusie’s definition of the de Rham-Witt complex to a relative situation, where X is a scheme over Spec(R) and R is a ℤ_{(p)}-algebra [LZ04]. If p is nilpotent in R and X is smooth, then they construct a functorial comparison isomorphism

H^{∗}(X, W_{n}Ω_{X/R}^{∗}) ≅ H_{crys}^{∗}(X/W_{n}(R)).

The big de Rham-Witt complex 𝕎Ω_{A}^{∗} was introduced, for any commutative ring A, by Hesselholt and Madsen [HM01]. The original construction relied on the adjoint functor theorem and has been replaced by a direct and explicit method due to Hesselholt [Hes].

1This work has been supported by the SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”

Again, it is a projective system of graded sheaves [S ↦ 𝕎_{S}Ω_{A}^{∗}], but the index set consists of finite truncation sets; that is, finite subsets S of ℕ_{>0} having the property that whenever n ∈ S, all (positive) divisors of n are also contained in S. For the ring of integers, 𝕎Ω_{ℤ}^{∗} has been computed by Hesselholt [Hes]. It vanishes in degree ≥ 2, but 𝕎Ω_{ℤ}^{1} is non-zero. Let X be an R-scheme. In this paper we will consider the relative version

S ↦ 𝕎_{S}Ω_{X/R}^{∗}

of the (big) de Rham-Witt complex, which is constructed from 𝕎Ω_{X}^{∗} by killing the ideal generated by 𝕎Ω_{R}^{1}. The relation with the de Rham-Witt complex of Langer-Zink is given in Proposition 1.2.7: if R is a ℤ_{(p)}-algebra then

𝕎_{{1,p,...,p^{n−1}}}Ω_{A/R}^{∗} = W_{n}Ω_{A/R}^{∗}.

In the following we will use the notation W_{n} = 𝕎_{{1,p,...,p^{n−1}}}, assuming that a prime p has been fixed. It is natural to consider 𝕎_{S}Ω_{X/R}^{∗} as a sheaf of complexes on the scheme 𝕎_{S}(X), which is obtained by gluing Spec(𝕎_{S}(A_{i})) for an affine covering X = ⋃_{i} Spec(A_{i}). Then the components 𝕎_{S}Ω_{X/R}^{q} form quasi-coherent sheaves, and are coherent under suitable finiteness conditions. Our purpose is to show that the de Rham-Witt cohomology

H_{dRW}^{i} (X/𝕎_{S}(R)) =^{def} H^{i}(𝕎_{S}(X), 𝕎_{S}Ω_{X/R}^{∗})

is as well-behaved as the usual de Rham cohomology. The main theorem of the paper is the following.

Theorem 1 (cf. Theorem 2.2.1). Let R be a smooth ℤ-algebra. Let X be a smooth and proper R-scheme. Suppose that the de Rham cohomology H_{dR}^{∗}(X/R) of X is a flat R-module. Then H_{dRW}^{∗} (X/𝕎_{S}(R)) is a finitely generated projective 𝕎_{S}(R)-module for all finite truncation sets S. Moreover, for an inclusion of finite truncation sets T ⊂ S, the induced map

(0.0.1)  H_{dRW}^{∗} (X/𝕎_{S}(R)) ⊗_{𝕎_{S}(R)} 𝕎_{T} (R) ⟶^{≅} H_{dRW}^{∗} (X/𝕎_{T} (R))

is an isomorphism.

If R is a smooth ℤ-algebra and X/R is smooth and proper, then there is a non-zero integer N such that the assumptions of the theorem hold for the base change X⊗_{R}R[N^{−1}]/R[N^{−1}]. Curves and abelian varieties are examples where the de Rham cohomology is flat (see Remark 2.2.2). In order to prove Theorem 1, we will construct for all maximal ideals 𝔪 of R and n, j > 0, a natural quasi-isomorphism:

RΓ(W_{n}Ω_{X/R}^{∗}) ⊗_{W_{n}(R)}^{𝕃} W_{n}(R/𝔪^{j}) ⟶^{q−iso} RΓ(W_{n}Ω_{X⊗R/𝔪^{j}/(R/𝔪^{j})}^{∗}),

where p = char(R/𝔪). The right hand side is RΓ of the de Rham-Witt complex defined by Langer and Zink. Thus it computes the crystalline cohomology, which in our case is a free W_{n}(R/𝔪^{j})-module. Taking the limit \underleftarrow{lim}_{j}, this will yield the flatness of

H_{dRW}^{∗} (X/W_{n}(R)) ⊗_{W_{n}(R)} W_{n}(\underleftarrow{lim}_{j} R/𝔪^{j})

as W_{n}(\underleftarrow{lim}_{j} R/𝔪^{j})-module for all maximal ideals 𝔪, which is sufficient in order to conclude the flatness of the de Rham-Witt cohomology. Concerning Poincaré duality we will show the following theorem.

Theorem 2 (cf. Corollary 3.3.7). Let R be a smooth ℤ-algebra. Let X ⟶ Spec(R) be a smooth projective morphism such that H_{dR}^{∗}(X/R) is a flat Rmodule. Suppose that X is connected of relative dimension d. If the canonical map

H_{dR}^{i}(X/R) ⟶ Hom_{R}(H_{dR}^{2d−i}(X/R), R)

is an isomorphism, then the same holds for the de Rham-Witt cohomology:

H_{dRW}^{i} (X/𝕎_{S}(R)) ⟶^{≅} Hom_{𝕎_{S}(R)}(H_{dRW}^{2d−i}(X/𝕎_{S}(R)), 𝕎_{S}(R)),

for all finite truncation sets S.

In fact, de Rham-Witt cohomology is equipped with a richer structure than the 𝕎(R)-module structure, coming from the Frobenius operators

ϕ_{n} : H_{dRW}^{∗} (X/𝕎_{S}(R)) ⟶ H_{dRW}^{∗} (X/𝕎_{S/n}(R)),

for all positive integers n, and where S/n := {s ∈ S | ns ∈ S}. These are Frobenius linear maps satisfying ϕ_{n} ∘ ϕ_{m} = ϕ_{nm}. The relationship with the Frobenius action on the crystalline cohomology of the fibers is as follows. Let 𝔪 be a maximal ideal of R, set k = R/𝔪 and p = char(k). If H_{dR}^{∗}(X/R) is torsion-free then there is a natural isomorphism

H_{dRW}^{i} (X/W_{n}(R)) ⊗_{W_{n}(R)} W_{n}(k) ≅ H_{crys}^{i}(X ⊗_{R} k/W_{n}(k)),

and ϕ_{p}⊗F_{p} corresponds via this isomorphism to the composition of H_{crys}^{i}(Frob) with the projection. As will be made precise in Section 3, the projective system

H_{dRW}^{i} (X/𝕎(R)) =^{def} [S ↦ H_{dRW}^{i} (X/𝕎_{S}(R))],

together with the Frobenius morphisms {ϕ_{n}}_{n∈ℕ_{>0}}, defines an object in a rigid ⊗-category C_{R}. Maybe the most important property of C_{R} is the existence of a conservative, faithful ⊗-functor

T : C_{R} ⟶ (R-modules), T (H_{dRW}^{i} (X/𝕎(R))) = H_{dR}^{i}(X/R).

Moreover, C_{R} has Tate objects 1(m), m ∈ ℤ, and the first step towards Poincaré duality will be to prove the existence of a natural morphism in C_{R}:

H_{dRW}^{2d} (X/𝕎(R)) ⟶ 1(−d) (d = relative dimension of X/R).

Then it will follow easily that

H_{dRW}^{i} (X/𝕎(R)) ⟶^{≅} \underline{Hom}(H_{dRW}^{2d−i}(X/𝕎(R)), 1(−d)),

provided that the assumptions of Theorem 2 are satisfied. Taking the underlying 𝕎(R)-modules one obtains Theorem 2.

Acknowledgements. After this manuscript had appeared on arXiv, we received a letter from professor James Borger who informed us that he had already obtained Theorem 1, for R = ℤ[N^{−1}], in a joint work with Mark Kisin by using similar methods. I thank Andreas Langer and Kay Rülling for several useful comments on the first version of the paper.

Contents

Introduction  567
Acknowledgements  570
1. Relative de Rham-Witt complexes  570
1.1. Witt vectors  570
1.2. Relative de Rham-Witt complex  574
1.3. Finiteness  579
2. De Rham-Witt cohomology  580
2.1. Reduction modulo an ideal  580
2.2. Flatness  582
3. Poincaré duality  585
3.1. A rigid ⊗-category  585
3.2. The tangent space functor  590
3.3. Proof of Poincaré duality  593
References  598

1. Relative de Rham-Witt complexes

1.1. Witt vectors. For the definition and the basic properties of the ring of Witt vectors we refer to [Hes, §1]. We briefly recall the notions in this section. A subset S ⊂ ℕ = {1, 2, … } is called a truncation set if n ∈ S implies that all positive divisors of n are contained in S. For a truncation set S and n ∈ S, we define S/n := {s ∈ S | sn ∈ S}. Let A be a commutative ring. For all truncation sets S we have the ring of Witt vectors 𝕎_{S}(A) at our disposal. The ghost map is the functorial ring homomorphism

gh = (gh_{n})_{n∈S} : 𝕎_{S}(A) ⟶ ∏_{n∈S} A, gh_{n}((a_{s})_{s∈S}) := ∑_{d|n} d · a_{d}^{n/d}.

It is injective provided that A is ℤ-torsion-free. For all positive integers n, there is a functorial morphism of rings

F_{n} : 𝕎_{S}(A) ⟶ 𝕎_{S/n}(A),

called the Frobenius. Moreover there is a functorial morphism of 𝕎_{S}(A)- modules, the Verschiebung,

V_{n} : 𝕎_{S/n}(A) ⟶ 𝕎_{S}(A),

where the source is a 𝕎_{S}(A)-module via F_{n}. For all coprime positive integers n, m ∈ ℕ we have

F_{n} ∘ V_{n} = n, F_{n} ∘ V_{m} = V_{m} ∘ F_{n} ((m, n) = 1).

We have a multiplicative Teichmüller map

[−] : A ⟶ 𝕎_{S}(A), a ↦ [a] := (a, 0, 0, …) ∈ 𝕎_{S}(A),

and if S is finite then every element a ∈ 𝕎_{S}(A) can be written as

a = ∑_{s∈S} V_{s}([a_{s}])

with unique elements (a_{s})_{s∈S} in A. Let T ⊂ A be a multiplicative set and suppose that S is a finite truncation set. We can consider T via the Teichmüller map as multiplicative set in 𝕎_{S}(A). Then the natural ring homomorphism

T^{−1}𝕎_{S}(A) ⟶ 𝕎_{S}(T^{−1}A).

is an isomorphism. If T ⊂ ℤ is a multiplicative set then

𝕎_{S}(A) ⊗_{ℤ} T^{−1}ℤ ⟶ 𝕎_{S}(T^{−1}A)

is an isomorphism. Let S be a truncation set, and let n be a positive integer; set T := S\{s ∈ S; n | s}. Then T is a truncation set and we have a short exact sequence of 𝕎_{S}(A)-modules:

(1.1.1)  0 ⟶ 𝕎_{S/n}(A) ⟶^{V_{n}} 𝕎_{S}(A) ⟶^{R_{T}^{S}} 𝕎_{T} (A) ⟶ 0.

Example 1.1.1. We have 𝕎_{S}(ℤ) = ∏_{n∈S} ℤ · V_{n}(1), and the product is given by V_{m}(1)·V_{n}(1) = c·V_{mn/c}(1), where c = (m, n) is the greatest common divisor [Hes, Proposition 1.6].

1.1.2. The following theorem will be very useful throughout the paper.

Theorem 1.1.3. (Borger-van der Kallen) Let S be a finite truncation set, and let n be a positive integer. Let ρ : A ⟶ B be an étale ring homomorphism. The following hold.

(1) The induced ring homomorphism 𝕎_{S}(A) ⟶ 𝕎_{S}(B) is étale.

(2) The morphism

𝕎_{S}(B) ⊗_{𝕎_{S}(A),F_{n}} 𝕎_{S/n}(A) ⟶ 𝕎_{S/n}(B), b ⊗ a ↦ F_{n}(b) · 𝕎_{S/n}(ρ)(a),

is an isomorphism.

The references for this theorem are [Bor11a, Theorem B] [Bor11b, Corollary 15.4] and [vdK86, Theorem 2.4] (cf. [Hes, Theorem 1.22]). By using Theorem 1.1.3, the exact sequence (1.1.1), and induction on the length of S, we easily obtain the following corollary.

Corollary 1.1.4. Let ρ : A ⟶ B be an étale ring homomorphism. Let S be a finite truncation set.

(i) For an inclusion of truncation sets T ⊂ S, the map

𝕎_{S}(B) ⊗_{𝕎_{S}(A)} 𝕎_{T} (A) ⟶ 𝕎_{T} (B)

is an isomorphism. (ii) Let n be a positive integer. For any A-algebra C, the natural ring

homomorphism

𝕎_{S/n}(C) ⊗_{F_{n},𝕎_{S}(A)} 𝕎_{S}(B) ⟶ 𝕎_{S/n}(C ⊗_{A} B), c ⊗ b ↦ c · F_{n}(b)

is an isomorphism.

Notation 1.1.5. If a prime p has been fixed then we set W_{n} := 𝕎_{{1,p,p^{2},...,p^{n−1}}}.

1.1.6. Let p be a prime. Let R be a ℤ_{(p)}-algebra. Since all primes different from p are invertible in R, the same holds in 𝕎_{S}(R). The category of 𝕎_{S}(R)- modules, for a finite truncation set S, factors in the following way. Set

ϵ_{1,S} := ∏_{\begin{matrix}primes ℓ ≠ p \\ S/ℓ≠∅ \end{matrix}}(1 − \frac{1}{ℓ}V_{ℓ}(1)) ∈ 𝕎_{S}(R),

and ϵ_{n,S} := \frac{1}{n}V_{n} (ϵ_{1,S/n}) for all positive integers n with (n, p) = 1. Of course, if S/n = ∅ then ϵ_{S,n} = 0. In the following we will simply write ϵ_{n} for ϵ_{n,S}. For all positive integers n ≠ n′ with (n, p) = 1 = (n′, p) the equalities

ϵ_{n}^{2} = ϵ_{n}, ϵ_{n}ϵ_{n′} = 0,

hold. Moreover, if (m, p) = 1 = (n, p) then

F_{m}(ϵ_{n}) = \begin{cases}ϵ_{n/m} & if m | n, \\ 0 & if m ∤ n. \end{cases}

Since ∑_{(n,p)=1} ϵ_{n} = 1 we obtain a decomposition of rings

(1.1.2)  𝕎_{S}(R) = ∏_{n≥1,(n,p)=1} ϵ_{n}𝕎_{S}(R).

Notation 1.1.7. For a finite truncation set S we denote by S_{p} the elements in S that are p-powers, that is S_{p} = S ∩ {p^{i} | i ≥ 0}.

The map R_{(S/n)_{p}}^{S/n} ∘ F_{n} : 𝕎_{S}(R) ⟶ 𝕎_{(S/n)_{p}}(R) induces an isomorphism ϵ_{n}𝕎_{S}(R) ≅ 𝕎_{(S/n)_{p}}(R). Thus

M ↦ ⨁_{n≥1,(n,p)=1} ϵ_{n}M

defines an equivalence of categories

(1.1.3)  (𝕎_{S}(R)-modules) ⟶^{≅} ∏_{n≥1,(n,p)=1}(𝕎_{(S/n)_{p}}(R)-modules).

1.1.8. The following two lemmas are concerned with maximal ideals in 𝕎_{S}(R).

Lemma 1.1.9. Let R be a ring. Let S be a finite truncation set. For every maximal ideal 𝔪 ⊂ 𝕎_{S}(R) there exists a maximal ideal 𝔭 ⊂ R such that 𝕎_{S}(R) ⟶ 𝕎_{S}(R)/𝔪 factors through 𝕎_{S}(R_{𝔭}).

Proof. Set k = 𝕎_{S}(R)/𝔪, we distinguish two cases:

(1) k has characteristic 0,

(2) k has characteristic p > 0.

In the first case we can factor

𝕎_{S}(R) ⟶ 𝕎_{S}(R) ⊗_{ℤ} ℚ ⟶^{=} 𝕎_{S}(R ⊗_{ℤ} ℚ) ⟶ k.

Since 𝕎_{S}(R ⊗_{ℤ} ℚ) ⟶^{gh,≅} ∏_{s∈S} R ⊗ ℚ, the claim follows. Suppose now that k has characteristic p > 0. We have a factorization

𝕎_{S}(R) ⟶ 𝕎_{S}(R) ⊗_{ℤ} ℤ_{(p)} ⟶^{=} 𝕎_{S}(R ⊗_{ℤ} ℤ_{(p)}) ⟶ k.

By decomposing

𝕎_{S}(R ⊗ ℤ_{(p)}) ⟶^{=} ∏_{n≥1,(n,p)=1} ϵ_{n}𝕎_{S}(R ⊗ ℤ_{(p)})

\underrightarrow{≅,∏_{n} R_{(S/n)p}^{S/n} ∘F_{n}} ∏_{n≥1,(n,p)=1} 𝕎_{(S/n)_{p}}(R ⊗ ℤ_{(p)}),

we can reduce to the case where S consists only of p-powers. Finally, V_{p}(a)^{2} = pV_{p}(a^{2}), for all a ∈ 𝕎_{S/p}(R ⊗ ℤ_{(p)}), hence V_{p}(a) maps to zero in k. Therefore 𝕎_{S}(R ⊗ ℤ_{(p)}) ⟶ k factors through 𝕎_{S}(R ⊗ ℤ_{(p)}) ⟶ 𝕎_{{1}}(R ⊗ ℤ_{(p)}) = R ⊗ ℤ_{(p)} ⟶^{ρ} k. In this case we can take 𝔭 = ker(R ⟶ R ⊗ ℤ_{(p)} ⟶^{ρ} k). □

Lemma 1.1.10. Let p be a prime. Let R be a ring such that every maximal ideal 𝔭 satisfies char(R/𝔭) = p > 0. Let S be a p-typical finite truncation set. Then

every maximal ideal 𝔪 of 𝕎_{S}(R) is of the form ker(𝕎_{S}(R) ⟶^{R_{{1}}^{S}} R ⟶ R/𝔭), for a unique maximal ideal 𝔭 of R.

Proof. Let 𝔪 be a maximal ideal of 𝕎_{S}(R), set k = 𝕎_{S}(R)/𝔪. We claim that char(k) = p. Suppose that char(k) ≠ p. From the commutative diagram

\xymatrix{𝕎_{S}(R) \ar[r] \ar[d]^{gh} & 𝕎_{S}(R) ⊗ ℤ[p^{−1}] \ar[r] \ar[d]^{gh}_{≅} & k \\ ∏_{s∈S} R \ar[r] & ∏_{s∈S} R ⊗ ℤ[p^{−1}] \ar[ur]}

we conclude that there is a factorization 𝕎_{S}(R) ⟶^{gh_{i}} R ⟶ k, but there are no epimorphism R ⟶ k to a field of characteristic ≠ p. Thus we may suppose that char(k) = p. Because V_{p}(a)^{2} = pV_{p}(a^{2}) for all

a ∈ 𝕎_{S/p}(R), we obtain a factorization 𝕎_{S}(R) ⟶^{R_{{1}}^{S}} R ⟶ k, which defines 𝔭 := ker(R ⟶ k). □

1.2. Relative de Rham-Witt complex. For every commutative ring A we have the absolute de Rham-Witt complex

S ↦ 𝕎_{S}Ω_{A}^{∗}

constructed by Hesselholt [Hes], at our disposal. The absolute de Rham-Witt complex is the initial object in the category of Witt complexes [Hes, §4]. In this section we will define the relative version, which is studied in this paper.

Definition 1.2.1. Let A be an R-algebra. Let S be a truncation set and q ≥ 0. We define

𝕎_{S}Ω_{A/R}^{q} = \underleftarrow{lim}_{\begin{matrix}T ⊂S \\ T finite \end{matrix}} 𝕎_{T} Ω_{A}^{q}/ (𝕎_{T} Ω_{R}^{1} · 𝕎_{T} Ω_{A}^{q−1})

For q = 0, the definition means 𝕎_{S}Ω_{A/R}^{0} = 𝕎_{S}(A).

We get an induced anti-symmetric graded algebra structure on 𝕎_{S}Ω_{A/R}^{∗}, that is, ω_{1} · ω_{2} = (−1)^{deg(ω_{1}) deg(ω_{2})}ω_{2} · ω_{1}. Recall that by construction of 𝕎_{S}Ω_{A}^{∗}, there is, for all finite truncation sets S, a surjective morphism of graded 𝕎_{S}(A)-algebras

(1.2.1)  π : T_{𝕎_{S}(A)}^{∗}Ω_{𝕎_{S}(A)}^{1} ⟶ 𝕎_{S}Ω_{A}^{∗},

such that π(da) = da for all a ∈ 𝕎_{S}(A).

Lemma 1.2.2. Let S be a finite truncation set.

(1) The morphism (1.2.1) induces a surjective morphism of anti-symmetric graded algebras

(1.2.2)  π : Ω_{𝕎_{S}(A)/𝕎_{S}(R)}^{∗} ⟶ 𝕎_{S}Ω_{A/R}^{∗},

which by abuse of notation is called π again.

(2) 𝕎_{S}Ω_{A/R}^{∗} is a differential graded algebra and (1.2.2) is compatible with the differential.

Proof. For (1). This follows from π(da ⊗ da) ∈ d log[−1] · 𝕎_{S}Ω_{A}^{1} [Hes, §3] and d log[−1] ∈ 𝕎_{S}Ω_{R}^{1}. For (2). The differential d : 𝕎_{S}Ω_{A/R}^{∗} ⟶ 𝕎_{S}Ω_{A/R}^{∗} is well-defined, because 𝕎_{S}Ω_{R}^{∗} is generated by 𝕎_{S}Ω_{R}^{1}. It satisfies d∘d = 0, because d log[−1] ∈ 𝕎_{S}Ω_{R}^{1}. The compatibility of π with d follows from π(da) = da for all a ∈ 𝕎_{S}(A). □

1.2.3. Induced from the absolute de Rham-Witt complex, we obtain for all positive integers n:

(1.2.3)  F_{n} :𝕎_{S}Ω_{A/R}^{q} ⟶ 𝕎_{S/n}Ω_{A/R}^{q},
(1.2.4)  V_{n} :𝕎_{S/n}Ω_{A/R}^{q} ⟶ 𝕎_{S}Ω_{A/R}^{q},

and S ↦ 𝕎_{S}Ω_{A/R}^{∗} forms a Witt complex. Note that, computed in the absolute de Rham-Witt complex, we have

V_{n}(da · ω) = V_{n}(F_{n}dV_{n}(a) · ω) = dV_{n}(a) · V_{n}(ω),

hence V_{n}(𝕎_{S/n}Ω_{R}^{1} · 𝕎_{S/n}Ω_{A}^{q−1}) ⊂ 𝕎_{S}Ω_{R}^{1} · 𝕎_{S}Ω_{A}^{q−1}. The following equalities hold for the maps (1.2.3), (1.2.4):

V_{n}F_{n}d = dV_{n}F_{n}, dV_{n}d = 0.

Proposition 1.2.4. The Witt complex S ↦ 𝕎_{S}Ω_{A/R}^{∗} is the initial object in the category of Witt complexes over A with 𝕎(R)-linear differential.

Proof. Let S ↦ E_{S}^{∗} be a Witt complex over A with 𝕎(R)-linear differential, that is, d(aω) = ad(ω) for a ∈ 𝕎_{S}(R) and ω ∈ E_{S}^{∗}. We only need to show that the canonical morphism

[S ↦ 𝕎_{S}Ω_{A}^{∗}] ⟶ [S ↦ E_{S}^{∗}]

factors through [S ↦ 𝕎_{S}Ω_{A/R}^{∗}]. It is enough to check this for finite truncation sets. Because π (1.2.1) is surjective, we conclude that 𝕎_{S}Ω_{R}^{1} is generated by elements of the form da with a ∈ 𝕎_{S}(R), which implies the claim. □

As a corollary we obtain the following statement.

Corollary 1.2.5. Let A be an R-algebra, let p be a prime, and set R′ := R ⊗_{ℤ} ℤ_{(p)}, A′ := A ⊗_{ℤ} ℤ_{(p)}. There is a unique isomorphism

[S ↦ 𝕎_{S}Ω_{A′/R′}^{∗}] ⟶ [S ↦ \underleftarrow{lim}_{\begin{matrix}T ⊂S \\ T finite \end{matrix}} 𝕎_{T} Ω_{A/R}^{∗} ⊗_{ℤ} ℤ_{(p)}]

of Witt complexes over A′.

Proposition 1.2.6. Let R be a ℤ_{(p)}-algebra and let A, B be R-algebras. Let S be a finite truncation set.

(1) Via the equivalence from (1.1.3) we have

(1.2.5)  𝕎_{S}Ω_{A/R}^{∗} ↦ ⨁_{n≥1,(n,p)=1} 𝕎_{(S/n)_{p}}Ω_{A/R}^{∗}.

(2) For a morphism f : A ⟶ B the induced morphism f_{S} : 𝕎_{S}Ω_{A/R}^{∗} ⟶

𝕎_{S}Ω_{B/R}^{∗} maps to

f_{S} ↦ ⨁_{n≥1,(n,p)=1} f_{(S/n)_{p}}

via the equivalence from (1.1.3).

Proof. For (1). The claim follows from [HM01, Proposition 1.2.5]. In the notation of loc. cit. the right hand side (1.2.5) equals i_{!}i^{∗}𝕎Ω_{A/R}^{∗}, and i^{∗}, i_{!} preserve initial objects, since both functors admit a right adjoint. For (2). Follows immediately from the construction in (1). □

Proposition 1.2.7. Let R be a ℤ_{(p)}-algebra, let A be an R-algebra. Then

n ↦ 𝕎_{{1,p,...,p^{n−1}}}Ω_{A/R}^{∗}

is the relative de Rham-Witt complex n ↦ W_{n}Ω_{A/R}^{∗} defined by Langer and Zink [LZ04].

Proof. We have a restriction functor

i^{∗} : (Witt systems over A with 𝕎(R)-linear differential) ⟶

(F -V -procomplexes over the R-algebra A),

where we use the definition of [Hes, §4] for the source category and the definition of [LZ04, Introduction] for the target category. The functor i^{∗} admits a right adjoint functor i_{!} defined in [HM01, §1.2]. Therefore i^{∗}([S ↦ 𝕎_{S}Ω_{A/R}^{∗}]) is the initial object in the category of F -V -procomplexes as is the relative de Rham-Witt complex constructed by Langer and Zink [LZ04]. □

1.2.8. Let S be a finite truncation set. Let A ⟶ B be an étale morphism of R-algebras. For all q ≥ 0 the induced morphism of 𝕎_{S}(B)-modules

(1.2.6)  𝕎_{S}(B) ⊗_{𝕎_{S}(A)} 𝕎_{S}Ω_{A/R}^{q} ⟶^{≅} 𝕎_{S}Ω_{B/R}^{q}

is an isomorphism. Indeed, this follows immediately from the analogous fact for the absolute de Rham-Witt complex [Hes, Theorem C].

Lemma 1.2.9. Let R′ ⟶ R be an étale ring homomorphism. Let A be an R-algebra. Then, for all truncation sets S,

𝕎_{S}Ω_{A/R′}^{∗} ⟶ 𝕎_{S}Ω_{A/R}^{∗}

is an isomorphism.

Proof. We may assume that S is finite. The assertion follows from

𝕎_{S}Ω_{R′}^{1} ⊗_{𝕎_{S}(R′)} 𝕎_{S}(A) ⟶^{=} 𝕎_{S}Ω_{R′}^{1} ⊗_{𝕎_{S}(R′)} 𝕎_{S}(R) ⊗_{𝕎_{S}(R)} 𝕎_{S}(A)

⟶^{≅} 𝕎_{S}Ω_{R}^{1} ⊗_{𝕎_{S}(R)} 𝕎_{S}(A).

□

1.2.10. For every truncation set S we have a functor

𝕎_{S} : (Schemes) ⟶ (Schemes), X ↦ 𝕎_{S}(X).

This functor has been studied by Borger [Bor11b], our notation differs slightly: the notation is W^{∗} in [Bor11b]. For an affine scheme U = Spec(A), we have 𝕎_{S}(U) = Spec(𝕎_{S}(A)). If X is separated and (U_{i})_{i∈I} is an affine covering of X, then 𝕎_{S}(X) is obtained by gluing 𝕎_{S}(U_{i}) along 𝕎_{S}(U_{i} ×_{X} U_{j}). In particular, (𝕎_{S}(U_{i}))_{i∈I} is an affine covering of 𝕎_{S}(X). The functor is extended to non-separated schemes in the usual way. If T ⊂ S is an inclusion of finite truncation sets then

ı_{T,S} : 𝕎_{T} (X) ⟶ 𝕎_{S}(X)

is a closed immersion and functorial in X.

Proposition 1.2.11. Let X be an R-scheme and let S be a finite truncations set. There is a unique quasi-coherent sheaf of 𝕎_{S}(O)-modules 𝕎_{S}Ω_{X/R}^{q} for the étale topology of 𝕎_{S}(X) such that Γ(𝕎_{S}(Spec(A)), 𝕎_{S}Ω_{X/R}^{q}) = 𝕎_{S}Ω_{A/R}^{q} for every étale map Spec(A) ⟶ X with the evident restriction maps.

Proof. Let us glue a quasi-coherent sheaf 𝕎_{S}Ω_{X/R}^{q} on 𝕎_{S}(X). Indeed, suppose first that X is separated. Let (Spec(A_{i}))_{i∈I} be an affine covering and set Spec(A_{ij}) = Spec(A_{i})×_{X} Spec(A_{j}). For every i, the 𝕎_{S}(A_{i})-module 𝕎_{S}Ω_{A_{i}/R}^{q} defines a quasi-coherent sheaf 𝕎_{S}Ω_{Spec(A_{i})/R}^{q} on 𝕎_{S}(Spec(A_{i})). Since

Γ(𝕎_{S}(Spec(A_{ij})), 𝕎_{S}Ω_{Spec(A_{i})/R}^{q}) = 𝕎_{S}Ω_{A_{i}/R}^{q}⊗_{𝕎_{S}(A_{i})}𝕎_{S}(A_{ij}) = 𝕎_{S}Ω_{A_{ij}/R}^{q},

by using (1.2.6), we can glue to a quasi-coherent sheaf 𝕎_{S}Ω_{X/R} on 𝕎_{S}(X). Independence of the covering and 𝕎_{S}(ȷ)^{∗}𝕎_{S}Ω_{X/R}^{q} = 𝕎_{S}Ω_{U/R}^{q}, for every étale map ȷ : U ⟶ X, can be checked by using (1.2.6) again. □

Proposition 1.2.12. If 𝕎_{S}(X) ⟶ 𝕎_{S}(Spec(R)) is of finite type and 𝕎_{S}(X) is noetherian, then 𝕎_{S}Ω_{X/R}^{q} is coherent.

Proof. We have a surjective morphism Ω_{𝕎_{S}(X)/𝕎_{S}(R)}^{j} ⟶ 𝕎_{S}Ω_{X/R}^{j} and the assumptions imply that Ω_{𝕎_{S}(X)/𝕎_{S}(R)}^{j} is coherent. □

1.2.13. If f : X ⟶ Y is a morphism of R-schemes then we get

𝕎_{S}Ω_{Y/R}^{q} ⟶ 𝕎_{S}(f)_{∗}𝕎_{S}Ω_{X/R}^{q}.

For an inclusion of truncation sets T ⊂ S, we obtain

𝕎_{S}Ω_{X/R}^{q} ⟶ ı_{T,S∗}𝕎_{T} Ω_{X/R}^{q}.

The following diagram is commutative:

\xymatrix{𝕎_{S}Ω_{Y/R}^{q} \ar[rr] \ar[d] &  & 𝕎_{S}(f)_{∗}𝕎_{S}Ω_{X/R}^{q} \ar[d] \\ ı_{T,S∗}𝕎_{T} Ω_{Y/R}^{q} \ar[r] & ı_{T,S∗}𝕎_{T} (f)_{∗}𝕎_{T} Ω_{X/R}^{q} \ar[r]^{=} & 𝕎_{S}(f)_{∗}ı_{T,S∗}𝕎_{T} Ω_{X/R}^{q}.}

The differential, the Frobenius and the Verschiebung operations are defined in the evident way:

d : 𝕎_{S}Ω_{X/R}^{q} ⟶ 𝕎_{S}Ω_{X/R}^{q+1},
F_{n} : 𝕎_{S}Ω_{X/R}^{q} ⟶ ı_{S/n,S∗}𝕎_{S/n}Ω_{X/R}^{q},
V_{n} : ı_{S/n,S∗}𝕎_{S/n}Ω_{X/R}^{q} ⟶ 𝕎_{S}Ω_{X/R}^{q}.

Definition 1.2.14. Let X be an R-scheme, let S be a finite truncation set. We define

H_{dRW}^{i} (X/𝕎_{S}(R)) := H^{i}(𝕎_{S}(X), 𝕎_{S}Ω_{X/R}^{∗}),

where the right hand side is the hypercohomology for the Zariski topology.

1.2.15. Note that F_{n} and V_{n} are not morphisms of complexes. For all positive integers n and all finite truncation sets we set

(1.2.7)  ϕ_{n} = n^{q}F_{n} : 𝕎_{S}Ω_{X/R}^{q} ⟶ ı_{S/n,S∗}𝕎_{S/n}Ω_{X/R}^{q},

to get a morphism of complexes

𝕎_{S}Ω_{X/R}^{∗} ⟶^{ϕ_{n}} ı_{S/n,S∗}𝕎_{S/n}Ω_{X/R}^{∗}.

Suppose that X is smooth over R of relative dimension d. Then we set

β_{n} = n^{d−q}V_{n} : ı_{S/n,S∗}𝕎_{S/n}Ω_{X/R}^{q} ⟶ 𝕎_{S}Ω_{X/R}^{q}

(we will prove 𝕎_{S}Ω_{X/R}^{q} = 0 if q > d in Proposition 1.2.17(ii)). We obtain a morphism of complexes

β_{n} : ı_{S/n,S∗}𝕎_{S/n}Ω_{X/R}^{∗} ⟶ 𝕎_{S}Ω_{X/R}^{∗},

satisfying the equalities:

ϕ_{n} ∘ β_{n} = n^{d+1},

β_{n}(λ · ϕ_{n}(x)) = n^{d}V_{n}(λ) · x for all x ∈ 𝕎_{S}Ω_{X/R}^{∗} and λ ∈ ı_{S/n,S∗}𝕎_{S/n}Ω_{X/R}^{∗}.

In Section 3 we will study the {ϕ_{n}}_{n≥1} operations induced on the de Rham-Witt cohomology.

1.2.16. Note that the Hodge to de Rham spectral sequence and the quasicoherence of 𝕎_{S}Ω_{X/R}^{q} imply the following fact. Assume X is separated and 𝕎_{S}(X) is a noetherian scheme. Let (U_{i}) be an open affine covering for X, we denote by 𝔘 = (𝕎_{S}(U_{i})) the induced covering of 𝕎_{S}(X). Then we can compute H_{dRW}^{i} (X/𝕎_{S}(R)) by using the Čech complex for 𝔘:

H^{i}(C(𝔘, 𝕎_{S}Ω_{X/R}^{∗})) ⟶^{≅} H_{dRW}^{i} (X/𝕎_{S}(R)).

In the derived category we have a quasi-isomorphism:

C(𝔘, 𝕎_{S}Ω_{X/R}^{∗}) ⟶^{q-iso} RΓ(𝕎_{S}Ω_{X/R}^{∗}).

Proposition 1.2.17. Let R be a flat ℤ-algebra. Let X be a smooth R-scheme. Let S be a finite truncation set.

(i) For all non-negative integers q, 𝕎_{S}Ω_{X/R}^{q} is ℤ-torsion-free, that is, multiplication by a non-zero integer is injective.

(ii) Let d be the relative dimension of X/R. Then 𝕎_{S}Ω_{X/R}^{q} = 0 for all q > d.

Proof. For (i) it suffices to prove that 𝕎_{S}Ω_{X/R}^{q}⊗ℤ_{(p)} = 𝕎_{S}Ω_{X′/R′}^{q} is p-torsion-free for all primes p, where X′ = X ⊗_{ℤ} ℤ_{(p)} and R′ = R ⊗_{ℤ} ℤ_{(p)}. For (ii) it suffices to show that 𝕎_{S}Ω_{X′/R′}^{q} vanishes. Via the decomposition 1.2.5 we may suppose that S = {1, p, …, p^{n−1}}. Certainly we may assume that X′ = Spec(B) and that there exists an étale ring homomorphism R′[x_{1}, …, x_{d}] ⟶ B. By using (1.2.6) we are reduced to the case B = R′[x_{1}, …, x_{d}]. The claim follows in this case from the explicit description of the de Rham-Witt complex in [LZ04, §2], more precisely [LZ04, Proposition 2.17]. □

1.3. Finiteness.

Proposition 1.3.1. Let R be a flat and finitely generated ℤ-algebra. Let X be a flat and proper scheme of relative dimension d over R. Let S be a finite truncation set. The following hold.

(i) For all non-negative integers i, j the cohomology group H^{i}(𝕎_{S}(X), 𝕎_{S}Ω_{X/R}^{j}) is a finitely generated 𝕎_{S}(R)-module. (ii) For all i > d and j ≥ 0, we have H^{i}(𝕎_{S}(X), 𝕎_{S}Ω_{X/R}^{j}) = 0. (iii) For all i, the de Rham-Witt cohomology H_{dRW}^{i} (X/𝕎_{S}(R)) (Defini-

tion 1.2.14) is a finitely generated 𝕎_{S}(R)-module. (iv) Suppose X/R is smooth. Then H_{dRW}^{i} (X/𝕎_{S}(R)) = 0 for all i > 2d.

Proof. For (i). We denote by f : X ⟶ Spec(R) the structure morphism. The scheme 𝕎_{S}(X) is noetherian, because it is of finite type over Spec(ℤ). By [Bor11b, Proposition 16.13] the induced morphism 𝕎_{S}(f) : 𝕎_{S}(X) ⟶ 𝕎_{S}(R) is proper. Moreover, 𝕎_{S}Ω_{X/R}^{j} defines a coherent sheaf on 𝕎_{S}(X) (see Proposition 1.2.12).

For (ii). The fibers of 𝕎_{S}(f) at closed points of Spec(𝕎_{S}(R)) have dimension d. In fact, as topological spaces they are disjoint unions of the corresponding fibers of f. This implies the claim. For (iii). Follows from (i) via the Hodge to de Rham spectral sequence. For (iv). Again this follows from the Hodge to de Rham spectral sequence, statement (ii), and Proposition 1.2.17(ii). □

2. De Rham-Witt cohomology

2.1. Reduction modulo an ideal.

2.1.1. Recall that W_{n} = 𝕎_{{1,p,...,p^{n−1}}} whenever a prime p has been fixed (Notation 1.1.5). The goal of this section is to prove the following theorem.

Theorem 2.1.2. Let R be a flat ℤ_{(p)}-algebra, let B be a smooth R-algebra, and let n be a positive integer. Let I ⊂ R be an ideal such that p^{m} ∈ I for some m. Choose a W_{n}(R)-free resolution

T := … ⟶ T^{−2} ⟶ T^{−1} ⟶ T^{0}

of W_{n}(R/I). There exists a functorial quasi-isomorphism of complexes of W_{n}(R)-modules

(2.1.1)  W_{n}Ω_{B/R}^{∗} ⊗_{W_{n}(R)} T ⟶ W_{n}Ω_{(B/IB)/(R/I)}^{∗}.

In particular, we obtain an isomorphism

(2.1.2)  W_{n}Ω_{B/R}^{∗} ⊗_{W_{n}(R)}^{𝕃} W_{n}(R/I) ⟶^{≅} W_{n}Ω_{(B/IB)/(R/I)}^{∗},

in the derived category of W_{n}(R)-modules.

More precisely, functoriality means that for any morphism A ⟶ B of smooth R-algebras, the diagram

\xymatrix{W_{n}Ω_{B/R}^{∗} ⊗_{W_{n}(R)} T \ar[r] & W_{n}Ω_{(B/IB)/(R/I)}^{∗} \\ W_{n}Ω_{A/R}^{∗} ⊗_{W_{n}(R)} T \ar[r] \ar[u] & W_{n}Ω_{(A/IA)/(R/I)}^{∗} \ar[u]}

is commutative.

Remark 2.1.3. The proof of Theorem 2.1.2 does not go beyond the methods of [LZ04], so that the theorem may be well-known but we couldn’t provide a reference.

Proof of Theorem 2.1.2. We define the morphism (2.1.1) by

W_{n}Ω_{B/R}^{∗} ⊗_{W_{n}(R)} T ⟶ W_{n}Ω_{B/R}^{∗} ⊗_{W_{n}(R)} W_{n}(R/I) ⟶ W_{n}Ω_{(B/I)/(R/I)}^{∗},

so that the functoriality of (2.1.1) is obvious. 1.Step: The first step is the reduction to B = R[x_{1}, …, x_{d}]. We can use the Čech complex (see 1.2.16) in order to reduce to the case where there exists an étale morphism A = R[x_{1}, …, x_{d}] ⟶ B.

Note that p^{nm} = 0 in W_{n}(R/I). Since W_{n}Ω_{B/R}^{∗} is p-torsion-free (Proposition 1.2.17), we see that

(2.1.3) W_{n}Ω_{B/R}^{∗} ⊗_{W_{n}(R)}^{𝕃} W_{n}(R/I) ⟶ W_{n}Ω_{B/R}^{∗}/p^{nm} ⊗_{W_{n}(R)/p^{nm}}^{𝕃} W_{n}(R/I)

is a quasi-isomorphism. Clearly, morphism (2.1.2) factors through (2.1.3). It will be easier to work modulo p^{nm}, because dF_{p}^{nm} = p^{nm}F_{p}^{nm}d vanishes modulo p^{nm}. Set c = nm + n, we claim that

(W_{c}(B)/p^{nm} ⊗_{W_{c}(A)/p^{nm}} W_{n}Ω_{A/R}^{∗}/p^{nm}, id ⊗ d) ⟶ (W_{n}Ω_{B/R}^{∗}/p^{nm}, d) (2.1.4)
b ⊗ ω ↦ F_{p}^{nm}(b) · ω,

is an isomorphism of complexes. Note that W_{c}(A) acts on W_{n}Ω_{A/ℤ}^{∗}/p^{nm} via

W_{c}(A) ⟶^{F_{p}^{nm}} W_{n}(A), and therefore (2.1.4) is a morphism of complexes. Theorem 1.1.3 implies that

W_{c}(B) ⊗_{W_{c}(A)} M ⟶^{≅} W_{n}(B) ⊗_{W_{n}(A)} M, b ⊗ m ↦ F_{p}^{nm}(b) ⊗ m,

is an isomorphism for all W_{n}(A)-modules M. Thus the claim follows from (1.2.6). On the other hand, Corollary 1.1.4 shows that for every W_{n}(A/I)-module M the map

W_{c}(B)/p^{nm} ⊗_{W_{c}(A)/p^{nm}} M ⟶ W_{n}(B/I) ⊗_{W_{n}(A/I)} M, b ⊗ m ↦ F_{p}^{nm}(b) ⊗ m,

is an isomorphism. This yields an isomorphism of complexes

(W_{c}(B)/p^{nm} ⊗_{W_{c}(A)/p^{nm}} W_{n}Ω_{(A/IA)/(R/I)}^{∗}, id ⊗ d) ⟶ (W_{n}Ω_{(B/IB)/(R/I)}^{∗}, d).

Finally, since W_{c}(B)/p^{nm} is étale over W_{c}(A)/p^{nm}, we are reduced to proving that

W_{n}Ω_{A/R}^{∗}/p^{nm} ⊗_{W_{n}(R)/p^{nm}}^{𝕃} W_{n}(R/I) ⟶ W_{n}Ω_{(A/IA)/(R/I)}^{∗}

is a quasi-isomorphism. 2.Step: Proof of the case B = R[x_{1}, …, x_{d}]. In this case it follows from [LZ04, §2] and the proof of [LZ04, Theorem 3.5] that

Ω_{W_{n}(R)[x_{1},...,x_{d}]/W_{n}(R)}^{∗} ⟶ Ω_{W_{n}(B)/W_{n}(R)}^{∗} ⟶^{π} W_{n}Ω_{B/R}^{∗}

is a quasi-isomorphism, where the first morphism is induced by x_{i} ↦ [x_{i}]. The same statement holds for R/I, hence the assertion follows from the quasiisomorphism

Ω_{W_{n}(R)[x_{1},...,x_{d}]/W_{n}(R)}^{∗} ⊗_{W_{n}(R)}^{𝕃} W_{n}(R/I) ⟶ Ω_{W_{n}(R/I)[x_{1},...,x_{d}]/W_{n}(R/I)}^{∗}.
□

Corollary 2.1.4. Let R be a flat and finitely generated ℤ-algebra, and let 𝔪 ⊂ R be a maximal ideal; set p = char(R/𝔪). Let X be a smooth and proper R-scheme, let n, j be positive integers. There is a natural quasi-isomorphism of complexes of W_{n}(R)-modules:

RΓ(W_{n}Ω_{X/R}^{∗}) ⊗_{W_{n}(R)}^{𝕃} W_{n}(R/𝔪^{j}) ⟶ RΓ(W_{n}Ω_{X⊗_{R}R/𝔪^{j}/(R/𝔪^{j})}^{∗}).

Proof. The claim follows from Theorem 2.1.2 by using Čech complexes (see 1.2.16). □

2.2. Flatness.

Theorem 2.2.1. Let R be a smooth ℤ-algebra. Let X be a smooth and proper R-scheme. Suppose that the de Rham cohomology H_{dR}^{∗}(X/R) of X is a flat R-module. Then H_{dRW}^{∗} (X/𝕎_{S}(R)) is a finitely generated projective 𝕎_{S}(R)- module for all finite truncation sets S. Moreover, for an inclusion of finite truncation sets T ⊂ S, the induced map

(2.2.1)  H_{dRW}^{∗} (X/𝕎_{S}(R)) ⊗_{𝕎_{S}(R)} 𝕎_{T} (R) ⟶^{≅} H_{dRW}^{∗} (X/𝕎_{T} (R))

is an isomorphism.

Remark 2.2.2. Let R be a smooth ℤ-algebra. Let X be a smooth and proper Rscheme. We know that H_{dR}^{∗}(X/R) is a coherent R-module and H_{dR}^{∗}(X/R)⊗_{ℤ}ℚ is a flat R⊗_{ℤ} ℚ-module. The first assertion follows from the Hodge to de Rham spectral sequence, the second assertion follows from the existence of a Gauss – Manin connection. Therefore we can find an integer N > 0 such that the base change X ⊗_{R} R[N^{−1}] has R[N^{−1}]-flat de Rham cohomology. If all H^{i}(X, Ω_{X/R}^{j}) are flat R-modules then H_{dR}^{∗}(X/R) is a flat R-module, because in this case the Hodge to de Rham spectral sequence degenerates at E_{1}. Examples include curves or abelian varieties over R.

Since 𝕎_{S}(R) is a noetherian ring and we know that H_{dRW}^{∗} (X/𝕎_{S}(R)) is a finitely generated 𝕎_{S}(R)-module (Proposition 1.3.1), it remains to show that it is flat. This is a local property and can be checked after localization at maximal ideals of 𝕎_{S}(R). Our proof relies on Theorem 2.1.2 or, more precisely, Corollary 2.1.4.

Lemma 2.2.3. Let R be a finitely generated ℤ-algebra. Let 𝔪 be a maximal ideal of R, let n be a positive integer, and set p = char(R/𝔪). Then W_{n}(R_{𝔪}) ⟶ W_{n}(\underleftarrow{lim}_{i} R/𝔪^{i}) is faithfully flat.

Proof. By Lemma 1.1.10, both rings are local. Thus we only need to prove flatness. We note that W_{n}(R) is a noetherian ring, because R is a finitely generated ℤ-algebra. Indeed, let x_{1}, …, x_{d} be generators for R, we claim that W_{n}(R) is a finitely generated module over the subring S_{n} of W_{n}(R) generated by [x_{1}], …, [x_{d}]. By using induction on n, we only need to show that the ideal V_{p}^{n−1}(R) is a finitely generated S_{n}-module. We have

[x_{i}] · V_{p}^{n−1}(r) = V_{p}^{n−1}(x_{i}^{p^{n−1}} · r) for all r ∈ R, hence {V_{p}^{n−1}(x_{1}^{i_{1}} ⋯ x_{d}^{i_{d}}) | 0 ≤ i_{k} ≤ p^{n−1} − 1 for all k} is a set of generators. Therefore W_{n}(R_{𝔪}), being a localization of W_{n}(R), is a noetherian ring. Obviously, we have the equalities

W_{n}(\underleftarrow{lim}_{i} R/𝔪^{i}) = \underleftarrow{lim}_{i} W_{n}(R/𝔪^{i}) = \underleftarrow{lim}_{i} W_{n}(R_{𝔪})/W_{n}(𝔪^{i}R_{𝔪}).

Moreover, it is easy to check that (W_{n}(𝔪^{i}R_{𝔪}))_{i} and (W_{n}(𝔪R_{𝔪})^{i})_{i} induce the same topology on W_{n}(R_{𝔪}). Therefore

(2.2.2)  \underleftarrow{lim}_{i} W_{n}(R_{𝔪})/W_{n}(𝔪R_{𝔪})^{i} ⟶^{≅} \underleftarrow{lim}_{i} W_{n}(R_{𝔪})/W_{n}(𝔪^{i}R_{𝔪}),

which implies flatness. □

Lemma 2.2.4. Let R be a finitely generated ℤ-algebra. Let 𝔪 be a maximal ideal of R, let n be a positive integer, and set p = char(R/𝔪). Let C be a bounded complex of W_{n}(R_{𝔪})-modules such that H^{i}(C) is a finitely generated W_{n}(R_{𝔪})-module for all i. Then, for all i,

H^{i}(C) ⊗_{W_{n}(R_{𝔪})} W_{n}(\underleftarrow{lim}_{j} R/𝔪^{j}) ≅ \underleftarrow{lim}_{j} H^{i} (C ⊗_{W_{n}(R_{𝔪})}^{𝕃} W_{n}(R/𝔪^{j})) .

Proof. Set R̂ := \underleftarrow{lim}_{j} R/𝔪^{j}. The map is induced by C ⟶ C ⊗_{W_{n}(R_{𝔪})}^{𝕃}W_{n}(R/𝔪^{j})

and the W_{n}(R̂)-module structure on the right hand side. As a first step we will prove that H^{i} (C ⊗_{W_{n}(R_{𝔪})}^{𝕃} W_{n}(R/𝔪^{j})) is a finite group. Clearly, we may assume that C = C_{0} is concentrated in degree 0. Since C_{0} is finitely generated we conclude that Tor_{i}^{W_{n}(R_{𝔪})}(C_{0}, W_{n}(R/𝔪^{j})) is a finitely generated W_{n}(R/𝔪^{j})-module for all i. The ring W_{n}(R/𝔪^{j}) contains only finitely many elements, hence

H^{−i}(C ⊗_{W_{n}(R_{𝔪})}^{𝕃} W_{n}(R/𝔪^{j})) = Tor_{i}(C_{0}, W_{n}(R/𝔪^{j}))

is finite. By using Lemma 2.2.3 and the first step (all R^{1} \underleftarrow{lim} vanish) we can reduce the assertion to the case of a complex C = C_{0} that is concentrated in degree zero (hence C_{0} is finitely generated). In this case we need to show:

(a) C_{0} ⊗_{W_{n}(R_{𝔪})} W_{n}(R̂) ⟶^{=} \underleftarrow{lim}_{j}(C_{0} ⊗_{W_{n}(R_{𝔪})} W_{n}(R/𝔪^{j})),

(b) \underleftarrow{lim}_{j} Tor_{i}(C_{0}, W_{n}(R/𝔪^{j})) = 0 for all i > 0.

Claim (a) follows from (2.2.2). Claim (b) follows from (a) and the flatness of W_{n}(R_{𝔪}) ⟶ W_{n}(R̂). □

Proposition 2.2.5. Assumptions as in Corollary 2.1.4. Set X_{j} := X⊗_{R}R/𝔪^{j}, R_{j} := R/𝔪^{j}, R̂ = \underleftarrow{lim}_{j} R_{j}.

(i) For all i and n, we have a functorial isomorphism

(2.2.3)  H_{dRW}^{i} (X/W_{n}(R)) ⊗_{W_{n}(R)} W_{n}(R̂) ⟶^{≅} \underleftarrow{lim}_{j} H^{i}(X_{j}, W_{n}Ω_{X_{j}/R_{j}}^{∗}).

(ii) Suppose furthermore that the following conditions are satisfied:

(1) There exists a lifting ϕ : R̂ ⟶ R̂ of the absolute Frobenius on R/𝔪; let ρ : R̂ ⟶ W_{n}(R̂) be the induced ring homomorphism. By abuse of notation we will denote the restriction of ρ to R by ρ again.

(2) The de Rham cohomology H_{dR}^{∗}(X/R) is a locally free R-module. Then there is an isomorphism

H^{i}(X_{j}, W_{n}Ω_{X_{j}/R_{j}}^{∗}) ≅ H_{dR}^{i}(X/R) ⊗_{R,ρ} W_{n}(R_{j})

which is natural in the following sense. For all l > j we have a commutative diagram

\xymatrix{H^{i}(X_{l}, W_{n}Ω_{X_{l}/R_{l}}^{∗}) \ar[r]^{≅} \ar[d] & H_{dR}^{i}(X/R) ⊗_{R,ρ} W_{n}(R_{l}) \ar[d]^{id⊗W_{n}(R_{l}⟶R_{j})} \\ H^{i}(X_{j}, W_{n}Ω_{X_{j}/R_{j}}^{∗}) \ar[r]^{≅} & H_{dR}^{i}(X/R) ⊗_{R,ρ} W_{n}(R_{j}).}

For a morphism of R-schemes f : X ⟶ Y, where Y/R satisfies the same assumptions as X, the following diagram is commutative:

\xymatrix{H^{i}(Y_{j}, W_{n}Ω_{Y_{j}/R_{j}}^{∗}) \ar[r]^{≅} \ar[d]^{f^{∗}} & H_{dR}^{i}(Y/R) ⊗_{R,ρ} W_{n}(R_{j}) \ar[d]^{f^{∗}⊗id} \\ H^{i}(X_{j}, W_{n}Ω_{X_{j}/R_{j}}^{∗}) \ar[r]^{≅} & H_{dR}^{i}(X/R) ⊗_{R,ρ} W_{n}(R_{j}).}

Proof. For (i). Set C = RΓ(W_{n}Ω_{X/R}^{∗}) ⊗_{W_{n}(R)} W_{n}(R_{𝔪}). In view of Proposition 1.3.1, the assumptions for Lemma 2.2.4 are satisfied. Applying the lemma and using Corollary 2.1.4 implies the claim. For (ii). Consider the following cartesian squares

\xymatrix{X_{j} \ar[r] \ar[d] & X_{n,j} \ar[r] \ar[d] & X ⊗_{R} R̂ \ar[d] \\ Spec(R_{j}) \ar[r]^{_{gh_{1}}} & Spec(W_{n}(R_{j})) \ar[r]^{ρ} & Spec(R̂),}

where X_{n,j} is by definition the fibre product. Note that R̂ ⟶^{ρ} W_{n}(R̂) ⟶^{gh_{1}} R̂ is the identity, which implies that the left hand square is cartesian. By the comparison theorem [LZ04, Theorem 3.1] we have a functorial isomorphism

H^{i}(X_{j}, W_{n}Ω_{X_{j}/R_{j}}^{∗}) ≅ H_{crys}^{i}(X_{j}/W_{n}(R_{j})).

By the comparison isomorphism of crystalline cohomology with de Rham cohomology due to Berthelot-Ogus we get

H_{crys}^{i}(X_{j}/W_{n}(R_{j})) ≅ H_{dR}^{i}(X_{n,j}/W_{n}(R_{j}))
≅ H_{dR}^{i}(X/R) ⊗_{R,ρ} W_{n}(R_{j}).

For the last isomorphism we have used condition (2) on the de Rham cohomology of X. □

Proof of Theorem 2.2.1. Without loss of generality we may assume that R is integral. It suffices to show the flatness of H_{dRW}^{i} (X/𝕎_{S}(R)) when considered as a 𝕎_{S}(R)-module. This can be checked after localizing at maximal ideals. By using Lemma 1.1.9 it suffices to prove that H_{dRW}^{i} (X/𝕎_{S}(R)) ⊗_{𝕎_{S}(R)} 𝕎_{S}(R_{𝔪}) is a flat 𝕎_{S}(R_{𝔪})-module for every maximal ideal 𝔪 ⊂ R. Similarly, it is sufficient to prove (2.2.1) after tensoring with 𝕎_{T} (R_{𝔪}). Let 𝔪 ⊂ R be a maximal ideal, and set p = char(R/𝔪). By using the decomposition of 𝕎_{S}Ω_{X/R}^{∗} ⊗ ℤ_{(p)} from Proposition 1.2.6 together with (1.1.3) we may assume that S is p-typical, say S = {1, p, …, p^{n−1}}, and hence T = {1, p, …, p^{m−1}}. Since R is a smooth ℤ-algebra, there is a lifting ϕ : R̂ ⟶ R̂ of the absolute Frobenius of R/𝔪, where R̂ = \underleftarrow{lim}_{j} R/𝔪^{j}. Therefore Proposition 2.2.5 implies

H_{dRW}^{i} (X/W_{n}(R)) ⊗_{W_{n}(R)} W_{n}(R̂) ⟶^{≅} \underleftarrow{lim}_{j} H^{i}(X_{j}, W_{n}Ω_{X_{j}/R_{j}}^{∗})

⟶^{≅} H_{dR}^{i}(X/R) ⊗_{R,ρ} W_{n}(R̂),

and we can prove the flatness by using Lemma 2.2.3. Tensoring (2.2.1) with W_{m}(R̂) (recall that T = {1, p, …, p^{m−1}}) and by using Proposition 2.2.5(ii), we see that (2.2.1) ⊗ W_{m}(R̂) is induced by the identity on the de Rham cohomology. Hence it is an isomorphism by Lemma 2.2.3. □

3. Poincaré duality

3.1. A rigid ⊗-category.

Definition 3.1.1. Let R be a ℤ-torsion-free ring and Q a non-empty truncation set. We denote by C′_{Q,R} the category with objects being contravariant functors S ↦ M_{S} from finite truncation sets contained in Q to sets, together with

• a 𝕎_{S}(R)-module structure on M_{S}, for all truncation sets S ⊂ Q, such that the maps M_{S} ⟶ M_{T}, for T ⊂ S, are morphisms of 𝕎_{S}(R)- modules when M_{T} is considered as a 𝕎_{S}(R)-module via the projection π_{T} : 𝕎_{S}(R) ⟶ 𝕎_{T} (R),

• for all positive integers n and all truncation sets S ⊂ Q, maps

ϕ_{n} : M_{S} ⟶ M_{S/n},

such that – ϕ_{n} ∘ ϕ_{m} = ϕ_{nm} for all n, m,

– ϕ_{n} is a morphism of 𝕎_{S}(R)-modules when M_{S/n} is considered as a 𝕎_{S}(R) module via F_{n} : 𝕎_{S}(R) ⟶ 𝕎_{S/n}(R), – for all truncation sets T ⊂ S ⊂ Q the following diagram is com-

mutative:

\xymatrix{M_{S} \ar[r]^{ϕ_{n}} \ar[d] & M_{S/n} \ar[d] \\ M_{T} \ar[r]^{ϕ_{n}} & M_{T/n}.}

The functor S ↦ M_{S} is required to satisfy the following properties.

• For all truncation sets S ⊂ Q, the 𝕎_{S}(R)-module M_{S} is finitely generated and projective.

• For all truncation sets T ⊂ S ⊂ Q:

𝕎_{T} (R) ⊗_{𝕎_{S}(R)} M_{S} ⟶ M_{T}

is an isomorphism.

• There is a positive integer a such that there exist morphisms

(3.1.1)  β_{n} : M_{S/n} ⟶ M_{S},

for all positive integers n and all finite truncation sets S ⊂ Q, satisfying the following properties:

– β_{n} is a morphism of 𝕎_{S}(R)-modules when M_{S/n} is considered as a 𝕎_{S}(R) module via F_{n} : 𝕎_{S}(R) ⟶ 𝕎_{S/n}(R),

– β_{n}(λ · ϕ_{n}(x)) = n^{a−1}V_{n}(λ) · x, for all x ∈ M_{S}, λ ∈ 𝕎_{S/n}(R),

– ϕ_{n} ∘ β_{n} = n^{a}.

Morphisms between two objects in C′_{Q,R} are morphism of functors that are compatible with the [S ↦ 𝕎_{S}(R)]-module structure and commute with ϕ_{n} for all positive integers n. We simply write C′_{R} for C′_{ℕ_{>0},R}.

Remark 3.1.2. Note that the β_{n} are not part of the datum; we can always change β_{n} ↦ n^{b}β_{n} for a non-negative integer b.

For an inclusion of truncation sets Q ⊂ Q′, we have an evident functor

C′_{Q′,R} ⟶ C′_{Q,R}.

Proposition 3.1.3. Let M ∈ ob(C′_{Q,R}). Let S ⊂ Q be a finite truncation set. Fix a > 0 and β_{n} as in 3.1.1.

(1) For all positive integers n, m with (n, m) = 1 we have

ϕ_{n} ∘ β_{m} = β_{m} ∘ ϕ_{n},

considered as morphisms M_{S/m} ⟶ M_{S/n}.

(2) For all positive integers n, m we have

β_{n} ∘ β_{m} = β_{nm},

considered as morphisms M_{S/nm} ⟶ M_{S}.

(3) For all truncation sets T ⊂ S the following diagram is commutative:

\xymatrix{M_{S/n} \ar[r]^{β_{n}} \ar[d] & M_{S} \ar[d] \\ M_{T/n} \ar[r]^{β_{n}} & M_{T} .}

Proof. The ring 𝕎_{S}(R) is ℤ-torsion-free, because it can be considered via the ghost map as a subring of ∏_{s∈S} R, and R is ℤ-torsion-free by assumption. Since M_{S} is a flat 𝕎_{S}(R)-module, it is ℤ-torsion-free. For (1). Since image(ϕ_{m}) ⊃ m^{a}M_{S/m} it is sufficient to prove

ϕ_{n} ∘ β_{m} ∘ ϕ_{m} = β_{m} ∘ ϕ_{n} ∘ ϕ_{m}.

This follows from β_{m} ∘ ϕ_{m} = V_{m}(1)m^{a−1} and ϕ_{n} ∘ ϕ_{m} = ϕ_{m} ∘ ϕ_{n}. For (2). We may argue as in (1) by composing with ∘ϕ_{nm}.

β_{n}∘β_{m}∘ϕ_{nm}(x) = β_{n}(V_{m}(1)m^{a−1}ϕ_{n}(x)) = m^{a−1}n^{a−1}V_{nm}(1)x = β_{nm}∘ϕ_{nm}(x).

For (3). We may argue as in (1) by composing with ∘ϕ_{n}. The computation is straightforward. □

Lemma 3.1.4. Let f : M ⟶ N be a morphism in C′_{Q,R}, and choose a positive integer a and β_{M,n}, β_{N,n} as in (3.1.1). Then f_{S} ∘ β_{M,n} = β_{N,n} ∘ f_{S/n} for all S, n. In particular, the choice of the β_{n} in Definition 3.1.1 depends only on the positive integer a.

Proof. Again, we may use that M_{S} is ℤ-torsion-free. Now,

n^{a}β_{n}f(x) = β_{n}(f(n^{a}x)) = β_{n}(f(ϕ_{n}β_{n}(x)))
= β_{n}ϕ_{n}f(β_{n}(x)) = n^{a−1}V_{n}(1)f(β_{n}(x))
= f(n^{a−1}V_{n}(1)β_{n}(x)) = f(β_{n}ϕ_{n}β_{n}(x)) = n^{a}f(β_{n}(x)).
□

Proposition 3.1.5 (Tensor products). For two objects M, N in C′_{Q,R} we set

(M ⊗ N)_{S} := M_{S} ⊗_{𝕎_{S}(R)} N_{S}, ϕ_{n} := ϕ_{M,n} ⊗ ϕ_{N,n}.

Then M ⊗ N defines an object in C′_{Q,R}.

Proof. This is a straightforward calculation. We can take β_{M⊗N,n} = β_{M,n} ⊗ β_{N,n}. □

The tensor product equips C′_{Q,R} with the structure of a ⊗-category with identity object 1, where

1_{S} := 𝕎_{S}(R), ϕ_{1,n} = F_{n}.

Definition 3.1.6. (Tate objects) Let b be a non-negative integer. We define the object 1(−b) in C′_{Q,R} by

1(−b)_{S} := 𝕎_{S}(R), ϕ_{1(−b),n} = n^{b}F_{n}.

For an object M in C′_{Q,R}, M_{S} is ℤ-torsion-free, hence we get an isomorphism:

Hom_{C′_{Q,R}}(M, N) ⟶^{≅} Hom_{C′_{Q,R}}(M ⊗ 1(−b), N ⊗ 1(−b))

Definition 3.1.7. We denote by C_{Q,R} the category with objects M(b), where M is an object in C′_{Q,R} and b ∈ ℤ. As morphisms we set

Hom_{C_{Q,R}}(M(b_{1}), N(b_{2})) = Hom_{C′_{Q,R}}(M ⊗ 1(b_{1} − c), N ⊗ 1(b_{2} − c)),

where c ∈ ℤ is such that b_{1} − c, b_{2} − c ≤ 0.

For two truncation sets Q ⊂ Q′, we have an obvious functor

C_{Q′,R} ⟶ C_{Q,R}.

The category C_{Q,R} is additive and via M ↦ M(0) the category C′_{Q,R} is a full subcategory of C_{Q,R}. For M ∈ C′_{Q,R}, we have M(−b) = M ⊗ 1(−b) if b is non-negative. For an integer b, the functor

C_{Q,R} ⟶ C_{Q,R}, M(n) ↦ M(n + b)

is an equivalence and has M(n) ↦ M(n − b) as inverse functor. For M(b_{1}), N(b_{2}) in C_{Q,R} we set

M(b_{1}) ⊗ N(b_{2}) := (M ⊗ N)(b_{1} + b_{2}).

The tensor product equips C_{Q,R} with the structure of a ⊗-category with identity object 1.

3.1.8. Internal Hom. The reason for introducing the new category C_{Q,R} is the internal Hom construction. Let M, N be two objects in C′_{Q,R}, fix positive integers a_{M}, a_{N} and β_{n,M}, β_{n,N} as in (3.1.1). In a first step we are going to define an object \underline{Hom}′(M, N) in C′_{Q,R} that depends on the choice of a_{M}. We set

\underline{Hom}′(M, N)_{S} := Hom_{𝕎_{S}(R)}(M_{S}, N_{S}).

We note that

Hom_{𝕎_{S}(R)}(M_{S}, N_{S}) ⊗_{𝕎_{S}(R)} 𝕎_{T} (R) ⟶^{≅} Hom_{𝕎_{T} (R)}(M_{T} , N_{T}),

since M_{S} is finitely generated and projective. We define

ϕ_{n} : Hom_{𝕎_{S}(R)}(M_{S}, N_{S}) ⟶ Hom_{𝕎_{S/n}(R)}(M_{S/n}, N_{S/n})
ϕ_{n}(f) := ϕ_{n} ∘ f ∘ β_{n}.

This definition depends on a_{M}. It is easy to check that \underline{Hom}′(M, N) is an object in C′_{Q,R} (take β_{n}(f) := β_{n} ∘ f ∘ ϕ_{n} and a = a_{M} + a_{N}). We set

(3.1.2)  \underline{Hom}(M, N) := \underline{Hom}′(M, N)(a_{M})

as an object in C_{Q,R}. In view of Lemma 3.1.4 this definition is independent of any choices. For two objects M(b_{1}), N(b_{2}) in C_{Q,R} we set

\underline{Hom}(M(b_{1}), N(b_{2})) := \underline{Hom}(M, N)(b_{2} − b_{1}).

3.1.9. For three objects M, N, P in C_{Q,R} we have an obvious natural isomorphism

\underline{Hom}(M ⊗ N, P) = \underline{Hom}(M, \underline{Hom}(N, P)).

Proposition 3.1.10. For objects M, N in C_{Q,R} we have a natural isomorphism

Hom(1, \underline{Hom}(M, N)) ⟶ Hom(M, N).

Proof. We may assume that M, N ∈ C′_{Q,R}. Fix a_{M} and β_{M,n} as in (3.1.1). We need to show that

Hom(1(−a_{M}), \underline{Hom}′(M, N)) = Hom(M, N),

and know that

Hom(1(−a_{M}), \underline{Hom}′(M, N)) =

= {[S ↦ f_{S}] | f_{S} ⊗_{𝕎_{S}(R)} 𝕎_{T} (R) = f_{T} for T ⊂ S ⊂ Q,

ϕ_{N,n} ∘ f_{S} ∘ β_{M,n} = n^{a_{M}} f_{S/n} for all n, S ⊂ Q.}

Since ϕ_{M,n}(M_{S}) ⊃ n^{a_{M}} M_{S/n} we have

ϕ_{N,n} ∘ f_{S} ∘ β_{M,n} = n^{a_{M}} f_{S/n} ⇔ ϕ_{N,n} ∘ f_{S} ∘ β_{M,n} ∘ ϕ_{M,n} = n^{a_{M}} f_{S/n} ∘ ϕ_{M,n}

⇔ ϕ_{N,n} ∘ f_{S} ∘ n^{a_{M} −1}V_{n}(1) = n^{a_{M}} f_{S/n} ∘ ϕ_{M,n}

⇔ n^{a_{M}} ϕ_{N,n} ∘ f_{S} = n^{a_{M}} f_{S/n} ∘ ϕ_{M,n}
⇔ ϕ_{N,n} ∘ f_{S} = f_{S/n} ∘ ϕ_{M,n}.
□

For M ∈ C_{Q,R} we define the dual by

M^{∨} := \underline{Hom}(M, 1).

It equips C_{Q,R} with the structure of rigid ⊗-category. We have

M^{∨} ⊗ N = \underline{Hom}(M, N).

3.1.11. Functoriality.

Proposition 3.1.12. Let R ⟶ A be a ring homomorphism between ℤ-torsion-free rings. The assignment

[S ↦ M_{S}] ↦ [S ↦ M_{S} ⊗_{𝕎_{S}(R)} 𝕎_{S}(A)], [n ↦ ϕ_{n}] ↦ [n ↦ ϕ_{n} ⊗ F_{n}],
[S ↦ f_{S}] ↦ [S ↦ f_{S} ⊗ id_{𝕎_{S}(A)}]

defines a functor

C′_{Q,R} ⟶ C′_{Q,A}.

The functor can be extended in the obvious way to a functor C_{Q,R} ⟶ C_{Q,A}.

Proof. Straightforward. □

3.1.13. Our motivation for introducing C_{Q,R} comes from geometry.

Proposition 3.1.14. Assumptions as in Theorem 2.2.1. Let Q be a non-empty truncation set. For all i ≥ 0 the assignment

S ↦ H_{dRW}^{i} (X/𝕎_{S}(R)), n ↦ ϕ_{n},

defines an object in C′_{Q,R}.

Proof. Theorem 2.2.1 implies that these modules are projective and finitely generated. For the construction of ϕ_{n} and β_{n} see Section 1.2.15. □

Definition 3.1.15. Let X ⟶ Spec(R) be a morphism such that the assumptions of Theorem 2.2.1 are satisfied. For all i, we denote by H_{dRW}^{i} (X/𝕎(R)) the object in C_{R} that is given by S ↦ H_{dRW}^{i} (X/𝕎_{S}(R)) (Proposition 3.1.14). We call H_{dRW}^{∗} (X/𝕎(R)) the de Rham-Witt cohomology of X.

3.1.16. Let X, Y be smooth proper schemes over R such that the assumptions of Theorem 2.2.1 are satisfied for X and Y. The multiplication

RΓ(𝕎_{S}Ω_{X/R}^{∗}) × RΓ(𝕎_{S}Ω_{Y/R}^{∗}) ⟶ RΓ(𝕎_{S}Ω_{X×_{R}Y/R}^{∗})

induces a morphism in C_{R}:

(3.1.3) H_{dRW}^{i} (X/𝕎(R)) ⊗ H_{dRW}^{j} (Y/𝕎(R)) ⟶ H_{dRW}^{i+j} (X ×_{R} Y/𝕎(R)).

3.2. The tangent space functor. We have a functor of rigid ⊗-categories

T : C_{Q,R} ⟶ (finitely generated and projective R-modules)

T (M(n)) := M_{{1}}.

Proposition 3.2.1. The functor T is conservative, i.e. if T (f) is an isomorphism then f is an isomorphism.

Proof. It is sufficient to consider a morphism f : M ⟶ N in C′_{Q,R}. We need to show that f_{S} : M_{S} ⟶ N_{S} is an isomorphism provided that f_{{1}} is an isomorphism. We may choose a positive integer a and β_{M,n}, β_{N,n} as in (3.1.1). By Lemma 3.1.4 the morphism f commutes with β_{n}. Let n := max{s | s ∈ S}; by induction we know that f_{T} is an isomorphism for T = S\{n}. Set I = ker(𝕎_{S}(R) ⟶ 𝕎_{T} (R)), we know that I = {V_{n}(λ) | λ ∈ R}. It suffices to show that

(3.2.1)  IM_{S} ⟶^{f_{S}} IN_{S}

is an isomorphism. If f_{S}(V_{n}(λ)x) = 0 then n^{a−1}V_{n}(λ)f_{S}(x) = 0 and therefore β_{n}(λ·ϕ_{n}f_{S}(x)) = β_{n}(λ·f_{{1}}(ϕ_{n}(x))) vanishes. Since β_{n} is injective, we conclude λ · ϕ_{n}(x) = 0, hence

0 = β_{n}(λ · ϕ_{n}(x)) = n^{a−1}V_{n}(λ)x,

which implies V_{n}(λ)x = 0.

For the surjectivity of (3.2.1) we note that, by induction, for every y ∈ N_{S} there is x ∈ M_{S} with f_{S}(x) − y ∈ IN_{S}. Therefore it suffices to show that I^{a}N_{S} is contained in the image of f_{S}. Now,

V_{n}(λ_{1}) ⋯ V_{n}(λ_{a}) = n^{a−1}V_{n}(λ_{1} ⋯ λ_{a}).

Thus

V_{n}(λ_{1}) ⋯ V_{n}(λ_{a})y = f_{S}(β_{n}f_{{1}}^{−1}(λ_{1} ⋯ λ_{a} · ϕ_{n}(y))).
□

Corollary 3.2.2. Let X, Y be smooth proper schemes over R such that the assumptions of Theorem 2.2.1 are satisfied for X and Y. If

⨁_{i+j=n} H_{dR}^{i}(Y/R) ⊗_{R} H_{dR}^{j}(X/R) ⟶ H_{dR}^{n}(X ×_{R} Y/R)

is an isomorphism then

⨁_{i+j=n} H_{dRW}^{i} (X/𝕎(R)) ⊗ H_{dRW}^{j} (Y/𝕎(R)) ⟶ H_{dRW}^{n} (X ×_{R} Y/𝕎(R))

(see (3.1.3)) is an isomorphism in C_{R}.

Proof. This is an application of Proposition 3.2.1, because

T (H_{dRW}^{i} (−/𝕎(R))) = H_{dR}^{i}(−/R).
□

Proposition 3.2.3. The functor T is faithful.

Proof. It is sufficient to consider a morphism f : M ⟶ N in C′_{Q,R}. We need to show that f_{S} : M_{S} ⟶ N_{S} vanishes provided that f_{{1}} is zero. We may choose a positive integer a and β_{M,n}, β_{N,n} as in (3.1.1). By Lemma 3.1.4 the morphism f commutes with β_{n}. Let n := max(S); by induction we know that f_{T} = 0 for T = S\{n}, so that for all x ∈ M_{S} the image f_{S}(x) is of the form f_{S}(x) = V_{n}(λ)y. Since

0 = f_{{1}} ∘ ϕ_{n}(x) = ϕ_{n} ∘ f_{S}(x) = n · λ · ϕ_{n}(y),

we conclude λ · ϕ_{n}(y) = 0 and n^{a−1}V_{n}(λ)y = 0, hence f_{S}(x) = 0. □

3.2.4. The following proposition shows that an object in C′_{Q,R}, where R is a ℤ_{(p)}-algebra, is determined by the p-typical part, that is, on its values for truncation sets consisting of p-powers. Recall the notation S_{p} from Notation 1.1.7.

Proposition 3.2.5. Let R a ℤ-torsion-free ring. Let Q be a truncation set. Suppose p is a prime such that ℓ^{−1} ∈ R for all primes ℓ ∈ Q\{p}. Let M, N be C′_{Q,R}-modules.

(1) Via the equivalence of (1.1.3):

M_{S} ↦ ⨁_{n∈S,(n,p)=1} M_{(S/n)_{p}}.

(2) If f : M ⟶ N is a morphism in C′_{Q,R} then f_{S} ↦ ⨁_{n∈S,(n,p)=1} f_{(S/n)_{p}} via the equivalence (1.1.3).

(3) The restriction functor

C′_{Q,R} ⟶ C′_{Q_{p},R}

is an equivalence of categories.

Proof. For (1). First one proves that the projection ϵ_{1}M_{S} ⟶ M_{S_{p}} is an isomorphism (see Notation 1.1.7 for S_{p}). The second step is the isomorphism

ϕ_{n} : ϵ_{n}M_{S} ⟶ ϵ_{1}M_{S/n},

with \frac{ϵ_{n}}{n^{a}} β_{n} as inverse. Statement (2) is obvious, and (3) follows from (1) and (2). □

Proposition 3.2.6. Let R a ℤ-torsion-free ring. Let Q be a truncation set. Suppose p^{−1} ∈ R for all primes p ∈ Q. Then

T : C′_{Q,R} ⟶ (finitely generated and projective R-modules)

defines an equivalence of categories.

Proof. Straightforward. □

3.2.7. Let P be a set of primes (maybe infinite). We set ℤ_{P} := ℤ[p^{−1} | p ∈ P]. Let A be a commutative ring. We denote by Mod_{A} the category of A-modules. We define the category Mod_{A,P} to be the category with objects

((M_{p})_{p∈P} , (α_{p,ℓ})_{p,ℓ∈P}),

where M_{p} is an A⊗_{ℤ}ℤ_{P \{p}}-module, and α_{p,ℓ} : M_{ℓ}⊗_{ℤ_{P \{ℓ}}} ℤ_{P} ⟶ M_{p}⊗_{ℤ_{P \{p}}} ℤ_{P} is an isomorphism of A ⊗ ℤ_{P} -modules such that

α_{p_{1},p_{1}} = id, α_{p_{1},p_{2}} ∘ α_{p_{2},p_{3}} = α_{p_{1},p_{3}} for all p_{1}, p_{2}, p_{3} ∈ P.

The morphisms of Mod_{A,P} are defined in the evident way. If P is finite and non-empty, then the evident functor

R_{P} : Mod_{A} ⟶ Mod_{A,P}

is an equivalence of categories, because we can glue quasi-coherent sheaves. If P is infinite then this may fail to be an equivalence, but we still have the following properties, whose proof is left to the reader.

Lemma 3.2.8. Suppose P ≠ ∅.

(i) R_{P} is faithful.

(ii) For every N ∈ Mod_{A} such that N ⟶ N ⊗_{ℤ} ℤ_{P} is injective, and every M ∈ Mod_{A} the following map is an isomorphism:

Hom_{Mod_{A}}(M, N) ⟶^{≅} Hom_{Mod_{A,P}} (R_{P} (M), R_{P} (N)).

(iii) Suppose that A ⟶ A ⊗ ℤ_{P} is injective. Let M̃ = ((M̃_{p}), (α_{p,ℓ})) ∈ Mod_{A,P} be such that M̃_{p} is a finitely generated and projective A ⊗_{ℤ} ℤ_{P \{p}}-module for all p ∈ P. Then there exists a finitely generated and projective M ∈ Mod_{A} such that R_{P} (M) ≅ M̃.

For a positive integer a, we denote by C′_{Q,R,a} the full subcategory of C′_{Q,R} consisting of objects such that there exist {β_{n}}_{n} as in (3.1.1) for a.

Definition 3.2.9. Let Q be a non-empty truncation set, and let P be the set of primes of Q. We denote by LC′_{Q,R,a} the category with objects

((M_{p})_{p∈P} , (α_{p,ℓ})_{p,ℓ∈P}),

where • M_{p} ∈ ob(C′_{Q_{p},R⊗ℤ_{P \{p}},a}) for all p ∈ P,

• α_{p,ℓ} : T (M_{ℓ}) ⊗_{ℤ_{P \{ℓ}}} ℤ_{P} ⟶^{≅} T (M_{p}) ⊗_{ℤ_{P \{p}}} ℤ_{P} is an isomorphism such

that

α_{p_{1},p_{1}} = id, α_{p_{1},p_{2}} ∘ α_{p_{2},p_{3}} = α_{p_{1},p_{3}} for all p_{1}, p_{2}, p_{3} ∈ P.

The morphisms are defined in the evident way.

Broadly speaking the next proposition shows that the category C′_{Q,R,a} is glued from the local components via the functor T.

Proposition 3.2.10. Let R be a ℤ-torsion-free ring, and let a be a positive integer. For every non-empty truncation set Q the evident functor

C′_{Q,R,a} ⟶ LC′_{Q,R,a}

is an equivalence of categories.

Proof. The claim follows easily from Proposition 3.2.5, Proposition 3.2.6, and Lemma 3.2.8. □

3.3. Proof of Poincaré duality.

3.3.1. Let f : X ⟶ Spec(R) be a smooth, projective morphism of relative dimension d between noetherian schemes such that H_{dR}^{∗}(X/R) is a flat Rmodule. Suppose furthermore that Spec(R) is integral and the field of fractions of R has characteristic zero. We know that H^{0}(X, O_{X}) is a finite étale R-algebra and

H_{dR}^{0}(X/R) = H^{0}(X, O_{X}).

Since H_{dR}^{∗}(X/R) is flat, we have

H_{dR}^{i}(X/R) ⊗_{R} k(y) ⟶^{≅} H_{dR}^{i}(X_{y}/k(y)),

for every point y ∈ Spec(R), and X_{y} being the fibre of y. In particular, we obtain

(3.3.1)  H^{0}(X, O_{X}) ⊗_{R} k(y) ⟶^{≅} H^{0}(X_{y}, O_{X_{y}}).

By Grothendieck-Serre duality we see that y ↦ dim_{k(y)} H^{d}(X_{y}, ω_{X_{y}}) is a constant function, thus H^{d}(X, ω_{X/R}) is a finitely generated projective R-module and we have

H^{d}(X, ω_{X/R}) ⊗_{R} k(y) ⟶^{≅} H^{d}(X_{y}, ω_{X_{y}})

for every point y ∈ Spec(R). Since the Hodge to de Rham spectral sequence degenerates at the generic point of Spec(R), we conclude:

H^{d}(X, ω_{X/R}) ⟶^{≅} H_{dR}^{2d}(X/R).

Recall that we have a trace map

Tr : H^{d}(X, ω_{X/R}) ⟶ R;

we will also denote by Tr the induced map H_{dR}^{2d}(X/R) ⟶ R. The duality pairing

H^{0}(X, O_{X}) × H^{d}(X, ω_{X/R}) ⟶ R

induces a duality pairing

H_{dR}^{0}(X/R) × H_{dR}^{2d}(X/R) ⟶ R.

Note that the fibres of f are connected if H^{0}(X, O_{X}) = R. Moreover, the equality H^{0}(X, O_{X}) = R implies that the fibres are geometrically connected by using (3.3.1). Suppose now that H^{0}(X, O_{X}) = R, and set c_{X} := Tr^{−1}(1) ∈ H^{d}(X, ω_{X/R}) = H_{dR}^{2d}(X/R). For a generically finite R-morphism g : X ⟶ Y, where Y satisfies the same assumptions as X (in particular, Y/R is of relative dimension d), we have a pull-back map

g^{∗} : H_{dR}^{2d}(Y/R) = H^{d}(Y, ω_{Y/R}) ⟶ H^{d}(X, ω_{X/R}) = H_{dR}^{2d}(X/R)

which is dual to the trace map

g_{∗} : H^{0}(X, O_{X}) ⟶ H^{0}(Y, O_{Y}), g_{∗}(1) = deg(g),

thus g^{∗}(c_{Y}) = deg(g) · c_{X}.

Proposition 3.3.2. Let R be a smooth ℤ-algebra. Let X be a smooth projective scheme over R such that H_{dR}^{∗}(X/R) is a projective R-module. Suppose that X is connected of relative dimension d. There is an isomorphism

H_{dRW}^{2d} (X/𝕎(R)) ≅ H_{dRW}^{0} (X/𝕎(R)) ⊗ 1(−d)

and a natural morphism in C_{R}:

H_{dRW}^{2d} (X/𝕎(R)) ⟶ 1(−d)

Proof. Certainly, we may suppose that Spec(R) is integral. 1.Step: Reduction to X/R has geometrically connected fibres. Set L = H^{0}(X, O_{X}), L is a finite étale R-algebra. It suffices to show the existence of an isomorphism

(3.3.2)  H_{dRW}^{2d} (X/𝕎(L)) ⟶^{τ} 1(−d)

in C_{L} such that τ_{{1}} is the trace map. In view of

H_{dRW}^{2d} (X/𝕎(L)) = H_{dRW}^{2d} (X/𝕎(R)),

(3.3.2) yields in C_{R}:

(3.3.3)  H_{dRW}^{2d} (X/𝕎(R)) ⟶^{≅} H_{dRW}^{0} (X/𝕎(R)) ⊗ 1(−d) ⟶^{tr⊗id} 1(−d),

with tr : H_{dRW}^{0} (X/𝕎(R)) ⟶ 1 being defined by the usual trace map

H_{dRW}^{0} (X/𝕎_{S}(R)) = 𝕎_{S}(L) ⟶ 𝕎_{S}(R).

The morphism (3.3.3) is functorial because it induces the usual trace map after evaluation at {1}. Therefore we may assume R = L in the following. 2.Step: Proposition 3.2.6 implies the existence of a unique isomorphism

e : 1(−d) ⊗ ℚ ⟶^{≅} H_{dRW}^{2d} (X/𝕎(R)) ⊗ ℚ

that induces Tr^{−1} after evaluation at {1}. In other words, there is a unique system (e_{S})_{S} with e_{S} ∈ H_{dRW}^{2d} (X/𝕎_{S}(R)) ⊗ ℚ such that

(1) π_{S,T} (e_{S}) = e_{T} for all T ⊂ S, where π_{S,T} is induced by the projection

H_{dRW}^{2d} (X/𝕎_{S}(R)) ⟶ H_{dRW}^{2d} (X/𝕎_{T} (R)),
(2) ϕ_{n}(e_{S}) = n^{d} · e_{S/n} for all n, S,
(3) e_{{1}} = Tr^{−1}(1).

Our goal is to show

(3.3.4)  e_{S} ∈ H_{dRW}^{2d} (X/𝕎_{S}(R))

for every finite truncation set S. The strategy of the proof will be to show this for X = ℙ_{R}^{d} first. The next step will be to prove that (3.3.4) is local in Spec(R). Locally on Spec(R) we can find generically finite morphisms to ℙ^{d}, which can be used together with the explicit description of de Rham-Witt cohomology after completion (Proposition 2.2.5) to prove the claim. 3.Step: Suppose X = ℙ_{R}^{d}. For any finite S, we get a morphism of 𝕎_{S}(R)- schemes

g_{S} : 𝕎_{S}(ℙ_{R}^{d}) ⟶ ℙ_{𝕎_{S}(R)}^{d}

induced by \frac{x_{i}}{x_{j}} ↦ [\frac{x_{i}}{x_{j}}] on the standard affine covering. The morphisms g_{S} are compatible with the Frobenius morphisms provided that the action on ℙ_{𝕎_{S}(R)}^{d} is given by ϕ_{n}^{∗}(x_{i}) = x_{i}^{n}. We obtain

g^{∗} : H^{d}(ℙ_{𝕎_{S}(R)}^{d}, ω_{ℙ_{𝕎_{S} (R)}^{d}/𝕎_{S}(R)}) ⟶ H^{d}(𝕎_{S}(ℙ_{R}^{d}), Ω_{𝕎_{S}(ℙ_{R}^{d})/𝕎_{S}(R)}^{d})

⟶ H^{d}(𝕎_{S}(ℙ_{R}^{d}), 𝕎_{S}Ω_{ℙ_{R}^{d}/R}^{d}) ⟶ H_{dRW}^{2d} (ℙ_{R}^{d}/𝕎_{S}(R)).

Note that Tr : H^{d}(ℙ_{𝕎_{S}(R)}^{d}, ω_{ℙ_{𝕎_{S} (R)}^{d}/𝕎_{S}(R)}) ⟶^{≅} 𝕎_{S}(R) and δ_{S} := Tr^{−1}(1) satis-

fies ϕ_{n}^{∗}(δ_{S}) = n^{d} · δ_{S/n}. Therefore e_{S} = g^{∗}(δ_{S}), which proves (3.3.4) in the case of a projective space. 4.Step: We claim that in order to prove (3.3.4) it is sufficient to prove

(3.3.5)  e_{S} ∈ H_{dRW}^{2d} (X/𝕎_{S}(R)) ⊗_{𝕎_{S}(R)} 𝕎_{S}(R_{𝔪})

for every maximal ideal 𝔪. Indeed, let ℱ be the coherent sheaf on Spec(𝕎_{S}(R)) associated to M := H_{dRW}^{2d} (X/𝕎_{S}(R)). For every 𝔪, we can choose an open affine neighborhood U_{𝔪} ⊂ Spec(R) and a section e_{𝔪} ∈ ℱ (𝕎_{S}(U_{𝔪})) mapping to e_{S} ∈ M ⊗_{𝕎_{S}(R)} 𝕎_{S}(R_{𝔪}). The section e_{𝔪} is unique and the sections (e_{𝔪})_{𝔪} glue to a section of ℱ on ⋃_{𝔪} 𝕎_{S}(U_{𝔪}) = 𝕎_{S}(Spec(R)), which proves the claim. Let R̂ be the completion \underleftarrow{lim}_{j} R/𝔪^{j}. For every integer n, we have

nR̂ ∩ R_{𝔪} = ⋂_{j=1}^{∞}(nR_{𝔪} + 𝔪^{j}) = nR_{𝔪},

and thus (R_{𝔪} ⊗_{ℤ} ℚ) ∩ R̂ = R_{𝔪} as intersection in R̂ ⊗_{ℤ} ℚ. Therefore

(3.3.6)  e_{S} ∈ H_{dRW}^{2d} (X/𝕎_{S}(R)) ⊗_{𝕎_{S}(R)} 𝕎_{S}(R̂)

implies (3.3.5). 5.Step: We will show (3.3.6). We may pass from Spec(R) to a neighborhood Spec(R′) of 𝔪. Let R′ be such that there exists a generically finite R′-morphism

f : X ×_{Spec(R)} Spec(R′) ⟶ ℙ_{R′}^{d}.

The existence is proved in Proposition 3.3.3 below. Then e_{S} = \frac{1}{deg(f)}f^{∗}(e_{S}), because the classes (\frac{1}{deg(f)}f^{∗}(e_{S}))_{S} satisfy the properties listed in the second step. Set p = char(R/𝔪). To prove (3.3.6) we may assume that S = {1, p, …, p^{n−1}}. Then Proposition 2.2.5(ii) yields the claim, because for the de Rham cohomology we know that f^{∗}(Tr^{−1}(1)) is divisible by deg(f). □

Proposition 3.3.3. Let Y be of finite type over Spec(ℤ). Let X/Y be smooth projective such that X is connected of relative dimension d. For every closed point y ∈ Y there is open neighborhood U of y, and a generically finite Umorphism X ×_{Y} U ⟶ ℙ_{U}^{d}.

In order to prove Proposition 3.3.3 we will need a sequence of lemmas.

Lemma 3.3.4. Let R be a local noetherian ring. Let X/R be a smooth projective R-scheme such that every connected component of X has relative dimension d ≥ 0 over Spec(R). Let ℒ be a relative ample line bundle. There is n > 0 satisfying the following property: for every k ≥ 1 there is a section s ∈ H^{0}(X, ℒ^{⊗kn}) such that V (s) is smooth of relative dimension d − 1 over R.

Proof. Let y ∈ Spec(R) denote the closed point. For n ≫ 0 we have H^{i}(X_{y}, ℒ_{|X_{y}}^{⊗n}) = 0 for all i > 0. By semicontinuity we get H^{i}(X, ℒ^{⊗n}) = 0 for all i > 0, and

H^{0}(X, ℒ^{⊗n}) ⟶ H^{0}(X_{y}, ℒ_{|X_{y}}^{⊗n})

is surjective. Replace ℒ by a power such that this holds for all n ≥ 1. If the residue field of R is infinite then we can find a section s_{y} ∈ H^{0}(X_{y}, ℒ_{|X_{y}}^{⊗n}) such that V (s_{y}) is smooth of dimension d−1. In the case of a finite residue field we have to use [Poo04] and may have to replace ℒ by a high enough power again. Let s be a lifting of s_{y} to H^{0}(X, ℒ^{⊗n}), set H := V (s). If d = 0 then H is empty, because it has empty intersection with the special fibre. For d ≥ 1, H is flat by the local criterion for flatness, because it has transversal intersection with the special fibre. Since H ⟶ Spec(R) is flat and the special fibre is smooth, we conclude that H is smooth. By Chevalley’s theorem, H is of relative dimension d − 1. □

Remark 3.3.5. H is empty if and only if d = 0.

Lemma 3.3.6. Assumptions as in Lemma 3.3.4. There is n ≥ 1 and sections s_{0}, …, s_{d} ∈ H^{0}(X, ℒ^{⊗n}) such that

(1) ⋂_{i=0}^{d} V (s_{i}) is empty,
(2) ⋂_{i=1}^{d} V (s_{i}) is finite over R and non-empty.

Proof. Let m and s ∈ H^{0}(X, ℒ^{⊗m}) such that H = V (s) is a smooth hypersurface as in Lemma 3.3.4. Without loss of generality m = 1. For k ≫ 0, we get a surjective map

H^{0}(X, ℒ^{⊗k}) ⟶ H^{0}(H, ℒ_{|H}^{⊗k}).

By induction on d we can find s_{H,0}, …, s_{H,d−1} ∈ H^{0}(H, ℒ_{|H}^{⊗k}), for some k ≥ 1, satisfying the desired properties for H. Note that s_{H,0}^{j}, …, s_{H,d−1}^{j}, for all j ≥ 1, also satisfy the properties, hence we may suppose k ≫ 0. Choose some liftings s_{0}, s_{1}, …, s_{d−1} ∈ H^{0}(X, ℒ^{⊗k}). Then s_{0}, s_{1}, …, s_{d−1}, s^{k} satisfy the required properties. □

Proof of Proposition 3.3.3. Let ℒ be a relative ample line bundle. Apply Lemma 3.3.6 to the local ring of Y at y. The sections s_{0}, …, s_{d} extend to X ×_{Y} Spec(U) for an open affine neighborhood U of y. After possibly shrinking U we have ⋂_{i=0}^{d} V (s_{i}) = ∅ so that

X ×_{Y} Spec(U) ⟶ ℙ_{U}^{d},

defined by s_{0}, …, s_{d}, is well-defined. The second property of Lemma 3.3.6 implies that the morphism is generically finite. □

Corollary 3.3.7. Let R be a smooth ℤ-algebra. Let X ⟶ Spec(R) be a smooth projective morphism such that H_{dR}^{∗}(X/R) is a projective R-module. Suppose that X is connected of relative dimension d. If the canonical map

(3.3.7)  H_{dR}^{i}(X/R) ⟶ Hom_{R}(H_{dR}^{2d−i}(X/R), R)

is an isomorphism, then

(3.3.8)  H_{dRW}^{i} (X/𝕎(R)) ⟶^{≅} \underline{Hom}(H_{dRW}^{2d−i}(X/𝕎(R)), 1(−d)).

Proof. In view of Proposition 3.3.2 and (3.1.3) we get a morphism in C_{R}:

H_{dRW}^{i} (X/𝕎(R)) ⊗ H_{dRW}^{2d−i}(X/𝕎(R)) ⟶ H_{dRW}^{2d} (X/𝕎(R)) ⟶ 1(−d)

inducing (3.3.8). Now, T (3.3.8) = (3.3.7) proves the claim. □

Remark 3.3.8. Note that the map

H_{dR}^{i}(X/R) ⟶ Hom_{R}(H_{dR}^{2d−i}(X/R), R)

induced by the pairing

H_{dR}^{i}(X/R) ⊗_{R} H_{dR}^{2d−i}(X/R) ⟶ H_{dR}^{2d}(X/R) ⟶ R

is an isomorphism if for every closed point y ∈ Spec(R) the Hodge-to-de-Rham spectral sequence for the fibre at y,

(3.3.9)  H^{j}(X_{y}, Ω_{X_{y}/k(y)}^{i}) ⇒ H_{dR}^{i+j}(X_{y}/k(y)),

degenerates. Indeed, since the de Rham cohomology is locally free, it is also stable under base change and it suffices to show that, for every closed point y ∈ Spec(R), the Poincaré pairing for the fibre at y,

H_{dR}^{i}(X_{y}/k(y)) ⊗_{k(y)} H_{dR}^{2d−i}(X_{y}/k(y)) ⟶ H_{dR}^{2d}(X_{y}/k(y)) ⟶ k(y),

is non-degenerate. This follows easily from the degeneration of the Hodge-to-de-Rham spectral sequence and Serre duality. The degeneration of the spectral sequence (3.3.9) is known in the following cases:

• H^{j}(X, Ω_{X/R}^{i}) is torsion-free for all i, j,
• dim X_{y} ≤ char(k(y)).

As an example, we have abelian schemes or curves over R.

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Andre Chatzistamatiou Fachbereich Mathematik Universität Duisburg-Essen 45117 Essen Germany a.chatzistamatiou@uni-due.de 600

Documenta Mathematica 19 (2014)