De Rham-Witt Cohomology and Displays

Andreas Langer and Thomas Zink

Received: July 21, 2006 Revised: March 7, 2007

Communicated by Peter Schneider

Abstract. Displays were introduced to classify formal p-divisible groups over an arbitrary ring R where p is nilpotent. We define a more general notion of display and obtain an exact tensor category. In many examples the crystalline cohomology of a smooth and proper scheme X over R carries a natural display structure. It is constructed from the relative de Rham-Witt complex. For this we refine the comparison between crystalline cohomology and de Rham-Witt cohomology of [LZ]. In the case where R is reduced the display structure is related to the strong divisibility condition of Fontaine [Fo].

2000 Mathematics Subject Classification: 14F30, 14F40

Keywords and Phrases: crystalline cohomology

1 Introduction

Displays of formal p-divisible groups were introduced in [Z2]. They are one possible extension of classical Dieudonné theory to more general ground rings. In [LZ] we gave a direct construction of a display for an abelian scheme by the relative de Rham-Witt complex. In the case where the p-divisible group of the abelian scheme is local the construction leads to the display of [Z2]. We define here a more general notion of display over a ring R, where a given prime number p is nilpotent. If R is a perfect field a display is just a finitely generated free W (R)-module M endowed with an injective Frobenius linear map F : M → M, while a display of [Z2] is a Dieudonné module, where V acts topologically nilpotent. Our category of displays is an exact tensor category which contains the displays of [Z2] as a full subcategory. There is also a good notion of base change for displays with respect to arbitrary ring morphisms R → R′. Neither the construction of the tensor product nor the construction of base change is straightforward. Special types of tensor products are related in [Z2] to biextensions of formal groups. Other operations of linear algebra as exterior products and duals up to Tate twist may be performed but we don’t discuss them here, since we don’t use them essentially and their construction requires just the same ideas. We add that the exact category of displays is Karoubian [T] and has a derived category. In many examples we have a display structure on the cohomology of a projective and smooth scheme which arises as follows: Let p be a fixed prime number and let R be a ring such that p is nilpotent in R. We denote by W (R) the ring of Witt vectors and we set I_{R} = V W (R). Let X be a projective and smooth scheme over R. Let W Ω_{X/R}^{·} be the de Rham-Witt complex. We define for m ≥ 0 the Nygaard complex N^{m}W Ω_{X/R}^{·} of sheaves of W (R)-modules:

(W Ω_{X/R}^{0})_{[F]} →^{d} … →^{d} (W Ω_{X/R}^{m−1})_{[F]} →^{dV} W Ω_{X/R}^{m} →^{d} W Ω_{X/R}^{m+1} →^{d} ….

Here F indicates restriction of scalars with respect to the Frobenius F : W (R) → W (R). We remark that N^{0}W Ω_{X/R}^{·} = W Ω_{X/R}^{·}. These complexes were considered by Nygaard, Illusie and Raynaud [I-R], and Kato [K] if R is a perfect field. Let m be a nonnegative integer and consider the hypercohomology groups

P_{i} = ℍ^{m}(X, N^{i}W Ω_{X/R})

for i ≥ 0. The structure of the de Rham-Witt complex gives naturally three sets of maps (compare: Definition 2.2):

1) A chain of morphisms of W (R)-modules

… → P_{i+1} →^{ι_{i}} P_{i} → … → P_{1} →^{ι_{0}} P_{0}.

2) For each i ≥ 0 a W (R)-linear map

α_{i} : I_{R} ⊗_{W (R)} P_{i} → P_{i+1}.

3) For each i ≥ 0 a Frobenius linear map

F_{i} : P_{i} → P_{0}.

The composition of ι and α is the multiplication I_{R} ⊗ P_{i} → P_{i}. Moreover we have the equation:

F_{i+1}(α_{i}( ^{V} η ⊗ x)) = ηF_{i}x, η ∈ I_{R}, x ∈ P_{i}  (1)

We will call a set of data P = (P_{i}, ι_{i}, α_{i}, F_{i}) with the properties above a predisplay. The predisplays form an abelian category. The equation (1) implies:

F_{i}(ι_{i}(y)) = pF_{i+1}(y)

i.e. the Frobenius F_{0} becomes more and more divisible by p if it is restricted to the Nygaard complexes. We are interested in predisplays, which are obtained by the following construction. We start with a set of data which are called standard:

A sequence L_{0}, …, L_{d} of finitely generated projective W (R)-modules.

A sequence of Frobenius linear maps for i = 0, … d:

Φ_{i} : L_{i} → L_{0} ⊕ … ⊕ L_{d}

We require that the map ⊕_{i}Φ_{i} is a Frobenius linear automorphism of L_{0} ⊕…⊕ L_{d}. From these data one defines a predisplay P = (P_{i}, ι_{i}, α_{i}, F_{i}), with

P_{i} = (I_{R} ⊗ L_{0}) ⊕ … ⊕ (I_{R} ⊗ L_{i−1}) ⊕ L_{i} ⊕ … ⊕ L_{d}

for i ∈ ℤ, i ≥ 0. The definition of the maps ι_{i}, α_{i}, F_{i} (compare Definition 2.2) is not obvious, but we skip it for the moment. We should warn the reader that the P_{i} for i > d are obviously isomorphic, but these isomorphisms are not canonical, i.e. they depend on our construction and not only on the predisplay P. Definition: A predisplay is called a display if it is isomorphic to a predisplay associated to standard data. A decomposition P_{0} = L_{0} ⊕ L_{1} ⊕ … ⊕ L_{d} which is given by standard data is called a normal decomposition. If we start with standard data for d = 1 we obtain exactly the 3n-displays of [Z2], which are called displays in [Me]. In this work we call them 1-displays. If we assume that the L_{i} are free the map ⊕Φ_{i} is represented by a block matrix (A_{ij}), where A_{ij} is the matrix of the Frobenius linear map L_{j} → L_{i} induced by ⊕Φ_{i}, where 0 ≤ i, j ≤ d. Conversely any block matrix (A_{ij}) from GL(W (R)) defines standard data for a display. Over a local ring R it would be possible to define the category of displays in terms of matrices. We note that the maps ι_{i} for a display P are generally not injective unless the ring R is reduced. In this case the whole display is uniquely determined by the Frobenius module (P_{0}, F_{0}). Indeed the display property implies that:

P_{i} = {x ∈ P_{0} | F_{0}(x) ∈ p^{i}P_{0}}  (2)

One has F_{i} = (1/p^{i})F_{0}. This makes sense because p is not a zero divisor in W (R) if R is reduced. Therefore over a reduced ring a display is a special kind of Frobenius module. If R = k is a perfect field a display is just the same as a Frobenius module (P_{0}, F_{0}). Indeed, consider the map F_{0} : P_{0} ⊗ℚ → P_{0} ⊗ℚ. We obtain inclusions of W (k)-modules:

P_{0} ⊂ F_{0}^{−1}P_{0} ⊂ P_{0} ⊗ ℚ.

By the theory of elementary divisors we find a decomposition by W (R)-modules P_{0} = L_{0} ⊕ L_{1} ⊕ … ⊕ L_{d}, such that

F_{0}^{−1}P_{0} = L_{0} ⊕ p^{−1}L_{1} ⊕ … ⊕ p^{−d}L_{d}.

Therefore the restriction of p^{−i}F_{0} to L_{i} defines a map Φ_{i} : L_{i} → P_{0}, for i = 0, …, d. These are the standard data for the display associated to the Frobenius module (P_{0}, F_{0}). If pR = 0 Moonen and Wedhorn [MW] introduced the structure of an F -zip. It is defined in terms of the de Rham cohomology of the scheme X/R. As one should expect any display gives rise to an F -zip (compare the remark after Definition 2.6.). For an arbitrary projective and smooth variety X/R we can’t expect that the crystalline cohomology H_{crys}^{m}(X/W (R)) has a display structure. Therefore we consider the following assumptions: There is a compatible system of smooth liftings X̃_{n}/W_{n}(R) for n ∈ ℕ of X/R such that the following properties hold:

(*) The cohomology groups H^{j}(X̃_{n}, Ω_{X̃_{n}/W_{n}(R)}^{i}) are for each n, i and j locally free W_{n}(R)-modules of finite type.

(**) The de Rham spectral sequence degenerates at E_{1}

E_{1}^{ij} = H^{j}(X̃_{n}, Ω_{X̃_{n}/W_{n}(R)}^{i}) ⇒ ℍ^{i+j}(X̃_{n}, Ω_{X̃_{n}/W_{n}(R)}^{·}).

Theorem: Let X be smooth and projective over a reduced ring R, such that the assumptions (*) and (**) are satisfied. Let d be an integer 0 ≤ m < p. Consider the Frobenius module P_{0} = H_{crys}^{m}(X/W (R)) and define P_{i} by the formula (2). Then the P_{i} form a display and P_{i} coincides with the hypercohomology of the Nygaard complex N^{i}W Ω_{X/R}^{·}. It would follow from the general conjecture made below that this theorem holds without the restriction m < p. Finally we indicate how to proceed if the ring R is not reduced. In order to overcome the problem with the p-torsion in W (R) we use frames [Z1]. A frame for R is a triple (A, σ, α), where A is a p-adic ring without p-torsion, σ : A → A is an endomorphism which lifts the Frobenius on A/pA, and α : A → R is a surjective ring homomorphism whose kernel has divided powers. Let us assume that X admits a lifting to a smooth formal scheme Y over Spf A, which satisfies assumptions analogous to (*) and (**). We define “displays” relative to A which we call windows (see [Z1]). Theorem 5.5 says that under the conditions made H_{crys}^{m}(X/A, O_{X/A}) has a window structure for m < p . There is a morphism A → W (A) → W (R) which allows to pass from windows to displays. We remark that because of this morphism the assumptions (*) and (**) for A are stronger than the original assumption for W (R). In equal characteristic we obtain e.g. the following: Theorem Let X be smooth and projective over a ring R, such that pR = 0. Let us assume that there is a frame A → R and a smooth p-adic lifting Y/ Spf A of X, which satisfies the conditions analogous to (*) and (**). Then there is a canonical display structure on H_{crys}^{m}(X/W (R)) for m < p, which does not depend on the lifting Y nor on the frame A.

We discuss three examples where the assumptions (∗) and (∗∗) hold. In these examples the assumptions made on X in the two preceding theorems are fullfilled. Let X be a K3-surface over R. We assume without restriction of generality that R is noetherian. We denote by T_{X/R} the tangent bundle of X. The cohomology group H^{2}(X, T_{X/R}) commutes with base change by [M] §5 Cor.3. From the case where R is an algebraically closed field, we deduce that this cohomology group vanishes. It follows that X has a formal lifting over Spf W (R) resp. Spf A. From the Hodge numbers of a K3-surface over an algebraically closed field [De1] one deduces that H^{1}(X, O_{X}) = 0, H^{0}(X, Ω_{X/R}^{1}) = 0, H^{2}(X, Ω_{X/R}^{1}) = 0, H^{1}(X, Ω_{X/R}^{2}) = 0. It follows that the cohomology of X commutes with arbitrary base change and is therefore locally free [M] loc.cit.. The degeneration of the de Rham spectral sequence follows now because the Hodge numbers above are zero, because there is no room for non-zero differentials. Let X be an abelian variety over R. In this case the assumptions (∗) and (∗∗) are fullfilled by [BBM] 2.5.2. Finally let X be a smooth relative complete intersection in a projective space over R. Then the conditions (∗) and (∗∗) are fullfilled by [De2] Thm.1.5.

Let p be a prime number. Let R be a ring such that p is nilpotent in R. In [LZ] Thm. 3.5 we proved a comparison between the crystalline cohomology and the hypercohomology of the de Rham-Witt complex extending a result of Illusie [I] if R is a perfect field. We show here a filtered version of this comparison, which is the key to the display structure. We conjecture a more precise comparison, which would lead to a wide generalization of the theorems above. Let W_{n}(R) be the truncated Witt vectors. We set I_{R,n} = V W_{n−1}(R). This ideal is 0 for n = 1. Let X/R be a smooth and projective scheme. We consider the crystalline site Crys(X/W_{n}(R)) with its structure sheaf O_{X/W_{n}(R)}. Let us denote by J_{X/W_{n}(R)} ⊂ O_{X/W_{n}(R)} the sheaf of pd-ideals. We denote by J_{X/W_{n}(R)}^{[m]} its m-th divided power. Let

u_{n} : Crys(X/W_{n}(R))^{∼} ⟶ X_{zar}^{∼}

be the canonical morphism of topoi. The comparison isomorphism [LZ] is an isomorphism in the derived category D(X_{zar}) of sheaves of W_{n}(R)-modules on X_{zar}:

Ru_{n∗}O_{X/W_{n}(R)} ⟶ W_{n}Ω_{X/R}^{·}

We will prove a filtered version of this. Let m be a natural number. Let I^{m}W_{n}Ω_{X/R}^{·} be the following subcomplex of the de Rham-Witt complex:

p^{m−1}V W_{n−1}Ω_{X/R}^{0} →^{d} p^{m−2}V W_{n−1}Ω_{X/R}^{1} … →^{d} V W_{n−1}Ω_{X/R}^{m−1} →^{d} W_{n}Ω_{X/R}^{m} …

The filtered comparison Theorem 4.6 says that for m < p we have an isomorphism in the derived category

Ru_{n∗}J_{X/W_{n}(R)}^{[m]} ⟶ I^{m}W_{n}Ω_{X/R}^{·}  (3)

We would like to have a similar comparison theorem for the truncated Nygaard complex N^{m}W_{n}Ω_{X/R}^{·} instead of I^{m}W_{n}Ω_{X/R}^{·}:

(W_{n−1}Ω_{X/R}^{0})_{[F]} →^{d} … →^{d} (W_{n−1}Ω_{X/R}^{m−1})_{[F]} →^{dV} W_{n}Ω_{X/R}^{m} →^{d} W_{n}Ω_{X/R}^{m+1} →^{d} …

The advantage of the Nygaard complex is that the restriction of the Frobenius from W Ω_{X/R}^{·} to N^{m}W Ω_{X/R}^{·} is in a natural way divisible by p^{m} even if p is a zero divisor. For a reduced ring R the Nygaard complex N^{m}W Ω_{X/R}^{·} is quasiisomorphic to I^{m}W Ω_{X/R}^{·}. Unfortunately in general we don’t know a definition for the Nygaard complex in terms of crystalline cohomology. Nevertheless we make the conjecture 4.1: Conjecture: Assume that X̃/W_{n}(R) is a smooth lifting of X. Then the Nygaard complex is in the derived category canonically isomorphic to the following complex F^{m}Ω_{X̃/W_{n}(R)}^{·}:

I_{R,n} ⊗_{W_{n}(R)} Ω_{X̃/W_{n}(R)}^{0} →^{pd} … →^{pd} I_{R,n} ⊗_{W_{n}(R)} Ω_{X̃/W_{n}(R)}^{m−1} →^{d} Ω_{X̃/W_{n}(R)}^{m} →^{d} ….

Assume that we have for varying n a compatible system of smooth liftings X̃_{n}/W_{n}(R). We obtain a formal scheme X = lim_{⟶} X̃_{n}. We set:

F^{m}Ω_{X /W (R)}^{·} = lim_{⟵_{n}} F^{m}Ω_{X̃_{n}/W_{n}(R)}^{·} N^{m}W Ω_{X/R}^{·} = lim_{⟵_{n}} N^{m}W_{n}Ω_{X/R}^{·}

We show the following weak form of the conjecture (Corollary 4.7): Theorem: Assume that R is reduced and that m < p. Then there is a natural isomorphism in the derived category of W (R)-modules on X_{zar}:

N^{m}W Ω_{X/R}^{·} ≅ F^{m}Ω_{X /W (R)}^{·}

Moreover we can show in support of our conjecture, that the complexes N^{m}W_{n}Ω_{X/R}^{·} and F^{m}Ω_{X̃_{n}/W_{n}(R)}^{·} are always locally quasi-isomorphic on X_{zar}. The last theorem is closely related to strong divisibility in the sense of [Fo] 1.3: Assume the assumptions (∗) and (∗∗) are satisfied. By the last theorem the splitting of the Hodge filtration of the formal scheme X defines a normal decomposition:

ℍ^{m}(X, F^{j}Ω_{X /W (R)}^{·}) = I_{R}L_{0} ⊕ … ⊕ I_{R}L_{j−1} ⊕ L_{j} ⊕ … ⊕ L_{d}

It is obvious from Definition 2.2 that the Frobenius F_{j} : H^{m}(X, N^{j}W Ω_{X/R}^{·}) → ℍ^{m}(X, W Ω_{X/R}^{·}) is bijective if j is bigger than the dimension. Therefore F_{0} ⊕ F_{1} ⊕ … ⊕ F_{d}: induces a bijection:

I_{R}L_{0} ⊕ … ⊕ I_{R}L_{d} → L_{0} ⊕ … ⊕ L_{d}

This is what strong divisibility asserts.

2 The Category of Displays

Let R be a ring, and let W (R) be the ring of Witt vectors. We set I_{R} = V W (R). If no confusion is possible we sometimes use the abbreviation I = I_{R}. Let Φ : M → N a Frobenius-linear homomorphism of W (R)-modules. We define a Frobenius-linear homomorphism Φ̃:

\begin{matrix}Φ̃ : I_{R} ⊗_{W (R)} M & → & N \\ ^{V} ξ ⊗ m & ↦ & ξΦ(m) \end{matrix}  (4)

Definition 2.1 A predisplay over R consists of the following data:

1) A chain of morphisms of W (R)-modules

… → P_{i+1} →^{ι_{i}} P_{i} → … → P_{1} →^{ι_{0}} P_{0}.

2) For each i ≥ 0 a W (R)-linear map

α_{i} : I_{R} ⊗_{W (R)} P_{i} → P_{i+1}.

3) For each i ≥ 0 a Frobenius linear map

F_{i} : P_{i} → P_{0}.

The following axioms should be fulfilled

(D1) For i ≥ 1 the diagram below is commutative and its diagonal I_{R} ⊗ P_{i} → P_{i} is the multiplication.

I_{R} ⊗ P_{i} ⟶^{α_{i}} P_{i+1}
I_{R}⊗ι_{i−1}↓  ^{ι_{i}}↓
I_{R} ⊗ P_{i−1} ⟶^{α_{i−1}} P_{i}

For i = 0 the following map is the multiplication:

I_{R} ⊗ P_{0} ⟶^{α_{0}} P_{1} ⟶^{ι_{0}} P_{0}

(D2) F_{i+1}α_{i} = F̃_{i} : I_{R} ⊗ P_{i}. → P_{0}

We will denote a predisplay as follows:

P = (P_{i}, ι_{i}, α_{i}, F_{i}), i ∈ ℤ_{≥0}.

Let X be a smooth and proper scheme over a scheme S. Then we obtain a predisplay structure on the crystalline cohomology through the Nygaard complexes N^{m}W_{n}Ω_{X/S} which are built from the de Rham-Witt complex as follows:

(W_{n−1}Ω_{X/S}^{0})_{[F]} →^{d} … →^{d} (W_{n−1}Ω_{X/S}^{m−1})_{[F]} →^{dV} W_{n}Ω_{X/S}^{m} →^{d} W_{n}Ω_{X/S}^{m+1} …

This is considered as a complex of W_{n}(O_{S})-modules. The index [F] means that we consider this term as a W_{n}(O_{S})-module via restriction of scalars F : W_{n}(O_{S}) → W_{n−1}(O_{S}). Let I_{S,n} = V W_{n−1}(O_{S}) ⊂ W_{n}(O_{S}) be the sheaf of ideals. We define three sets of maps:

\begin{aligned}α̂_{m} : & I_{S,n} ⊗_{W_{n}(O_{S})} N^{m}W_{n}Ω_{X/S}^{·} & → & N^{m+1}W_{n}Ω_{X/S}^{·} \\ ι̂_{m} : & N^{m+1}W_{n}Ω_{X/S}^{·} & → & N^{m}W_{n}Ω_{X/S}^{·} \\ F̂_{m} : & N^{m}W_{n}Ω_{X/S}^{·} & → & W_{n−1}Ω_{X/S}^{·} \end{aligned}  (5)

These maps are given in this order by the maps between the following vertically written procomplexes (the index n is omitted):

I_{S} ⊗ (W Ω_{X/S}^{0})_{[F]} ⟶ (W Ω_{X/S}^{0})_{[F]} ⟶^{p} (W Ω_{X/S}^{0})_{[F]} ⟶^{id} W Ω_{X/S}^{0}
I_{S}⊗d↓  ^{d}↓  ^{d}↓  ^{d}↓
…  …  …  …
I_{S} ⊗ (W Ω_{X/S}^{m−1})_{[F]} ⟶ (W Ω_{X/S}^{m−1})_{[F]} ⟶^{p} (W Ω_{X/S}^{m−1})_{[F]} ⟶^{id} W Ω_{X/S}^{m−1}
id ⊗dV ↓  ^{d}↓  ^{dV} ↓  ^{d}↓
I_{S} ⊗ (W Ω_{X/S}^{m}) ⟶^{F̃} (W Ω_{X/S}^{m})_{[F]} ⟶^{V} W Ω_{X/S}^{m}  ⟶^{F} W Ω_{X/S}^{m}
id ⊗d↓  ^{dV} ↓  ^{d}↓  ^{d}↓
I_{S} ⊗ W Ω_{X/S}^{m+1}  ⟶^{mult} W Ω_{X/S}^{m+1}  ⟶^{id} W Ω_{X/S}^{m+1}  ⟶^{pF} W Ω_{X/S}^{m+1}
id ⊗d↓  ^{d}↓  ^{d}↓  ^{d}↓
I_{S} ⊗ W Ω_{X/S}^{m+2}  ⟶^{mult} W Ω_{X/S}^{m+2}  ⟶^{id} W Ω_{X/S}^{m+2}  ⟶^{p^{2}F} W Ω_{X/S}^{m+2}
…  …  …  …

The first unlabeled arrows on the left hand side denote the maps ^{V} ξ⊗ω ↦ ξω, where the product is taken in W Ω_{X/S}^{i} (without restriction of scalars).

Definition 2.2 Let S = Spec R be an affine scheme. Let X/S be a smooth and proper scheme. Then we associate a predisplay. We set:

P_{i} = ℍ^{d}(X, N^{i}W Ω_{X/S})

The predisplay structure on the P_{i} is easily obtained by taking the cohomology of the maps (5).

Here we write N^{m}W Ω_{X/R}^{·} = lim_{⟵_{n}} N^{m}W_{n}Ω_{X/R}^{·}. The P_{i} coincide with the coho-

mology of R lim_{⟵_{n}} RΓ(X, N^{i}W_{n}Ω_{X/S}) by the proof of [LZ] Prop. 1.13 (compare

[BO] Appendix).

Remark: Let S = Spec k be the spectrum of a perfect field. Then I(k) is isomorphic to W (k) as W (k)-module. The maps of complexes which define α̂_{i} and ι̂_{i} are in this case the maps F̃ and Ṽ used by Kato in his definition of the F -gauges GH^{d}(X/S).

Let A/S be an abelian scheme. Then the predisplay structure on the crystalline cohomology H^{1}(A/W (R), O_{A/W (R)}) is in fact a 3n-display structure in the sense of [Z2]. We will introduce additional properties of predisplay structures which arise in the crystalline cohomology of smooth and proper varieties. Let P be a predisplay. Then we have a commutative diagram:

\begin{matrix}P_{i} & ⟶^{F_{i}} & P_{0} \\ ^{ι_{i}}↑ & & ^{p}↑ \\ P_{i+1} & ⟶^{F_{i+1}} & P_{0} \end{matrix}  (6)

Indeed, let y ∈ P_{i+1}. Then we obtain from (D1) that

α_{i}( ^{V} 1 ⊗ ι_{i}(y)) = ^{V} 1y

If we apply F_{i+1} to the last equation and use (D2), we obtain:

F_{i}(ι_{i}(y)) = pF_{i+1}(y)

Definition 2.3 A predisplay P = (P_{i}, ι_{i}, α_{i}, F_{i}) is called separated if the map of P_{i+1} to the fibre product induced by the commutative diagram (6) is injective.

Remark: Predisplays form obviously an abelian category. To each predisplay P we can associate a separated predisplay P^{sep} and a canonical surjection P → P^{sep}. This is defined inductively: P_{0}^{sep} = P_{0} and P_{i+1}^{sep} is the image of P_{i+1} in the fibre product of:

P_{i}^{sep} ⟶^{F_{i}^{sep}} P^{0} ⟵^{p} P^{0}

The functor P ↦ P^{sep} to the category of separated displays is left adjoint to the forgetful functor, but it is not exact.

It is not difficult to prove that a separated predisplay has the following property: Consider the iteration of the maps α:

I^{⊗k} ⊗ P_{i} ⟶^{α_{i}} I^{⊗k−1} ⊗ P_{i+1} ⟶^{α_{i+1}} … ⟶^{α_{i+k−1}} P_{i+k}  (7)

Here the maps α pick up the last factor of I^{⊗}. The following map is called the “Verjüngung”:

\begin{aligned}ν^{(k)} : & I^{⊗k} & → & I \\ & ^{V} ξ_{1} ⊗ … ⊗ ^{V} ξ_{k} & ↦ & ^{V} (ξ_{1} · . . . · ξ_{k}) \end{aligned}  (8)

For a separated display the iteration (7) factors through the Verjüngung:

I^{⊗k} ⊗ P_{i} ⟶^{ν^{(k)}} I ⊗ P_{i} ⟶ P_{i+k}

The last arrow will be called α_{i}^{(k)}. In particular this shows that the iteration (7) is independent of the factors we picked up, when forming α_{j}. For a separated display the data α_{i}, i ≥ 0 are uniquely determined by the remaining data. This is seen by the following commutative diagram:

[figure: I ⊗ P_{i}; P_{i+1}; P_{i}; P_{0}; P_{0}; F_{i}; ι_{i}; F̃_{i}; p; F_{i+1}; α_{i}; (∗)]

For a predisplay P the cokernel E_{i+1} := Coker α_{i} is annihilated by I. It is therefore an R-module.

Definition 2.4 We say that a predisplay is of degree d (or a d-predisplay), if the maps α_{i} are surjective for i ≥ d.

A separated predisplay of degree d is already uniquely determined by the data:

P_{0}, … P_{d}, ι_{0}, … ι_{d−1}, F_{0}, …, F_{d}, α_{0}, …, α_{d−1}  (9)

For this consider the diagram (∗) above for i = d. The data already given determine a map of I ⊗ P_{d} to the fibre product. This map is α_{d} and the image is P_{d+1}. Thus inductively all data of the display are uniquely determined. Conversely assume that we have data (9) which satisfy all predisplay axioms reasonable for these data. Then we define P_{d+1} by the diagram (∗) above. We obtain also the maps α_{d}, ι_{d}, and F_{d+1}. The axioms for the extended data are trivially satisfied, except for the requirement that

I ⊗ P_{d+1} → I ⊗ P_{d} → P_{d+1}

is the multiplication. But this follows easily by composing the diagram (∗) for i = d, with the arrow id ⊗ι_{d} : I ⊗ P_{i+1} → I ⊗ P_{i}. Inductively we see that a set of data (9) satisfying the predisplay axioms may be extended uniquely to a predisplay of degree d. We may define the twist of a predisplay. Let

P = (P_{i}, ι_{i}, α_{i}, F_{i})

be a predisplay. Then we define its Tate-twist

P(1) = (P′_{i}, ι′_{i}, α′_{i}, F′_{i})  (10)

as follows: For i ≥ 1 we set P′_{i} = P_{i−1}, ι′_{i} = ι_{i−1}, α′_{i} = α_{i−1}, F′_{i} = F_{i−1}. We set P′_{0} = P_{0} = P′_{1}, F′_{0} = pF_{0}, ι′_{0} = id_{P_{0}}. Finally α′_{0} : I ⊗ P_{0} → P_{0} is defined to be the multiplication. If we repeat this n-times we write P(n). We define a predisplay U = (P_{i}, ι_{i}, α_{i}, F_{i}) called the unit display as follows: P_{0} = W (R), P_{i} = I for i ≥ 1. The chain of the maps ι is as follows:

… I →^{p} I … →^{p} I → W (R),  (11)

where the last map ι_{0} is the natural inclusion. The maps F_{i} : I = P_{i} → W (R) for i ≥ 1 coincide with the map

V^{−1} : I → W (R), ^{V} ξ ↦ ξ.

The map F_{0} is the Frobenius on W (R). The map α_{0} : I ⊗ W (R) → I is the multiplication. The maps α_{i} : I ⊗ I → I are the Verjüngung ν^{(2)}. Since the “Verjüngung” is surjective the unit display has degree zero. A 3n-display (P, Q, F, V^{−1}) as defined in [Z2] gives naturally rise to data of type (9) with P_{0} = P, P_{1} = Q, F_{0} = F, F_{1} = V^{−1} and therefore extends naturally to a predisplay of degree 1 as we explained above. We will make this explicit later on. Let R be a reduced ring. Then the multiplication by p is injective on W (R). Let M be a projective W (R)-module, and F : M → M be a Frobenius linear map. Then we set:

P_{i} = {x ∈ M | F (x) ∈ p^{i}M}

We obtain maps

F_{i} = (1/p^{i})F : P_{i} → P_{0} = M

For ι_{i} we take the natural inclusion P_{i+1} → P_{i}. For α_{i} we take the maps I ⊗ P_{i} → IP_{i} ⊂ P_{i+1} induced by multiplication. The data (P_{i}, ι_{i}, α_{i}, F_{i}) constructed in this way are a separated predisplay. The predisplays we are interested in arise from a construction which we explain now.

Definition 2.5 The following set of data we will call standard data for a display of degree d.

A sequence L_{0}, …, L_{d} of finitely generated projective W (R)-modules.

A sequence of Frobenius linear maps for i = 0, … d:

Φ_{i} : L_{i} → L_{0} ⊕ … ⊕ L_{d}

We require that the map ⊕_{i}Φ_{i} is a Frobenius linear automorphism of L_{0} ⊕…⊕ L_{d}

From these data we obtain a predisplay in the following manner: We set:

P_{i} = (I ⊗ L_{0}) ⊕ … ⊕ (I ⊗ L_{i−1}) ⊕ L_{i} ⊕ … ⊕ L_{d}

for i ∈ ℤ, i ≥ 0.

We note that P_{i} = P_{d+1} for i > d. But these identifications are not part of the predisplay structure we are going to define. They depend on the standard data!

We define “divided” Frobenius maps:

F_{i} : P_{i} → P_{0}

The restriction of F_{i} to I ⊗ L_{k} for k < i is Φ̃_{k}, and to L_{i+j} for j ≥ 0 is p^{j}Φ_{i+j}.

The map ι_{i} : P_{i+1} → P_{i} is given by the following diagram:

\begin{matrix}(I ⊗ L_{0}) & ⊕ … ⊕ & (I ⊗ L_{i−1}) & ⊕ & (I ⊗ L_{i}) & ⊕ & L_{i+1} & ⊕ … ⊕ & L_{d} \\ ^{p}↓ & ^{p}↓ & ^{mult}↓ & ^{id}↓ & ^{id}↓ \\ (I ⊗ L_{0}) & ⊕ … ⊕ & (I ⊗ L_{i−1}) & ⊕ & L_{i} & ⊕ & L_{i+1} & ⊕ … ⊕ & L_{d} \end{matrix}  (12)

The map α_{i} : I ⊗ P_{i} → P_{i+1} is given by the following diagram:

\begin{matrix}I ⊗ (I ⊗ L_{0}) & ⊕ … ⊕ & I ⊗ (I ⊗ L_{i−1}) & ⊕ & I ⊗ L_{i} & ⊕ & I ⊗ L_{i+1} & ⊕ … ⊕ & I ⊗ L_{d} \\ ^{ν}↓ & ^{ν}↓ & ^{id}↓ & ^{mult}↓ & ^{mult}↓ \\ (I ⊗ L_{0}) & ⊕ … ⊕ & (I ⊗ L_{i−1}) & ⊕ & (I ⊗ L_{i}) & ⊕ & L_{i+1} & ⊕ … ⊕ & L_{d} \end{matrix} (13)

Here ν = ν^{(2)} is the Verjüngung. We leave the verification that P = (P_{i}, ι_{i}, α_{i}, F_{i}) is a separated predisplay to the reader.

Definition 2.6 A predisplay is called a display if it is isomorphic to a predisplay associated to standard data.

Remark: Let us assume that pR = 0. There is the notion of an F -zip by Moonen and Wedhorn. The relation to displays is as follows. Let P = (P_{i}, ι_{i}, α_{i}, F_{i}) be a display over R. We define an F -zip structure on M = P_{0}/I_{R}P_{0} by the following two filtrations. Let C^{i} as the image of P_{i} in P_{0}/I_{R}P_{0} given by the composite of the maps ι_{k}. This gives the decreasing “Hodge filtration”:

… ⊂ C^{d} ⊂ C^{d−1} ⊂ … ⊂ C^{1} ⊂ C^{0} = M.

We set D_{i} = W (R)F_{i}P_{i} +I_{R}P_{0}/I_{R}P_{0} and obtain an increasing filtration, called the “conjugate filtration”:

0 = D_{−1} ⊂ D_{0} ⊂ D_{1} ⊂ D_{2} ⊂ … ⊂ D_{d} ⊂ … ⊂ M.

The morphisms F_{i} for i ≥ 0 induce Frobenius linear morphisms:

F_{i} : C_{i}/C_{i+1} → D_{i}/D_{i−1}  (14)

These are Frobenius linear isomorphisms of R-modules. Indeed, if we choose a normal decomposition {L_{i}} we obtain identification:

C^{i}/C^{i+1} ≅ L_{i}/I_{R}L_{i} and D_{i}/D_{i−1} ≅ W (R)F_{i}L_{i}/I_{R}W (R)F_{i}L_{i}

The two filtrations C^{·} and D_{·} together with the operators (14) form an F -zip [MW] Def. 1.5. Let P be the display associated to the standard data (L_{i}, Φ_{i}) as above. Let Q = (Q_{i}, ι_{i}, α_{i}, F_{i}) be a predisplay. Assume we are given homomorphisms ρ_{i} : L_{i} → Q_{i}. Then we define maps τ_{i}:

P_{i} = (I ⊗ L_{0}) ⊕ … ⊕ (I ⊗ L_{i−1}) ⊕ L_{i} ⊕ … ⊕ L_{d} ⟶ Q_{i}

On the summand (I ⊗ L_{i−k}) the map τ_{i} is the composite:

I ⊗ L_{i−k} ⟶^{id ⊗ρ_{i−k}} I ⊗ Q_{i−k} ⟶^{α^{(k)}} Q_{k}

On the summand L_{i+j} the map τ_{i} is the composite:

L_{i+j} ⟶^{ρ_{i+j}} Q_{i+j} ⟶^{ι^{(j)}} Q_{i},

where the last arrow is a compositions of ι′s.

Proposition 2.7 The maps τ_{i} define a homomorphism of predisplays P → Q if and only if the following diagrams are commutative:

L_{i} ⟶^{ρ_{i}} Q_{i}
Φ_{i}↓  ^{F_{i}}↓
P_{0} ⟶^{τ_{0}} Q_{0}

We omit the verification. If P = (P, Q, F, V^{−1}) is 3n-display in the sense of [Z2], then any normal decomposition P = L_{0} ⊕ L_{1}, Q = IL_{0} ⊕ L_{1} defines standard data, which determine this display. We will now define the tensor product of displays: Assume that P = (P_{i}, ι_{i}, α_{i}, F_{i}) and P′ = (P′_{i}, ι′_{i}, α′_{i}, F′_{i}) are displays over R.

A tensor product T = (T_{i}, ι_{i}^{∘}, α_{i}^{∘}, F_{i}^{∘}) may be constructed as follows. We choose normal decompositions

P_{0} = ⊕_{n≥0}L_{n}, P′_{0} = ⊕_{n≥0}L′_{n}.

More precisely this means, that we fix isomorphisms of P resp. P′ with standard displays. We obtain:

P_{i} = I ⊗ L_{0} ⊕ ⋯ ⊕ I ⊗ L_{i−1} ⊕ L_{i} ⊕ …

We denote the restriction of F_{i} : P_{i} ⟶ P_{0} to the direct summand L_{i} by Φ_{i}.

We obtain data for a standard display K_{l}, Φ^{∘}_{l}, l ≥ 0, if we set

K_{l} = ⊕_{n+m=l}(L_{n} ⊗ L′_{m}).

Then ⊕_{l}K_{l} = P_{0} ⊗ P′_{0}, and we define Frobenius linear maps

Φ_{l}^{∘} : K_{l} ⟶ P_{0} ⊗ P′_{0},

by

Φ_{l}^{∘} = ∑_{n+m=l} Φ_{n} ⊗ Φ′_{m}

From the standard data K_{l}, Φ_{l}^{∘} we obtain a display

T = (T_{i}, ι^{∘}_{i}, α_{i}^{∘}, F_{i}^{∘})  (15)

We will show that T is up to canonical isomorphism independent of the normal decompositions of P resp. P′. In order to do this we define bilinear forms of displays. Let T be an arbitrary predisplay. A bilinear form

λ : P × P′ ⟶ T .

consists of the following data. λ is a sequence of maps of W (R)-modules

λ_{ij} : P_{i} ⊗ P′_{j} ⟶ T_{i+j}.

De Rham-Witt Cohomology and Displays 161

We require that the following diagrams are commutative:

P_{i} ⊗ P′_{j} ⟶ T_{i+j}
F_{i}⊗F′_{j}↓ ↓F^{∘}_{i+j}
P_{0} ⊗ P′_{0} ⟶_{id} T_{0}

\begin{matrix}P_{i} ⊗ P′_{j} & ⟶ & T_{i+j} \\ ^{ι⊗id}↑ & & ι^{∘}↑ \\ P_{i+1} ⊗ P′_{j} & ⟶ & T_{i+j+1} \end{matrix}  \begin{matrix}P_{i} ⊗ P′_{j} & ⟶ & T_{i+j} \\ ^{id ⊗ι′}↑ & & ↑ι^{∘} \\ P_{i} ⊗ P′_{j+1} & ⟶ & T_{i+j+1} \end{matrix}

\begin{matrix}I_{R} ⊗ P_{i} ⊗ P′_{j} & ⟶ & I_{R} ⊗ T_{i+j} \\ ^{α_{i}⊗id}↓ & & ↓α^{∘}_{i+j} \\ P_{i+1} ⊗ P′_{j} & ⟶ & T_{i+j+1} \end{matrix}  \begin{matrix}I_{R} ⊗ P_{i} ⊗ P′_{j} & ⟶ & I ⊗ T_{i+j} \\ ^{id ⊗α′_{j}} ↓ & & ↓α^{∘}_{i+j} \\ P_{i} ⊗ P′_{j+1} & ⟶ & T_{i+j+1}. \end{matrix}

Remark: We will consider also the maps

P_{i} ⊗ P_{j} ⟶ T_{k}, for i + j ≥ k,

which are the compositions of λ_{ij} and T_{i+j} ⟶ T_{k}, the iteration of ι. If i + j > k we obtain a commutative diagram:

\begin{aligned}& P_{i−1} ⊗ P_{j} & ⟶ & T_{k} \\ ι ⊗ id & ↑ & & ↑ \\ & P_{i} ⊗ P_{j} & ⟶ & T_{k+1}. \end{aligned}  (16)

We will denote the set of bilinear forms of displays in this sense by

Bil(P × P′, T ).

We return to the display T given by the standard data K_{l}, Φ_{l}^{∘}. We will now define maps λ_{ij} : P_{i} ⊗ P′_{j} ⟶ T_{i+j}. For this we write P_{i} ⊗ P′_{j} according to the normal decompositions:

Documenta Mathematica 12 (2007) 147–191

\begin{matrix}P_{i} ⊗ P′_{j} & = ( ⨁_{\begin{matrix}n<i \\ m<j \end{matrix}} I ⊗ I ⊗ L_{n} ⊗ L′_{m}) ⊕ ( ⨁_{\begin{matrix}n<i \\ m≥j \\ m+n<i+j \end{matrix}} I ⊗ L_{n} ⊗ L′_{m}) \\ ⊕ & ( ⨁_{\begin{matrix}n≥i \\ m<j \\ n+m<i+j \end{matrix}} (I ⊗ L_{n} ⊗ L′_{m})) ⊕ ( ⨁_{\begin{matrix}n<i \\ m≥j \\ n+m≥i+j \end{matrix}} I ⊗ L_{n} ⊗ L′_{m}) \\ ⊕ & ( ⨁_{\begin{matrix}m≥i \\ m≤j \\ n+m≥i+j \end{matrix}} (I ⊗ L_{n} ⊗ L′_{m}) ⊕ ( ⨁_{\begin{matrix}n≥i \\ m≥j \end{matrix}} L_{n} ⊗ L′_{m}). \end{matrix} (17)

We have six direct sums in brackets, which we denote by Z_{i}, i = 1, …, 6 in the order as above. By definition T_{i+j} has the decomposition

T_{i+j} = ( ⨁_{n+m<i+j} I ⊗ L_{n} ⊗ L′_{m}) ⊕ ( ⨁_{n+m≥i+j} L_{n} ⊗ L′_{m}).  (18)

We define λ_{ij} : P_{i} ⊗ P′_{j} ⟶ T_{i+j} as a bigraded map with respect to n, m ≥ 0, which is on the homogeneous components as follows. Case Z_{1}: n < i, m < j

\begin{matrix}I ⊗ I ⊗ L_{n} ⊗ L′_{m} & ⟶ I ⊗ L_{n} ⊗ L′_{m} \\ ^{V} ξ ⊗ ^{V} η ⊗ l_{n} ⊗ l′_{m} & ⟼ ^{V} (ξη) ⊗ l_{n} ⊗ l′_{m} \end{matrix}

Case Z_{2}: n < i, m ≥ j, n + m < i + j

p^{m−j} id : I ⊗ L_{n} ⊗ L′_{m} ⟶ I ⊗ L_{n} ⊗ L′_{m}

Case Z_{3}: n ≥ i, m < j, n + m < i + j

p^{n−i} id : I ⊗ L_{n} ⊗ L′_{m} ⟶ I ⊗ L_{n} ⊗ L′_{m}

Case Z_{4}: n < i, m ≥ j, n + m ≥ i + j

p^{i−n−1} id : I ⊗ L_{n} ⊗ L′_{m} ⟶ I ⊗ L_{n} ⊗ L′_{m}

Case Z_{5}: n ≥ i, m < j, n + m ≥ i + j

p^{j−m−1} id : I ⊗ L_{n} ⊗ L′_{m} ⟶ I ⊗ L_{n} ⊗ L′_{m}

Case Z_{6}: n ≥ i, m ≥ j

id : L_{n} ⊗ L′_{m} ⟶ L_{n} ⊗ L′_{m}.

Documenta Mathematica 12 (2007) 147–191

Proposition 2.8 The homomorphism λ_{ij} : P_{i}⊗P′_{j} ⟶ T_{i+j} defined by Z_{1}−Z_{6} above define a bilinear form of displays.

Proof: We omit the tedious but simple verification.

Lemma 2.9 The homomorphism

⊕_{i+j=k}P_{i} ⊗ P′_{j} ⟶ T_{k}

given by the sum of λ_{ij} is surjective.

Proof: We have to show that all summand of (18) are in the image. Consider the submodule L_{n} ⊗ L′_{m} ⊂ T_{k} where n + m ≥ k. We set i = n and j = k − i = k − n ≤ m. By Z_{6} this submodule is in the image of P_{i} ⊗ P′_{j} ⟶ T_{k}. Next we consider a submodule I ⊗ L_{n} ⊗ L′_{m} ⊂ T_{k}, where n + m < k. We set i = n and j = k − i = k − n > m. Thus we are in the case Z_{3} with factor p^{n−i} = 1. Again the submodule is in the image of P_{i} ⊗ P′_{j} ⟶ T_{k}. Q.E.D.

Proposition 2.10 Let P and P′ be displays. Let T = (T_{i}, ι^{∘}_{i}, α_{i}^{∘}, F_{i}^{∘}) be the display (15). Let Q be a separated predisplay. There is a canonical isomorphism of abelian groups

Bil(P × P′, Q) ≅ Hom(T , Q).

Proof: Assume that we are given a bilinear form. We set T = P ⊗ P′. The maps T_{i} ⟶ Q_{i} are constructed inductively. For i = 0 this map is λ_{00}, where λ denotes the bilinear form. For the induction step to i + 1 we consider the diagram

\begin{matrix}T_{i} & ⟶ & Q_{i} & ⟶^{F_{i}} & Q_{0} \\ ↑ & & & & ^{p}↑ \\ T_{i+1} & ⟶^{F_{i+1}} & T_{0} & ⟶ & Q_{0} \end{matrix}  (19)

We claim that (19) is commutative. By Lemma 2.9 it suffices to show the commutativity if we compose the diagram with the maps P_{s} ⊗ P′_{r} ⟶ T_{i+1}, for s + r = i + 1. This amounts to the commutativity of the following diagram

[figure: P_{r} ⊗ P′_{s}; Q_{i+1}; Q_{i}; Q_{0}; Q_{0}; F_{i}; ι; F_{r} ⊗ F_{s}; p]

But the diagram is commutative by the definition of a bilinear form. Now the commutativity of (19) gives a map: T_{i+1} ⟶ Q_{i}×_{F_{i},Q_{0},p}Q_{0}. It is clear from the diagram above and Lemma 2.9 that this map factors through Q_{i+1}. Q.E.D.

Corollary 2.11 The display (15)

T = (T_{i}, ι^{∘}_{i}, α^{∘}_{i}, F^{∘}_{i})

does not depend up to canonical isomorphism on the normal decompositions of P and P′. We write

T = P ⊗ P′

This is clear because of the universal property of T proved in the last proposition. Q.E.D. Remark: Assume that P and P′ are given by standard data (L_{i}, Φ_{i}) and (L′_{i}, Φ′_{i}). Assume we are given bilinear forms of W (R)-modules:

β_{ij} : L_{i} ⊗ L′_{j} → Q_{i+j}.

Composing this with the iteration of ι, Q_{i+j} → Q_{0}, we obtain a bilinear form

P_{0} ⊗ P′_{0} = (⊕_{i}L_{i}) ⊗ (⊕_{j}L′_{j}) → Q_{0}

Let us assume that the following diagrams are commutative:

L_{i} ⊗ L′_{j} ⟶^{Φ_{i}⊗Φ′_{j}} P_{0} ⊗ P′_{0}
β_{ij} ↓  ↓
Q_{i+j} ⟶^{F_{i+j}} Q_{0}

Then the β_{ij} extend uniquely to a bilinear form

P × P′ → Q

In [Z2] Definition 18 the notion of a bilinear form of 1-displays was defined. It is obvious from the formulas there, that a bilinear form on two 1-displays in the sense of loc.cit. is the same as a bilinear form

P × P′ → U(1),

where the right hand side is the twisted unit display (11). Starting from the normal decomposition of a display P it is easy to write down the standard data of a candidate for the exterior power ⋀^{k} P. It comes with an alternating map ⊗^{k}P → ⋀^{k} P. One proves as above that ⋀^{k} P has the universal property. We will now define the base change for displays. Let R ⟶ S be a homomorphism of rings. Let P = (P_{i}, ι_{i}, α_{i}, F_{i}) be a display over R. We will define a display P_{S} = (Q_{i}, ι_{i}, α_{i}, F_{i}) over S, with the following properties. There are W (R)-linear maps

P_{i} ⟶ Q_{i},

such that the following diagrams are commutative

\begin{matrix}P_{i} & ⟶ & Q_{i} \\ ^{ι_{i}}↑ & & ↑^{ι_{i}} \\ P_{i+1} & ⟶ & Q_{i+1} \end{matrix}  \begin{matrix}Q_{i} & ⟶^{F_{i}} & Q_{0} \\ ↑ & & ↑ \\ P_{i} & ⟶^{F_{i}} & P_{0} \end{matrix}  \begin{matrix}I_{R} ⊗ Q_{i} & ⟶^{α_{i}} & Q_{i+1} \\ ↑ & & ↑ \\ I_{R} ⊗ P_{i} & ⟶^{α_{i}} & P_{i+1} \end{matrix}

(20)

Proposition 2.12 There is a unique display P_{S} as above which enjoys the following universal property. If T = (T_{i}, ι_{i}, α_{i}, F_{i}) is an arbitrary display over S and

P_{i} ⟶ T_{i}

is a set of W (R)-linear morphisms, such that the diagrams above, with Q_{i} replaced by T_{i} are commutative, then there is a unique morphism of displays over S

P_{S} ⟶ T ,

such that the following diagrams are commutative:

\xymatrix{Q_{i} \ar[rr] &  & T_{i} \\  & P_{i} \ar@{-->}[ul] \ar@{-->}[ur]}

Documenta Mathematica 12 (2007) 147–191

The display P_{S} may be constructed using a normal decomposition of P. Let P_{0} = ⊕L_{i} be such a decomposition, and let Φ_{i} : L_{i} ⟶ P_{0} be the maps induced by F_{i}. Then L_{i}, Φ_{i} are standard data for a display over R. We can define P_{S} to be the display over S associated to the standard data W (S) ⊗_{W (R)} L_{i}, with the Frobenius linear maps F ⊗_{W (R)} Φ_{i} = Φ′_{i}. We will now see that this definition is up to canonical isomorphism independent of the normal decomposition chosen. It suffices to see that P_{S} has the universal property Proposition 2.12. The obvious maps P_{i} ⟶ Q_{i} make the diagrams (20) commutative.

Lemma 2.13 The following W (S)-module homomorphism is surjective

W (S) ⊗_{W (R)} P_{i} ⊕ I_{S} ⊗_{W (S)} Q_{i−1} ⟶ Q_{i}.

Proof: This is clear from the definitions. Assume that P_{i} ⟶ T_{i} is a set of maps as in Proposition 2.12. We construct inductively maps Q_{i} ⟶ T_{i}, which are compatible with F_{i}, ι_{i}, α_{i}. Therefore we obtain the desired morphism of displays P_{S} ⟶ T . Since P_{0} ⟶ T_{0} is W (R)-linear, we obtain a map

Q_{0} = W (S) ⊗_{W (R)} P_{0} ⟶ T_{0}.

Assume we have already constructed W (S)-module homomorphisms

Q_{j} ⟶ T_{j},

which are compatible with F, ι and α for j ≤ i. Consider the diagram

\begin{matrix}T_{i} & ⟶^{F_{i}} & T_{0} \\ ↑ & & ↑^{p} \\ Q_{i+1} & ⟶ & T_{0}. \end{matrix}  (21)

The arrow Q_{i+1} ⟶ T_{i} is the composition Q_{i+1} ⟶^{ι} Q_{i} ⟶ T_{i} and the arrow

Q_{i+1} ⟶ T_{0} is the composition Q_{i+1} ⟶^{F_{i+1}} Q_{0} ⟶ T_{0}. By Lemma 2.13 we deduce that (21) is commutative. Thus it induces a map

Q_{i+1} ⟶ T_{i} ×_{F_{i},T_{0},p} T_{0}.  (22)

It suffices to show that the last map factors through T_{i+1}. This is seen easily by composing (22) with the morphism of the lemma. The uniqueness of the constructed morphism P_{S} ⟶ T is obvious. This proves the proposition. Q.E.D.

3 Degeneracy of some Spectral Sequences

Proposition 3.1 Let π : X → Y be a separated and quasicompact morphism. Let K^{·} be a complex of of flat π^{−1}O_{Y} -modules on X which is bounded above. We assume that each K^{i} is a quasicoherent O_{X}-module. Then for each m the hypercohomology groups ℝ^{m}π_{∗}K^{·} are quasicoherent O_{Y} -modules. If M is a quasicoherent O_{Y} -module there is a canonical isomorphism

ℝπ_{∗}(K^{·} ⊗_{π^{−1}(O_{Y} )}^{𝕃} π^{−1}M) ≅ ℝπ_{∗}K^{·} ⊗_{O_{Y}}^{𝕃} M  (23)

Proof: We may assume that Y is affine. Let U = {U_{i}} be a finite affine covering of X. Let F^{·} = C^{·}(U, K^{·}) be the Czech complex. It is the complex of global sections of a sheafified Czech complex on Y: F^{·} = C^{·}(U, K^{·}). The sheaves in this complex are acyclic with respect to π_{∗} because the cohomology of an affine scheme vanishes. One concludes [EGA III] Prop. 1.4.10 that ℝπ_{∗}K^{m} are quasicoherent O_{Y} -modules namely the sheaves associated to the cohomology of F^{·}. Since the modules and sheaves involved are flat with respect to Y the projection formula reduces to the trivial equation:

C^{·}(U, K^{·} ⊗_{O_{Y}} M) ≅ F^{·} ⊗_{Γ(Y,O_{Y} )} Γ(Y, M)
Q.E.D.

Let π : X → S be a proper morphism of schemes, such that S is affine. In this section we consider a bounded complex K^{·} of flat π^{−1}(O_{S})-modules. We assume that each K^{i} is a quasicoherent O_{X}-module. Moreover we assume that the following conditions are satisfied:

(i) R^{j}π_{∗}K^{i} is a locally free O_{S}-module of finite type for any i and j.

(ii) the spectral sequence of hypercohomology degenerates:

E_{1}^{ij} = R^{j}π_{∗}K^{i} ⇒ ℝ^{n}π_{∗}K^{·}

One can easily see that with these assumptions the simple complex associated to C^{·}(U, K^{·}) as above is quasi-isomorphic to the direct sum of its cohomology groups. It follows that ℝ^{m}π_{∗}K^{·} commutes with arbitrary base change for any m. For the same reason the cohomology groups ℝ^{j}π_{∗}K^{i} commute with arbitrary base change. The degeneration of this spectral sequence may be reformulated as follows. Let us denote the by σ^{≥m}K^{·} and σ^{<m}K^{·} the truncated complexes with respect to the naive truncation. Then the cohomology sequence of

0 → σ^{≥m}K^{·} → K^{·} → σ^{<m}K^{·} → 0,

splits into short exact sequences:

0 → ℝ^{q}π_{∗}(σ^{≥m}K^{·}) → ℝ^{q}π_{∗}K^{·} → ℝ^{q}π_{∗}(σ^{<m}K^{·}) → 0.  (24)

Indeed, take a Cartan-Eilenberg resolution K^{·} → I^{·} by injective sheaves of abelian groups. Let L^{·} = π_{∗}I^{·}. This complex comes with a filtration F il^{m}L^{·} which is induced by the naive filtration of K. The spectral sequence in question is the spectral sequence of this filtered complex. The condition (24) is equivalent to the requirement that the maps

H^{q}(F il^{m+1}L^{·}) → H^{q}(F il^{m}L^{·})

are injective for each q and m, as one may see easily from the exact cohomology sequence. This injectivity may be restated as follows:

d(F il^{m}L^{q−1}) ∩ F il^{m+1}L^{q} = d(F il^{m+1}L^{q−1}).

We conclude by [De3] Prop. 1.3.2. The observation shows that the spectral sequences of hypercohomology of the truncated complexes σ^{≥m}K^{·} and σ^{<m}K^{·} degenerate too.

Proposition 3.2 Let π : X → S and K^{·} be as in Proposition 3.1. Let … → M_{0} → M_{1} → M_{2} → … be a sequence of O_{S}-modules (not necessarily a complex). We consider the complex

L^{·} : … → K^{0} ⊗_{O_{S}} M_{0} → K^{1} ⊗_{O_{S}} M_{1} → K^{2} ⊗ M_{2} → …

Then the spectral sequence

E_{1}^{ij} : R^{j}π_{∗}L^{i} ⇒ ℝ^{p+q}π_{∗}L^{·}

degenerates.

Proof: We assume without loss of generality that K^{i} = 0 for i < 0. We say that a sequence M_{0} → M_{1} → … is m-stationary if it is isomorphic to a sequence of the form:

M_{0} → … → M_{m−1} → M_{m} = M_{m} = …

Because K^{·} is bounded it suffices to show the theorem for m-stationary sequences. We argue by induction. For m = 0 this is clear from the projection formula (23). Assume that the proposition holds for r-stationary sequences with r < m. For an m-stationary sequence we consider the following morphism of complexes:

L^{·} → I^{·}  (25)
K^{0} ⊗ M^{0} … K^{m−2} ⊗ M^{m−2} ⟶ K^{m−1} ⊗ M^{m−1} ⟶ K^{m} ⊗ M^{m} …
id↓  ^{id}↓  ↓  ^{id}↓
K^{0} ⊗ M^{0} … K^{m−2} ⊗ M^{m−2} ⟶ K^{m−1} ⊗ M^{m} ⟶ K^{m} ⊗ M^{m} …

If we apply the induction assumption to I^{·} we obtain an exact sequence for each q and the given m.

0 → ℝ^{q}π_{∗}(σ^{≥m}I^{·}) → ℝ^{q}π_{∗}I^{·} → ℝ^{q}π_{∗}(σ^{<m}I^{·}) → 0.  (26)

The morphism of complexes (25) induces a commutative diagram:

ℝ^{q}π_{∗}σ^{≥m}L^{·} ⟶ ℝ^{q}π_{∗}L^{·}
id↓  ↓
ℝ^{q}π_{∗}σ^{≥m}I^{·} ⟶ ℝ^{q}π_{∗}I^{·}

By our induction assumption (26) it follows that the upper horizontal arrow is injective. We have to prove that the following sequences are exact for arbitrary integers q and n.

0 → ℝ^{q}π_{∗}(σ^{≥n}L^{·}) → ℝ^{q}π_{∗}L^{·} → ℝ^{q}π_{∗}(σ^{<n}L^{·}) → 0.

We have seen this for n = m. For n > m we have to consider the maps.

ℝ^{q}π_{∗}(σ^{≥n}L^{·}) → ℝ^{q}π_{∗}(σ^{≥m}L^{·}) → ℝ^{q}π_{∗}L^{·}

It suffices to show that the first arrow is injective. But this follows from the beginning of our induction. Finally we consider the case n < m. By the cohomology sequence it is sufficient to see that the map

ℝ^{q}π_{∗}L^{·} → ℝ^{q}π_{∗}(σ^{<n}L^{·})

is surjective. But this map factors as:

ℝ^{q}π_{∗}L^{·} → ℝ^{q}π_{∗}(σ^{<m}L^{·}) → ℝ^{q}π_{∗}(σ^{<n}L^{·})

We need to show that the second map is surjective. But the complex σ^{<m}L^{·} is the tensor product of σ^{<m}K^{·} with an (m − 1)-stationary sequence of modules. Therefore the map is surjective by induction assumption and we are done. Q.E.D.

Proposition 3.3 Let T : C → D be a left exact functor of abelian categories such that C has enough injective objects. Let K^{·} be a complex in C which is bounded below. We assume that the spectral sequences in hypercohomology

E_{1}^{ij} = R^{j}T K^{i} ⇒ ℝ^{i+j}T K^{·}

degenerates. Let f^{·} : K^{·} → K^{·} be a homomorphism of complexes. Then for each integer m the corresponding spectral sequence of hypercohomology associated to the complex

K(m, f) : →^{d} K^{m−2} →^{d} K^{m−1} →^{f^{m}d} K^{m} →^{d} K^{m+1} → …

degenerates.

We omit the proof because it uses exactly the same arguments as above.

4 Filtered Comparison Theorems for the de Rham-Witt complex

Let R be a ring such that p is nilpotent in R. We consider a smooth scheme X over R. We will fix a natural number n. Assume we are given a smooth lifting X̃/W_{n}(R). If X̃ admits a Witt-lift ([LZ] Def.3.3) OX̃ ⟶ W_{n}(O_{X}) we obtain a morphism of complexes

Ω_{X̃/W_{n}(R)}^{·} ⟶ Ω_{W_{n}(X)/W_{n}(R)}^{·} ⟶ W_{n}Ω_{X/R}^{·}.  (27)

It is shown in [LZ] 3.2 and 3.3, that even if X̃ admits no Witt lift, there is a natural isomorphism in the derived category D^{+}(X_{zar}, W_{n}(R)) of sheaves of W_{n}(R)-modules on X:

Ω_{X̃/W_{n}(R)}^{·} ⟶ W_{n}Ω_{X/R}^{·}.

The aim of this section is to prove a filtered version of this isomorphism. For typographical reasons we use the abbreviations:

Ω̃_{n}^{·} = Ω_{X̃/W_{n}(R)}^{·}, W_{n}Ω^{·} = W_{n}Ω_{X/R}^{·}.

Let us denote by F^{m}Ω_{X̃/W (R)}^{·} the complex

I_{R,n} ⊗_{W_{n}(R)} Ω̃_{n}^{0} →^{pd} … →^{pd} I_{R,n} ⊗_{W_{n}(R)} Ω̃_{n}^{m−1} →^{d} Ω̃_{n}^{m} →^{d} Ω̃_{n}^{m+1} → …. (28)

Conjecture 4.1 There is a canonical isomorphism in the derived category D^{+}(X_{zar}, W_{n}(R)) between the Nygaard complex and the complex (28):

N^{m}W_{n}Ω_{X/R}^{·} ≅ F^{m}Ω_{X̃/W_{n}(R)}^{·}

This question is closely related to the work of Deligne and Illusie [DI]. We will now see that the complexes in question are always locally isomorphic. Let us assume we are given a Witt-lift. It induces a map

κ : Ω̃_{n}^{·} ⟶ W_{n}Ω^{·}.

By composition with the Frobenius F : W_{n}Ω^{·} → W_{n−1}Ω_{[F]}^{·} we obtain a map

\begin{matrix}F̃ : & I_{R,n} ⊗_{W_{n}(R)} Ω̃_{n}^{·} ⟶ W_{n−1}Ω_{[F]}. \\ ^{V} ξ ⊗ ω ⟼ ξ ^{F} κ(ω) \end{matrix}

Using F̃ we obtain a morphism of complexes of F^{m}Ω̃^{·} ⟶ N^{m}W_{n}Ω:

\begin{matrix}I_{R} ⊗ Ω̃_{n}^{0} & ⟶^{pd} & … & ⟶^{pd} & I_{R} ⊗ Ω̃_{n}^{m−1} & ⟶^{d} & Ω̃_{n}^{m} & ⟶^{d} & … \\ F̃↓ & & & & F̃↓ & & ↓ \\ W_{n−1}Ω_{[F]}^{0} & ⟶^{d} & … & ⟶^{d} & W_{n−1}Ω_{[F]}^{m−1} & ⟶^{dV} & W_{n}Ω^{m} & ⟶^{d} & … \end{matrix} (29)

Let us consider the morphism (29) in the following simple situation: Let A = R[T_{1}, …, T_{d}] and X = Spec A. We set Ã = W_{n}(R)[T_{1}, … T_{d}] and X̃ = Spec Ã. We consider the Witt-lift:

\begin{matrix}Ã ⟶ W_{n}(A) \\ T_{i} ⟶ [T_{i}]. \end{matrix}  (30)

It is the unique map of W_{n}(R)-algebras, which maps T_{i} to its Teichmüller representative in W_{n}(A).

Proposition 4.2 For the Witt-lift (30) the induced morphism

F^{m}Ω_{X̃/W_{n}(R)}^{·} ⟶ N^{m}W_{n}Ω_{X/R}^{·}  (31)

is for any m ≥ 0 a quasi-isomorphism.

Proof: We use the W_{n}(R)-basis of Ω_{Ã/W_{n}(R)}^{l} given by p-basic differential forms. For each weight function k : [1, d] → ℤ_{≥0} we fix an order on the set

Supp k = {i_{1}, …, i_{r}}, such that
ord_{p}k_{i_{1}} ≤ ⋯ ≤ ord_{p}k_{i_{r}}.

For any ascending partition of Supp k into disjoint intervals

P : Supp k = I_{0} ⊔ I_{1} ⊔ ⋯ ⊔ I_{l},

such that I_{t} ≠ ∅ for 1 ≤ t ≤ l, we have the p-basic differential

ẽ(k, P) = T^{k_{I_{0}}} (p^{−ord_{p}k_{I_{1}}} dT^{k_{I_{1}}} ) · · · · · (p^{−ord_{p}k_{I_{l}}} dT^{I_{l}}) .  (32)

The order on Supp k is fixed once for all and therefore not indicated in the notation (compare [LZ] 2.1). In [LZ] 2.2 we have defined the basic Witt differentials

e_{n}(ξ, k, P) ∈ W_{n}Ω_{A/R}^{l}.

They are defined for functions k : [1, d] → ℤ_{≥0}[\frac{1}{p}], and ξ ∈ V^{u(k)}W_{n−u(k)}(R), where u(k) is the minimal nonnegative integer, such that the weight p^{u(k)}k takes integral values. In our case the map (27) is the unique W_{n}(R)-linear map given by

\begin{matrix}Ω_{Ã/W_{n}(R)}^{l} ⟶ W_{n}Ω_{A/R}^{l}. \\ ẽ(k, P) ⟼ e_{n}(1, k, P). \end{matrix}  (33)

The map F̃ looks as follows

\begin{matrix}F̃ : & I_{R} ⊗_{W_{n}(R)} Ω_{Ã/W_{n}(R)}^{l} ⟶ W_{n−1}Ω_{A/R,[F]}^{l} \\ ^{V} ξ ⊗ ẽ(k, P) ⟼ e_{n−1}(ξ, pk, P). \end{matrix}

For each weight k : [1, d] ⟶ ℤ_{≥0}[\frac{1}{p}], we consider the subgroup W_{n}Ω_{A/R}^{l}(k) of W_{n}Ω_{A/R}^{l}, which is generated by basic Witt-differentials e_{n}(ξ, k, P) of fixed weight k. The complex N^{m}W_{n}Ω splits into a direct sum of subcomplexes N^{m}(k):

W_{n−1}Ω_{[F]}^{0}(pk) →^{d} ⋯ →^{d} W_{n−1}Ω_{[F]}^{m−1}(pk) →^{dV} W_{n}Ω_{[F]}^{m}(k) → · · · .

Similarly let Ω_{Ã/W_{n}(R)}^{l}(k) ⊂ Ω_{Ã/W_{n}(R)}^{l} the W_{n}(R)-submodule generated by the

p-basic differentials ẽ(k, P) of fixed integral weight k. Then F^{m}Ω̃^{·} is the direct sum of the following subcomplexes F^{m}(k):

I_{R} ⊗_{W_{n}(R)} Ω̃_{n}^{0}(k) →^{pd} ⋯ →^{pd} I_{R} ⊗_{W_{n}(R)} Ω̃_{n}^{m−1}(k) →^{d} Ω̃_{n}^{m}(k) → · · · .

It is obvious that for integral weight k the map

F^{m}(k) ⟶ N^{m}(k)  (34)

is an isomorphism of complexes. Therefore the proposition follows if we show that for k not integral the complexes N^{m}(k) are acyclic. This follows in degrees not equal to m−1 or m from the corresponding statement for the de Rham-Witt complex (see [LZ] Proof of thm. 3.5). For non-integral k consider a cycle ω ∈ W_{n−1}Ω_{[F]}^{m−1}(k), i.e. dV ω = 0. Because of the relation F dV = d, it follows that ω is also a cycle in the de Rham-Witt complex W_{n−1}Ω^{·} and consequently a boundary, because k is not integral. Finally consider a cycle ω ∈ W_{n}Ω^{m}(k). It may be uniquely written as a sum

ω = ∑_{P} e_{n}(ξ_{P}, k, P).

By [LZ] Prop. 2.6 ω is a cycle, iff P = ∅ ⊔ P′, i.e. iff the first interval I_{0} of the partition P is empty, for all e_{n}(ξ_{P}, k, P) ≠ 0 which appear in the sum. Since k is not integral the coefficient ξ_{P} is of the form ξ_{P} = ^{V} τ_{P} and

d ^{V} e_{n−1}(τ_{P}, pk, P) = e_{n}(ξ_{P}, k, P).
Q.E.D.

We make n variable. We set A = R[T_{1}, …, T_{d}], A_{n} = W_{n}(R)[T_{1} … T_{d}]. We extend the Frobenius homomorphism F : W_{n}(R) ⟶ W_{n−1}(R) to a map

\begin{matrix}ϕ_{n} : & A_{n} ⟶ A_{n−1}, \\ T_{i} ⟼ T_{i}^{p}. \end{matrix}  (35)

We denote δ_{n} : A_{n} ⟶ W_{n}(A) the W_{n}(R)-algebra homomorphism, such that δ_{n}(T_{i}) = [T_{i}]. Assume we are given an étale homomorphism A ⟶ B of R-algebras. Then we find a unique set of lifting B_{n} of B, which are étale over A_{n} and morphisms

ψ_{n} : B_{n} ⟶ B_{n−1} and ε_{n} : B_{n} ⟶ W_{n}(B),

which are compatible with ϕ_{n} and δ_{n}, compare [LZ] Prop. 3.2.

Corollary 4.3 The map ε_{n} defines a quasi-isomorphism of complexes:

I_{R} ⊗ Ω_{B_{n}/W_{n}(R)}^{0} ⟶^{pd} … ⟶^{pd} I_{R} ⊗ Ω_{B_{n}/W_{n}(R)}^{m−1} ⟶^{d} Ω_{B_{n}/W_{n}(R)}^{m} …
F̃↓  F̃↓  ↓
W_{n−1}Ω_{B/R,[F]}^{0} ⟶^{d} … ⟶^{d} W_{n−1}Ω_{B/R,[F]}^{m−1} ⟶^{dV} W_{n}Ω_{B/R}^{m} …

Proof: For the given number n, we find a number m such that p^{m}W_{n}(R) = 0. Let us denote by ϕ^{m} : A_{m+n} ⟶ A_{n} the composite of m morphisms of type (35). It is clear from the definition that

dϕ^{m} : A_{m+n} ⟶ Ω_{A_{n}/W_{n}(R)}^{1}

is zero. Consider the commutative diagram

B_{m+n} →^{dψ^{m}} Ω_{B_{n}/W_{n}(R)}^{1}
↑  ↑
A_{m+n} →^{dϕ^{m}} Ω_{A_{n}/W_{n}(R)}^{1}.

The derivation A_{m+n} ⟶ Ω_{B_{n}/W_{n}(R)}^{1} is zero. Since B_{m+n}/A_{m+n} is étale, the extension dψ^{m} is zero too. Consider the commutative diagram

B_{m+n} ⟶^{ψ^{m}} B_{m}
↑  ↑
A_{m+n} →^{ϕ^{m}} A_{n}.

It induces a morphism of algebras which are étale over A_{n}:

B_{m+n} ⊗_{A_{m+n},ϕ^{m}} A_{n} ⟶ B_{n}.  (36)

This is an isomorphism. Indeed since A_{n} ⟶ A/pA has nilpotent kernel it is enough to show that (36) becomes an isomorphism after tensoring with ⊗_{A_{n}}A/pA. But then we obtain the well-known isomorphism

\begin{matrix}B/pB ⊗_{A/pA,F rob^{m}} A/pA & ⟶ & B/pB \\ b ⊗ a & ⟼ & b^{p^{m}} · a. \end{matrix}

From the isomorphism (36) we deduce an isomorphism

\begin{matrix}B_{m+n} ⊗_{A_{m+n,ϕ^{m}}} Ω_{A_{n}/R}^{·} & ˜ ⟶ & Ω_{B_{n}/R}^{·} \\ b ⊗ ω & ⟼ & ψ^{m}(b) · ω. \end{matrix}  (37)

We note that (37) becomes an isomorphism of complexes if we take 1 ⊗ d as a differential on the left hand side. Hence the first row of (4.3) is obtained by tensoring the corresponding complex for B_{n} = A_{n} with B_{n+m}. Let us consider the complex

W_{n−1}Ω_{A/R,[F]}^{0} →^{d} ⋯ →^{d} W_{n−1}Ω_{A/R,[F]}^{m−1} →^{dV} W_{n}Ω_{A/R}^{m} →^{d} · · · .  (38)

We consider it as a complex of W_{n+m}(A)-modules via F^{m} : W_{n+m}(A) ⟶ W_{n}(A). Then all differentials become linear (compare [LZ] Remark 1.8). This shows that we obtain the second row of diagram of Corollary 4.3 if we tensorize (38) with W_{n+m}(B)⊗_{W_{n+m}(A),F^{m}}. Because of the obvious isomorphism ([LZ] (3.2))

B_{n+m} ⊗_{A_{n+m},δ} W_{n+m}(A) ˜→W_{n+m}(B),

the result is the same if we tensorize (38) by

B_{n+m} ⊗_{A_{n+m},δϕ^{m}} .

Therefore the whole diagram of Corollary 4.3 is obtained from the corresponding diagram for B = A by tensoring with B_{n+m}⊗_{A_{n+m,ϕ^{m}}}. Since this tensor product is an exact functor we obtain the corollary from the proposition. Q.E.D.

Let X/R be a smooth scheme. We assume that R is reduced and p·R = 0. Then we consider still another complex derived from the de Rham-Witt complex. We set W Ω^{l} = W Ω_{X/R}^{l} and define I^{m}W_{n}Ω_{X/R} starting in degree 0.

p^{m−1}V W_{n−1}Ω^{0} →^{d} p^{m−2}V W_{n−1}Ω^{1} →^{d} ⋯ →^{d} V W_{n−1}Ω^{m−1} →^{d} W_{n}Ω …. (39)

We recall the relation pd ^{V} ω = ^{V} dω of [LZ] 1.17. For varying n we obtain a procomplex I^{m}W_{·}Ω_{X/R}.

Proposition 4.4 Let R be a reduced ring of char p. The procomplexes I^{m}W_{·}Ω and N^{m}W_{·}Ω are isomorphic in the pro-category of the category of complexes of abelian sheaves on X_{zar}.

Proof: We have an obvious morphism of procomplexes

N^{m}W_{·}Ω ⟶ I^{m}W_{·}Ω  (40)
W_{n−1}Ω_{[F]}^{0}  →^{d}  W_{n−1}Ω_{[F]}^{1}  … W_{n−1}Ω_{[F]}^{m−1}  →^{dV} W_{n}Ω^{m}  →^{d} …
p^{m−1}V ↓  p^{m−2}V ↓  V ↓  id ↓
p^{m−1}V W_{n−1}Ω^{0} →^{d} p^{m−2}V W_{n−1}Ω^{1} … V W_{n−1}Ω^{m−1} →^{d} W_{n}Ω^{m} →^{d} ….

We have to prove that this induces an isomorphism of proobjects. We set W Ω = lim_{⟵} W_{n}Ω. On W Ω the multiplication by p and the Verschiebung are

injective. Therefore we have an inverse p^{i}V W Ω ⟶^{p^{−i}V^{−1}} W Ω_{[F]}.

Lemma 4.5 Let n > k ≥ i + 1. Then there is a map p^{i}V W_{n}Ω^{l} ⟶ W_{n−k}Ω^{l}, which makes the following diagram commutative

\begin{matrix}p^{i}V W_{n}Ω_{X/R}^{l} & ⟶ & W_{n−k}Ω_{X/R,[F]}^{l} \\ ↑ & & ↑ \\ p^{i}V W Ω_{X/R}^{l} & ⟶^{p^{−i}V^{−1}} & W Ω_{X/R,[F]}^{l}. \end{matrix}  (41)

Proof of the lemma: Let n > k ≥ i. For ξ ∈ W_{n}(R) we denote by ξ̄ its restriction to W_{n−k}(R). Then we have a well-defined map

\begin{matrix}p^{i}V W_{n}(R) & ⟶ & W_{n−k}(R) \\ p^{i} ^{V} ξ & ⟼ & ξ̄. \end{matrix}  (42)

Indeed, write ξ = (x_{0}, …, x_{n−1}). Then

p^{i} ^{V} ξ = (0, …, 0, x_{0}^{p^{i}}, …, x_{n−i−1}^{p^{i}}) ∈ W_{n+1}(R).

Therefore the vector (x_{0}, …, x_{n−i−1}) ∈ W_{n−i}(R) is uniquely determined by p^{i} ^{V} ξ. We view W_{n−i}(R) as a W_{n+1}(R)-module via

W_{n+1}(R) ⟶^{F} W_{n}(R) ⟶^{Res} W_{n−i}(R).

Then we obtain a morphism of W_{n+1}(R)-modules because of the following commutative diagram

p^{i}V W (R) ⟶^{p^{−i}V^{−1}} W (R)
↓  ↓
p^{i}V W_{n}(R) ⟶ W_{n−i}(R).

The existence of the diagram (41) is clearly local for the Zariski-topology on X. We begin with the case, where X = Spec A and A = R[T_{1}, …, T_{d}] is a polynomial algebra. In this case an element of p^{i}V W Ω_{A/R}^{l} may be expressed, in terms of basic Witt-differentials:

ω = ∑ p^{i} ^{V} e_{n}(ξ_{P,k}, k, P), ξ_{P,k} ∈ V^{u(k)}W_{n−u(k)}(R).  (43)

Note that e_{n}(ξ_{P,k}, k, P) = 0, when u(k) ≥ n. The terms of the sum (43) are uniquely determined by [LZ] Prop.2.5 because of the direct decomposition

W_{n+1}Ω_{A/R}^{l} = ⊕_{k,P}W_{n+1}Ω_{A/R}^{l}(\frac{k}{p} , P).

Using loc. cit. we find:

p^{i} ^{V} e(ξ_{P,k}, k, P) = p^{i} ^{V} e(ξ′_{P,k}, k, P),  (44)

iff p^{i} ^{V} ξ_{P,k} = p^{i} ^{V} ξ′_{P,k}, except in the case where k/p is not integral and I_{0} = ∅. In the latter case the equality (44) holds, iff p^{i+1} ^{V} ξ_{P,k} = p^{i+1} ^{V} ξ′_{P,k}. With the lemma above this shows that the following map is well-defined:

\begin{matrix}p^{i}V W_{n}Ω^{l} & ⟶ & W_{n−(i+1)}Ω^{l} \\ ω & ⟼ & ∑ e_{n−(i+1)}(ξ̄_{k,P}, k, P). \end{matrix}

This proves the lemma in the case of a polynomial algebra A. Assume now that A ⟶ B is a étale morphism. The image of the canonical injection

W_{n+1}(B)⊗_{W_{n+1}(A)} p^{i}V W_{n}Ω_{A/R} → W_{n+1}(B)⊗_{W_{n+1}(A)} W_{n}Ω_{A/R} ≃ W_{n+1}Ω_{B/R}

coincides with p^{i}V W_{n}Ω_{B/R}. This follows from the following commutative diagram

W_{n+1}(B) ⊗_{W_{n+1}(A),F} W_{n}Ω_{A/R} ⟶^{∼} W_{n}Ω_{B/R}
id⊗p^{i}V ↓  ^{p^{i}V} ↓
W_{n+1}(B) ⊗_{W_{n+1}(A)} W_{n+1}Ω_{A/R} ⟶^{∼} W_{n+1}Ω_{B/R}.

The top horizontal arrow is given by ξ ⊗ ω ↦^{F} ξω and the lower horizontal arrow is multiplication. Now we find the desired map by tensoring p^{i}V W_{n}Ω_{A/R} ⟶ W_{n−(i+1)}Ω_{A/R}:

W_{n+1}(B) ⊗_{W_{n+1}(A)} p^{i}V W_{n}Ω_{A/R} ⟶ W_{n−i}(B) ⊗_{W_{n−i}(A),F} W_{n−(i+1)}Ω_{A/R}
≀ ↓  ≀ ↓
p^{i}V W_{n}Ω_{B/R}  ⟶  W_{n−(i+1)}Ω_{B/R}.

The composition of the last map with p^{i}V : W_{n−(i+1)}Ω_{B/R} ⟶ W_{n−i}Ω_{B/R} is just the restriction. This proves the lemma. Q.E.D. The proposition follows immediately because we obtain an inverse to the map (40):

p^{m−1}V W_{n−1}Ω^{0} →^{d} p^{m−2}V W_{n−1}Ω^{1} … V W_{n−1}Ω^{m−1}  →^{d}  W_{n}Ω^{m} …
↓  ↓  ↓  ↓Res
W_{n−m−1}Ω_{[F]}^{0}  →^{d} W_{n−m−1}Ω_{[F]}^{1} … W_{n−m−1}Ω_{[F]}^{m−1} →^{dV} W_{n−m−1}Ω^{m} …

The first m vertical maps defined by the lemma are equivariant with respect to

W_{n}(R) →^{Res} W_{n−m}(R) →^{F} W_{n−m−1}(R)

The remaining maps are equivariant with respect to W_{n}(R) → W_{n−m}(R). The commutativity of the diagram follows, since it is a homomorphic image of a corresponding diagram for W Ω without level. This proves the proposition. Q.E.D. Let X/R be a smooth scheme. Let us denote by J_{X/W_{n}(R)} ⊂ O_{X/W_{n}(R)} the sheaf of pd-ideals. We denote by J_{X/W_{n}(R)}^{[m]} its m-th divided power. Let

u_{n} : Crys(X/W_{n}(R)) ⟶ X_{zar}

be the canonical morphism of sites. We are going to define a morphism in D(X_{zar}) the derived category of abelian sheaves on X_{zar} for m < p:

Ru_{n∗}J_{X/W_{n}(R)}^{[m]} ⟶ I^{m}W_{n}Ω_{X/R}^{·}  (45)

In order to define (45) we begin with the case, where X admits an embedding in a smooth scheme Y/R, such that Y has a Witt-lift: Ỹ /W_{n}(R) and OỸ ⟶ W_{n}(O_{Y}). The left hand side of (45) may be computed with the filtered Poincaré lemma [BO] Theorem 7.2: Let D be the divided power hull of X in Ỹ. Let I_{D} ⊂ O_{D} be the pd-ideal. The pd-de Rham-complex Ω̆_{D/W_{n}(R)} has the following subcomplex F il^{m}Ω̆_{D/W_{n}(R)}:

I_{D}^{[m]}Ω̆_{D/W_{n}(R)}^{∘} →^{d} I_{D}^{[m−1]}Ω̆_{D/W_{n}(R)}^{1} →^{d} … I_{D}Ω̆_{D/W_{n}(R)}^{m−1} →^{d} Ω̆_{D/W_{n}(R)} … (46)

Then the left hand side of (45) is isomorphic to the hypercohomology of (46). The Witt-lift defines a morphism

OỸ ⟶ W_{n}(O_{Y}) ⟶ W_{n}(O_{X}).

It maps the ideal sheaf of X ⊂ Ỹ to the ideal sheaf I_{X} = V W_{n−1}(O_{X}) ⊂ W_{n}(O_{X}). Since I_{X} is endowed with divided powers, we obtain

O_{D} ⟶ W_{n}(O_{X}),  (47)

mapping I_{D} to I_{X}. The homomorphism (47) induces a map of the pd-de Rham complexes

Ω̆_{D/W_{n}(R)} ⟶ Ω̆_{W_{n}(X)/W_{n}(R)} ⟶ W_{n}Ω_{X/R}.

Taking into account that I_{X}^{[h]} = p^{h−1}I_{X} for h < p, we obtain the desired morphism from (46) to the complex I^{m}W_{n}Ω if m < p:

p^{m−1}I_{X}W_{n}Ω_{X/R}^{0} ⟶ … ⟶ I_{X}W_{n}Ω_{X/R}^{m−1} →^{d} W_{n}Ω_{X/R} → ….

We note that I_{X}W_{n}Ω_{X/R}^{l} = V W_{n−1}Ω_{X/R}^{l} follows from the formula

V (ηdω_{1} … dω_{r}) = ^{V} ηd ^{V} ω_{1} … d ^{V} ω_{r}.

Hence we obtain a morphism

Ru_{n∗}J_{X/W_{n}(R)}^{[m]} ˜→F il^{m}Ω̆_{D/W_{n}(R)} ⟶ I^{m}W_{n}Ω_{X/R}.  (48)

The independence of the last arrow from the embedding of X into a Witt lift (Y, Ỹ) is proved in a standard manner: Let X ↪ Y′ be an embedding into a second Witt lift (Y′, Ỹ′). Then we obtain a Witt lift of the product Y ×_{Spec R} Y′ : Indeed, Ỹ ×_{Spec W_{n}(R)} Ỹ′ is a lifting of Y × Y′ and the two given Witt lifts induce a morphism:

OỸ ⊗_{W_{n}(R)} OỸ′ ⟶ W_{n}(O_{Y}) ⊗_{W_{n}(R)} W_{n}(O_{Y′}) ⟶ W_{n}(O_{Y} ⊗ O_{Y′}).

If P denotes the pd-hull of X in Ỹ ×_{Spec W_{n}(R)} Ỹ′. We obtain a commutative diagram

\xymatrix{F il^{m}Ω̆_{D/W_{n}(R)} \ar[dd] \ar[dr] \\  & I^{m}W_{n}Ω_{X/R} \\ F il^{m}Ω̆_{P/W_{n}(R)} \ar@{-->}[ur]}

Since the vertical arrow induces by [BO] the identity on Ru_{n∗}J_{X/W_{n}(R)}^{[m]} the independence of (45) of the chosen Witt lift follows. If X admits no embedding in a smooth scheme Y which has a Witt lift, one can proceed by simplicial methods [I] or [LZ] §3.2, but we omit the details here.

Theorem 4.6 For each m < p and n the map in D^{+}(X_{zar}, W_{n}(R))

Ru_{n∗}J_{X/W_{n}(R)}^{[m]} ⟶ I^{m}W_{n}Ω_{X/R}  (49)

is a quasi-isomorphism.

Proof: Clearly the question is local for the Zariski-topology on X. We may therefore assume that X = Spec B, where the R-algebra B is étale over R[T_{1}, …, T_{d}]. From the discussion above we know that any Witt-lift of B leads to the same morphism (49). We choose a Frobenius lift {B_{n}}_{n∈ℕ} of the algebra B as in the corollary 4.3. We begin with the reduction to the case B = R[T_{1}, …, T_{d}]. Let J be the kernel of B_{n} ⟶ B. Then J^{[i]} = p^{i−1}I_{R}B_{n}, where I_{R} = V W_{n−1}(R) ⊂ W_{n}(R). Hence we have to show that the following morphism of complexes induces a quasi-isomorphism:

p^{m−1}I_{R}Ω_{B_{n}/W_{n}(R)}^{0} ⟶^{d} … I_{R}Ω_{B_{n}/W_{n}(R)}^{m−1} ⟶^{d} Ω_{B_{n}/W_{n}(R)}^{m} ⟶^{d}
↓  ↓  ↓
p^{m−1}V W_{n−1}Ω_{B/R}^{0} ⟶^{d} … V W_{n−1}Ω_{B/R}^{m−1} ⟶^{d} W_{n}Ω_{B/R}^{m} ⟶^{d}

(50)

We choose a number s, such that p^{s}W_{n}(R) = 0. We consider the groups in the first complex as B_{n+s} modules via ψ^{s} : B_{n+s} → B_{n}. As shown in the proof of Corollary 4.3 we obtain a complex of B_{n+s}-modules. The same is true if we consider the groups in the second complex as B_{n+s}-modules by ψ^{s} : B_{n+s} → B_{n} → W_{n}(B). We obtain the diagram above from the corresponding diagram for B = A by tensoring with B_{n+s}⊗_{A_{n+s}}. Since B_{n+s} is étale over A_{n+s}, we have reduced our statement to the case where B = R[T_{1}, …, T_{d}] and where the Witt-lift is a standard one. In the case of a polynomial algebra we have a decomposition of the de Rham Witt complex according to weights [LZ] 2.17. Because the operator V is homogeneous, we have a similar decomposition for the complex I^{m}W_{n}Ω_{A/R}. In fact, by [LZ] Prop. 2.5 an element of p^{m−l−1}V W_{n−1}Ω^{l}, for l ≤ m − 1 may be uniquely written as a sum of elements of the following types e_{n}(p^{m−l−1} ^{V} ξ, k, I_{0}, …, I_{l}) for k integral e_{n}(p^{m−l−1} ^{V} ξ, k, I_{0}, …, I_{l}) for I_{0} ≠ ∅, k not integral e_{n}(p^{m−l} ^{V} ξ, k, I_{0}, …, I_{l}) for I_{0} = ∅, k not integral. Here ξ ∈ W_{n−1}(R) for k integral and ξ ∈ V^{u(k)−1}W_{n−u(k)}(R) for k nonintegral. Clearly the elements of the first type span a subcomplex of I^{m}W_{n}Ω_{A/R} which is isomorphic to the complex in the first row of (50). Indeed, the p-basic differentials of this complex are mapped to basic Witt-differentials of the first type above. The last two types of Witt-differentials above span an acyclic subcomplex because of the formula

de_{n}(p^{m−l−1} ^{V} ξ, k, I_{0}, …, I_{l}) = e_{n}(p^{m−l−1} ^{V} ξ, k, ϕ, I_{0}, …, I_{l}),

for I_{0} ≠ ∅ and k not integral. The exactness of the non integral part at W_{n}Ω_{B/R}^{m} follows in the same way. Q.E.D. Let X_{n}/W_{n}(R) be a compatible system of smooth liftings of X/R for n ∈ ℕ. The Theorem 4.6 provides an isomorphism in the derived category between I^{m}W_{n}Ω_{X/R} and

p^{m−1}I_{R}Ω_{X_{n}/W_{n}(R)}^{0} → p^{m−2}I_{R}Ω_{X_{n}/W_{n}(R)}^{1} → … I_{R}Ω_{X_{n}/W_{n}(R)}^{m−1} → Ω_{X_{n}/W_{n}(R)}^{m−1} ….

(51)

We know by Proposition 4.4 that {I^{m}W_{n}Ω_{X/R}} is isomorphic to the procomplex {N^{m}W_{n}Ω_{X/R}}. The same argument shows that the procomplex (51) is quasi-isomorphic to {F^{m}Ω_{X_{n}/W_{n}(R)}^{·}}_{n∈N}. Passing to the projective limit we obtain:

Corollary 4.7 Let R be a reduced ring. Let X/R be a smooth and proper scheme. Assume that X_{n}/W_{n}(R) is a compatible system of smooth liftings of X. Then there is for each number m < p a natural isomorphism in the derived category D^{+}(X_{zar,W (R)}):

N^{m}W Ω_{X/R}^{·} ≅ F^{m}Ω_{X /W (R)}^{·},

where X = lim_{⟶} X_{n} in the sense of EGA I Prop. 10.6.3.

This is a weak form of the Conjecture 4.1 which asserts this for every level separately.

5 Display Structure on crystalline cohomology

Let R be a ring such that p is nilpotent in R. Let (A, σ, α) be a frame for R [Z1]. This means that A is a torsion free a p-adic ring with an endomorphism σ : A → A, which induces the Frobenius endomorphism A/pA → A/pA. The map α : A → R is a surjective ring homomorphism, such that the ideal 𝔞 = Ker α has divided powers.

Definition 5.1 An A-window consists of

1) A finitely generated projective A-module P_{0}.

2) A descending filtration of P_{0} by A-submodules

… P_{i+1} ⊂ P_{i} ⊂ ⋯ ⊂ P_{2} ⊂ P_{1} ⊂ P_{0}.  (52)

3) σ-linear homomorphisms

F_{i} : P_{i} → P_{0}.

The following conditions are required.

(i) 𝔞P_{i} ⊂ P_{i+1} and the factor module P_{i+1}/𝔞P_{i} is a finitely generated projective R-module E_{i+1} for i ≥ 0. We set E_{0} = P_{0}/𝔞P_{0}.

(ii) The inclusions P_{i+1} → P_{i} induce injective R-module morphisms

⋯ → E_{i+1} → E_{i} → ⋯ → E_{0},

such that E_{i+1} is a direct summand of E_{i}.

(iii) 𝔞P_{i} = P_{i+1} if i is big enough.

(iv) F_{i}(x) = pF_{i+1}(x) for x ∈ P_{i+1}.

(v) The union of the images F_{i}(P_{i}) for i ∈ ℤ_{≥0} generate P_{0} as an A-module.

A window is called standard if it arises in the following way. Let L_{0}, …, L_{d} be finitely generated projective A-modules. Let

Φ_{i} : L_{i} → ⨁_{j=0}^{d} L_{j}

be σ-linear homomorphisms, such that the determinant of Φ_{0} ⊕ ⋯ ⊕ Φ_{d} is a unit. Then we set for i ≥ 0

P_{i} = 𝔞^{i}L_{0} ⊕ 𝔞^{i−1}L_{1} ⊕ … ⊕ 𝔞L_{i−1} ⊕ L_{i} ⊕ ⋯ ⊕ L_{d}.

We define F_{i} on this direct sum as follows: The restriction of F_{i} to 𝔞^{i−k}L_{k} for k < i resp. L_{k} for k ≥ i to is defined by

F_{i}(ax) = \frac{σ(a)}{p^{i−k}} Φ_{k}(x) for 0 ≤ k < i, x ∈ L_{k}, a ∈ 𝔞^{i−k}
F_{i}(x) = p^{k−i}Φ_{k}(x) for i ≤ k x ∈ L_{k}.

It is clear that (P_{i}, F_{i}) form a window. Each window is isomorphic to a standard window. Indeed, let E_{0} = ⊕L̄_{j} be a splitting of the filtration (52) in the definition:

E_{i} = ⊕_{j≥i}L̄_{j}.

Let L_{i} be a finitely generated projective A-module which lifts L̄_{i}. We find homomorphisms L_{i} → P_{i} which make the following diagrams commutative:

L̄_{i} ⟶ E_{i}
↑  ↑
L_{i} ⟶ P_{i}.

It follows from the lemma of Nakayama that ⊕L_{i} → P_{0} is an isomorphism, since it is modulo 𝔞. By induction we obtain

P_{i} = 𝔞^{i}L_{0} ⊕ ⋯ ⊕ 𝔞L_{i−1} ⊕ L_{i} ⊕ ⋯ ⊕ L_{d}.  (53)

We set Φ_{i} = F_{i}|L_{i}. The condition (v) implies that ⊕Φ_{i} : ⊕L_{j} → ⊕L_{j} is a σ-linear epimorphism and therefore an isomorphism. Remark: A window (P_{i}) is of degree d, if P_{i+1} = 𝔞P_{i} for i ≥ d. To give a window of degree d it is enough to give only the modules P_{0}, …, P_{d}. The axioms may be formulated in the same way for this finite chain of modules. The axiom (v) then requires that the union of F_{0}(P_{0}), F_{1}(P_{1}) …, F_{d}(P_{d}) generates P_{0} as an A-module. We will now see that an A-window induces a display over R. There is a natural ring homomorphism δ : A → W (A), such that for the Witt-polynomials w_{n} there is the identity

w_{n}(δ(a)) = σ^{n}(a), a ∈ A.

Consider the composite ring homomorphism.

κ : A → W (A) → W (R).

We have by [Z1] Prop. 1.5:

\begin{aligned}κ(σ(a)) & = & ^{F} κ(a) & for a ∈ A \\ κ(\frac{σ(a)}{p}) & = & ^{V^{−1}}κ(a) & for a ∈ 𝔞. \end{aligned}

The last equation makes sense because κ(a) ∈ V W (R) for a ∈ 𝔞. It is clear that a datum (L_{i}, Φ_{i}) for a standard window over A induces the datum (W (R) ⊗_{W (A)} L_{i}, F ⊗ Φ_{i}) for a standard display over R. We will show that the resulting display does not depend on the decomposition P_{0} = ⊕L_{i} we have used. We give an invariant construction of a display (Q_{i}, ι_{i}, α_{i}, F_{i}) from a window (P_{i}, F_{i}). The display comes with morphisms τ_{i} : P_{i} → Q_{i} such that the following diagrams commute

\begin{matrix}P_{i} & ⟶^{τ_{i}} & Q_{i} \\ ↑ & & ^{ι_{i}}↑ \\ P_{i+1} & ⟶^{τ_{i+1}} & Q_{i+1} \end{matrix}  \begin{matrix}P_{i+1} & ⟶^{τ_{i+1}} & Q_{i+1} \\ ↑ & & ^{α_{i}}↑ \\ 𝔞 ⊗ P_{i} & ⟶ & I_{R} ⊗_{W (R)} Q_{i} \end{matrix}  \begin{matrix}P_{0} & ⟶^{τ_{0}} & Q_{0} \\ ^{F_{i}}↑ & & ^{F_{i}}↑ \\ P_{i+1} & ⟶^{τ_{i}} & Q_{i}. \end{matrix}

(54)

We construct Q_{i} and τ_{i} inductively, such that the diagrams (54) commute. We set Q_{0} = W (R) ⊗_{κ,A} P_{0} and we let τ_{0} : P_{0} → Q_{0} be the canonical map. Assume that τ_{k} : P_{k} → Q_{k} was constructed for k ≤ i. Then we consider the following commutative diagrams:

\xymatrix{P_{i} \ar[r]^{τ_{i}} & Q_{i} \ar[r]^{F_{i}} & Q_{0} \\  &  & Q_{0} \ar[u]^{p} & Q_{i} \ar[r]^{F_{i}} & Q_{0} \\ P_{i+1} \ar[uu]^{ι} \ar[rr]^{F_{i+1}} &  & P_{0} \ar[u]^{τ_{0}} & I_{R} ⊗ Q_{i} \ar[u] \ar[r]^{F̃_{i}} & Q_{0} \ar[u]^{p}}

We obtain a morphism to the fibre product

(W (R) ⊗_{A} P_{i+1}) ⊕ (I_{R} ⊗ Q_{i}) → Q_{i} ×_{F_{i},Q_{0},p} Q_{0}.  (55)

We define Q_{i+1} as the image of (55). This gives a map P_{i+1} →^{τ_{i+1}} Q_{i+1}. We define ι : Q_{i+1} → Q_{i} and F_{i+1} : Q_{i+1} → Q_{0} and α_{i} : I_{R} ⊗ Q_{i} → Q_{i+1} as the canonical maps determined by these data. A routine verification shows that this construction gives the same result as the construction via standard windows.

Moreover the following universal property holds. Let (Q′_{i}, ι′_{i}, α′_{i}, F′_{i}) be a display over R and let τ′_{i} : P_{i} → Q′_{i} be maps such that the diagrams (54) for τ′_{i} commute. Then the maps τ′_{i} are the composition of τ_{i} and a morphism of displays (Q_{i}, ι_{i}, α_{i}F_{i}) → (Q′_{i}, ι′_{i}, α′_{i}, F′_{i}).

Let A →^{α} R, σ, 𝔞 as before. Let X → Spec R be a scheme which is projective and smooth. Let Y →^{f} Spf A be a smooth pA-adic formal scheme, which lifts X. We set A_{n} = A/p^{n} and Y_{n} = Y ×_{Spf A} Spec A_{n}. For big n the map α factors through A_{n} →^{α_{n}} R. The kernel 𝔞_{n} inherits a pd-structure. We consider the crystalline topos (X/A)_{crys}. Let J_{X/A_{n}} ⊂ O_{X/A_{n}} be the pd-ideal sheaf. We are interested in the cohomology groups:

H^{i}(X, J_{X/A}^{[m]}) = lim_{⟵_{n}} H_{crys}^{i}(X/A_{n}, J_{X/A_{n}}^{[m]}).  (56)

Remark: It would be more accurate to consider the cohomology groups of R lim_{⟵_{n}} RΓ(X/A_{n}, J_{X/A_{n}}^{[m]}). But under the Assumptions 5.2 and 5.3 we are going

to make these groups will coincide.

By [BO] 7.2 the groups H_{crys}^{i}(X/A_{n}, J_{X/A_{n}}^{[m]}) are the hypercohomology groups of the following complex F il^{[m]}Ω_{Y_{n}/A_{n}}^{·}:

𝔞_{n}^{[m]}⊗_{A_{n}}Ω_{Y_{n}/A_{n}}^{0} → 𝔞_{n}^{[m−1]}⊗_{A_{n}}Ω_{Y_{n}/A_{n}}^{1} ⋯ → 𝔞_{n}⊗_{A_{n}}Ω_{Y_{n}/A_{n}}^{m−1} → Ω_{Y_{n}/A_{n}}^{m} … (57)

We will make the following assumptions:

Assumption 5.2 The cohomology groups H^{q}(Y_{n}, Ω_{Y_{n}/A_{n}}^{p}) are for each n locally free A_{n}-modules of finite type.

Assumption 5.3 The de Rham spectral sequence degenerates at E_{1}

E_{1}^{pq} = H^{q}(Y_{n}, Ω_{Y_{n}/A_{n}}^{p}) ⇒ ℍ^{p+q}(Y_{n}, Ω_{Y_{n}/A_{n}}^{·}).

Since Y_{n} is quasicompact and separated by assumption the cohomology sheafs ℝ^{m}f_{n∗}Ω_{Y_{n}/A_{n}}^{·} are quasicoherent. From the assumption we see that these sheaves are locally free of finite type. Hence the complex ℝf_{n∗}Ω_{Y_{n}/A_{n}}^{·} is quasiisomorphic to the direct sum of its cohomology groups. This implies that the cohomology groups ℝ^{m}f_{n∗}Ω_{Y_{n}/A_{n}}^{·} commute with arbitrary base change. The same applies to the cohomology groups R^{q}f_{n∗}Ω_{Y_{n}/A_{n}}^{p}. By Proposition 3.2 and the projection formula (Proposition 3.1) we obtain a degenerating spectral sequence

E_{1}^{ij} = H^{j}(Y_{n}, Ω_{Y_{n}/A_{n}}^{i}) ⊗_{A_{n}} 𝔞^{[m−i]} ⇒ℍ^{i+j}(Y_{n}, F il^{[m]}Ω_{Y_{n}/A_{n}}^{·})
∥
H_{crys}^{i+j}(X/A_{n}, J_{X/A_{n}}^{[m]})

If we pass to the projective limit we obtain a degenerating spectral sequence

E_{1}^{ij} = 𝔞^{[m−i]} ⊗ H^{j}(Y, Ω_{Y/A}^{i}) ⇒ H_{crys}^{i+j}(X/A, J_{X/A}^{[m]}).  (58)

The groups involved have no p-torsion. We set X̄ = X ×_{Spec R} Spec R̄, where R̄ = R/pR. By [BO] 5.17 there is a canonical isomorphism

H_{crys}^{i}(X/A, O_{X/A}) ≃ H_{crys}^{i}(X̄/A, O_{X̄/A}).  (59)

The absolute Frobenius on X̄ and σ on A induce an endomorphism on the right hand side of (59) and therefore an endomorphism

F : H_{crys}^{i}(X/A, O_{X/A}) → H_{crys}^{i}(X/A, O_{X/A}).

Lemma 5.4 Let p^{[m]} be the maximal power of p which divides p^{m}/m! Then the image of the following composition

H_{crys}^{i}(X/A, J_{X/A}^{[m]}) → H_{crys}^{i}(X/A), O_{X/A}) →^{F} H_{crys}^{i}(X/A, O_{X/A})

is contained in p^{[m]}H_{crys}^{i}(X/A, O_{X/A}).

Proof: The argument is well known [K], but we repeat it in the generality we need. We may replace A by A_{n}. We embed X into a smooth and projective A_{n}-scheme Z, such that there is an endomorphism σ : Z → Z which lifts the absolute Frobenius modulo p and which is compatible with σ on A_{n}. We may take for Z the projective space. Consider the pd-hull D of X in Z. It is also the pd-hull of X̄ in Z. Therefore σ extends to D/A_{n} and to the pd-differentials Ω̆_{D/A_{n}}. We obtain by [BO] an isomorphism

ℍ^{i}(X, Ω̆_{D/A_{n}}^{·}) ⟶^{∼} H_{crys}^{i}(X/A, O_{X/A_{n}}),

which is equivariant with respect to the action of σ on the left hand side and F on the right hand side. Consider the morphisms

X̄ → D → Z.

Let I(X̄) be the ideal of X̄ in Z and J̄_{D} be the ideal of X̄ in D. Consider the diagram

\xymatrix{(O_{Z}, I(X̄)) \ar[r] \ar[d]^{σ} \ar[dr]^{κ} & (O_{D}, J̄_{D}) \ar[d]^{σ_{D}} \\ (O_{Z}, I(X̄)) \ar[r] & (O_{D}, J̄_{D})}

The composite κ maps I(X̄) to p · O_{D}. This follows because

σ(z) ≡ z^{p} mod p for z ∈ O_{Z}.  (60)

If z ∈ I(X̄) the image of z^{p} in J̄_{D} becomes divisible by p, because we have divided powers. Therefore the induced map σ_{D} on the divided power envelope maps J̄_{D} to pO_{D}. Therefore

σ(J̄_{D}^{[m]}) ⊂ p^{[m]}O_{D}.

For z ∈ O_{Z} we find from (60) that in Ω̆_{D/A_{n}}^{1}:

dσ(z) ≡ 0 mod p.

The composite map of the lemma is induced by a map of complexes:

\begin{matrix}J_{D}^{[m]}Ω̆_{D/A_{n}}^{∘} & ⟶ & … & ⟶ & J_{D}^{[m−i]}Ω̆_{D/A_{n}}^{i} & ⟶ & … \\ ^{σ}↓ & & & & ^{σ}↓ \\ Ω̆_{D/A_{n}}^{∘} & ⟶ & … & ⟶ & Ω̆_{D/A_{n}}^{i} & ⟶ & …. \end{matrix}  (61)

The image of this map lies in p^{[m]} · Ω̆_{D/A_{n}}^{·} = p^{[m]}A_{n} ⊗_{A_{n}}^{𝕃} Ω̆_{D/A_{n}}. The last equality follows since by [BO] 3.32 the sheaf O_{D} is flat over A_{n}. The hypercohomology of the last complex is by the projection formula

\begin{aligned}p^{[m]}A_{n} ⊗^{𝕃} RΓ(X, Ω̆_{D/A_{n}}) & = p^{[m]}A_{n} ⊗^{𝕃} ℝΓ_{crys}(X/A_{n}, O_{X/A_{n}}) \\ & = p^{[m]}A_{n} ⊗^{𝕃} RΓ(Y_{n}, Ω_{Y_{n}/A_{n}}^{·}) \end{aligned}

But the cohomology of the last complex is p^{[m]}ℍ^{i}(Y_{n}, Ω_{Y_{n}/A_{n}}^{·}), since we assumed that the cohomology is locally free. This shows that (61) factors on the hypercohomology through p^{[m]}ℍ_{crys}(X/A_{n}, O_{X/A_{n}}) = p^{[m]}ℍ^{i}(Y_{n}, Ω_{Y_{n}/A_{n}}^{·}). Q.E.D.

Theorem 5.5 Let R be a ring, such that p is nilpotent in R. Let X be a scheme which is projective and smooth over R. Let A → R be a frame. We assume that X lifts to a projective and smooth p-adic formal scheme Y/ Spf A such that the assumptions 5.2 and 5.3 are fullfilled. Then for each number n < p the canonical maps

H_{crys}^{n}(X/A, J_{X/A}^{[m]}) → H_{crys}^{n}(X/A, J_{X/A}^{[m−1]}) → ⋯ → H_{crys}^{n}(X/A, O_{X/A})

are injective. The A-modules P_{m} = H_{crys}^{n}(X/A, J_{X/A}^{[m]}) for m ≤ n together with the maps

\frac{1}{p^{m}} F = F_{m} : P_{m} → P_{0}

given by Lemma 5.4 form a window of degree n.

Proof: We consider a number m ≤ n. Then we have J_{X/A}^{m} = J_{X/A}^{[m]}, 𝔞^{m} = 𝔞^{[m]}. We write F il^{[m]}Ω_{Y/A}^{·} = lim_{⟵_{n}} F il^{[m]}Ω_{Y_{n}/A_{n}}^{·}. Then we find a canonical

isomorphism

P_{m} = ℍ^{n}(X, F il^{[m]}Ω_{Y/A}) =^{∼} H_{crys}^{n}(X/A, J_{X/A}^{m})  (62)

From the degenerating spectral sequence (58) we obtain the injectivity of P_{m} → P_{m−1}, since we have injectivity on the associated graded groups. In the following considerations m, n can be arbitrary natural number, without the restriction m ≤ n < p. Then F il_{Y/A}^{[m]} will be the complex F il_{Y/A}^{m}

𝔞^{m}Ω_{Y/A}^{0} → 𝔞^{m−1}Ω_{Y/A}^{1} → ⋯ → 𝔞Ω_{Y/A}^{m−1} → Ω_{Y/A}^{m} → …

Consider the following morphism:

𝔞 ⊗ ℍ^{n}(X, F il^{m}Ω_{Y/A}^{·}) → ℍ^{n}(X, 𝔞F il^{m}Ω_{Y/A}^{·}).  (63)

We have for 𝔞F il^{m}Ω_{Y/A}^{·} a degenerating spectral sequence as (58). Therefore the right hand side of (63) is a subgroup of H^{n}(X, F il^{m}Ω_{Y/A}^{·}). We claim that the induced inclusion is an equality

𝔞ℍ^{n}(X, F il^{m}Ω_{Y/A}^{·}) = ℍ^{n}(X, 𝔞F il^{m}Ω_{Y/A}^{·}).  (64)

This equality holds for m = 0 by the projection formula. Indeed, consider the canonical map:

F il^{m}Ω_{Y/A} → 𝔞^{m}Ω_{Y/A}^{0} → 0.

The kernel is the following complex C:

0 → 𝔞^{m−1}Ω_{Y/A}^{1} → ⋯ → 𝔞Ω_{Y/A}^{m−1} → Ω_{Y/A}^{m} → ….

This complex C is of the same nature as F il^{m}Ω_{Y/A}^{·} but with less ideals involved. By an induction we may assume that

𝔞ℍ^{n}(X, C) = ℍ^{n}(X, 𝔞C).

By the projection formula we find

𝔞H^{n}(X, 𝔞^{m}Ω_{Y/A}^{0}) = 𝔞^{m+1}H^{n}(X, Ω_{Y/A}^{0}).

The assertion (64) follows from the diagram

\begin{matrix}ℍ^{n}(X, 𝔞C) & ⟶ & ℍ^{n}(X, 𝔞F il^{m}Ω_{Y/A}^{·}) & ⟶ & H^{n}(X, 𝔞^{m+1}Ω_{Y/A}^{0}) \\ ^{∥}↑ & & ^{∪}↑ & & ^{∥}↑ \\ 𝔞ℍ^{n}(X, C) & ⟶ & 𝔞ℍ^{n}(X, F il^{m}Ω_{Y/A}^{·}) & ⟶ & 𝔞H^{n}(X, 𝔞^{m}Ω_{Y/A}^{0}) \end{matrix} (65)

The upper line is a short exact sequence by a spectral sequence argument as above. The lower line is a complex. The first arrow is injective and the second surjective but it is a priori not exact in the middle term. One sees that the upper and lower line in (63) must be isomorphic. This proves (65). We have already seen that the following maps are injective

ℍ^{n}(X, 𝔞F il^{m}Ω_{Y/A}^{·}) → ℍ^{n}(X, F il^{m+1}Ω_{Y/A}^{·}) → ℍ^{n}(X, F il^{m}Ω_{Y/A}^{·}).

Therefore we obtain an exact sequence

0 → ℍ^{n}(X, 𝔞F il^{m}Ω_{Y/A}^{·}) → ℍ^{n}(X, F il^{m+1}Ω_{Y/A}^{·}) → ℍ^{n}(X, σ^{≥m+1}Ω_{X/R}^{·}) → 0.

Since by (64) the map 𝔞 ⊗ ℍ^{n}(X, F il^{m}Ω_{Y/A}) → ℍ^{n}(X, 𝔞F il^{m}Ω_{Y/A}^{·}) is surjective, we see that

P_{m} = ℍ^{n}(X, F il^{m}Ω_{Y/A}^{·}) and E_{m} = ℍ^{n}(X, σ^{≥m}Ω_{X/R}^{·})

fulfill the conditions (i)-(iii) for a window without any restriction on m and n. We note that for fixed n we have P_{m+1} = 𝔞P_{m} for m ≥ n. As explained after the definition of a window, we can obtain a decomposition

P_{m} = 𝔞^{m}L_{0} ⊕ 𝔞^{m−1}L_{1} ⊕ ⋯ ⊕ 𝔞^{m−n}L_{n},

with the convention that 𝔞^{k} = A if k ≤ 0. Concretely we can find the liftings L_{i} as follows. We consider the maps:

ℍ^{n}(X, F il^{m}Ω_{Y/A}^{·}) → ℍ^{n}(X, σ^{≥m}Ω_{Y/A}^{·}) → H^{(n−m)}(X, Ω_{Y/A}^{m})

Then L_{m} is obtained by splitting the last surjection. This construction gives isomorphisms:

L_{m} ≅ H^{(n−m)}(X, Ω_{Y/A}^{m})

We now impose the condition m ≤ n < p of the theorem. By lemma 5.4 and (62) the Frobenius endomorphism F : P_{0} → P_{0} is divisible by p^{m} when restricted to P_{m}. We set

Φ_{m} = \frac{1}{p^{m}} F_{|L_{m}}.

The assertion that {P_{m}} is a window is then equivalent with the condition that

⊕_{i=0}^{n}Φ_{i} : ⊕_{i=0}^{n}L_{i} → ⊕_{i=0}^{u}L_{i}

is a σ-linear isomorphism, or in other words that det(⊕_{i=0}^{n}Φ_{i}) is a unit in W (A). Clearly it suffices to show that for any homomorphism R → k to a perfect field k the image of det(⊕Φ_{i}) by the morphism

A →^{κ} W (R) → W (k) → k

is a nonzero. The compositum map A → W (k) respects the Frobenius and induces a map on crystalline cohomology

H_{crys}^{n}(X/A, O_{X/A}) → H_{crys}^{n}(X_{k}/W (k), O_{X_{k}/W (k)})

which respects the Frobenius. It is induced by the base change map for de Rham cohomology.

ℍ^{n}(X, Ω_{Y/A}^{·}) → ℍ^{n}(X_{k}, Ω_{Y ⊗_{A}W (k)/W (k)}^{·}).

The special decomposition we have chosen

ℍ^{n}(X, Ω_{Y/A}^{·}) = ⊕L_{i},

induces a similar decomposition

ℍ^{n}(X_{k}, Ω_{Y_{W (k)}/W (k)}^{·}) = ℍ^{n}(X, Ω_{Y/A}^{·}) ⊗_{A} W (k) = ⊕L_{i} ⊗_{A} W (k).

Therefore we have reduced our assertion to the case R = k a perfect field and A = W (k). This case was proved by Mazur (Compare [Fo] p.91 and Kato [K] Prop.2.5). We give an argument in the case n < p − 2 which is based on the comparison Corollary 4.7 but doesn’t use gauges. For any complex A^{·} of abelian sheaves on X consider the exact sequence induced by the naive filtration.

0 → σ_{>i}A^{·} → A^{·} → σ_{≤i}A^{·} → 0,

where i is an arbitrary integer. If n + 1 ≤ i we obtain an isomorphism

ℍ^{n}(X, A^{·}) ≅ ℍ^{n}(X, σ_{≤i}A^{·}).

We apply this to the Nygaard complex N^{m}W Ω_{X/k}^{·} and to the de Rham-Witt complex W Ω_{X/k}^{·}. For i ≤ m−1 the operator F̂_{m} (5) induces clearly a bijection of the truncated complexes

F̂_{m} : σ_{≤i}N^{m}W Ω_{X/k}^{·} → σ_{≤i}W Ω_{X/k}^{·}

Therefore if n + 1 ≤ i ≤ m − 1 we obtain a bijection

F_{m} : ℍ^{n}(X, N^{m}W Ω_{X/k}^{·}) → ℍ^{n}(X, W Ω_{X/k}^{·})

We set m = n + 2. Since m < p by assumption (and because k is reduced) there are canonical isomorphisms in the derived category:

N^{m}W Ω_{X/k}^{·} ≅ F^{m}Ω_{Y/W (k)} ≅ F il^{m}Ω_{Y/W (k)}

But since m > n the map F_{m} is identified with the linearization of ⊕Φ_{i}. This says that the last map is a Frobenius linear isomorphism. Q.E.D. Remark: The proof shows that H_{DR}^{n}(Y) with its Hodge filtration is strongly divisible (compare [Fo] 1.2 Prop.) for n < p − 2. If we knew that N^{m}W Ω_{X/k}^{·} and F^{m}Ω_{Y/W (k)} are quasi-isomorphic, the last argument would imply that H_{DR}^{n}(Y) is strongly divisible without restriction on n. We note also that the last argument works directly over any reduced ring k.

Corollary 5.6 Let X be a smooth and projective scheme over a ring R such that p is nilpotent in R. Let us assume that there is a frame A → R and a smooth and projective p-adic lifting Y/ Spf A of X, which satisfies the conditions of the theorem. Then we obtain for n < p by base change a display structure of degree n on H_{crys}^{n}(X/W (R), O_{X/W (R)}). This display structure is independent of the frame A and the formal lifting Y we have chosen if p · R = 0.

Proof: For a given frame A the independence of the lifting Y is clear, because the window structure is purely defined in terms of the crystalline cohomology of X/A. If we have a morphism of frames B → A and a formal lifting Z of X to B, then we set Y = Z_{A}. Then the window associated to Y is obtained from the window associated to Z by base change (one should think in terms of decompositions (53)). Therefore the induced displays are the same. If p · R = 0 and A′ and A′′ are 2 frames, we obtain a new frame A′ ×_{R} A′′ → R. Then σ′ × σ′′ is an endomorphism of A′ ×_{R} A′′ because σ′ and σ′′ induce the same endomorphism on R. If Y′/ Spf A′ and Y′′/ Spf A′′ are formal liftings, we obtain a formal lifting Y′ ×_{κ} Y′′ of X over A′ ×_{R} A′′. Therefore we obtain the same display structure by base change.

Theorem 5.7 Let R be a reduced ring of characteristic p. Let X/R be a smooth projective scheme. Assume that there is a compatible system of smooth and projective liftings Y_{n}/W_{n}(R). We assume that the assumptions 5.2 and 5.3 are satisfied with A_{n} = W_{n}(R) Then there is a display structure on H_{crys}^{n}(X/W (R), O_{X/W (R)}) for n < p, where

P_{m} = ℍ^{n}(X, N^{m}W Ω_{X/R}) = H_{crys}^{n}(X/W (R), J_{X/W (R)}^{[m]}).

Proof: The second equality is the filtered comparison theorem. If we had a p-adic lifting Y/ Spf W (R), the theorem would follow from the last one because W (R) → R is a frame. The slightly more general statement follows by the same reasoning as the last theorem. Q.E.D. We make the following conjecture:

Conjecture 5.8 Let R be a ring such that p is nilpotent in R. Let X/R be a smooth projective scheme. Let us assume that the crystalline cohomology groups H_{crys}^{i}(X/W_{n}(R)) are locally free W_{n}(R)-modules for i ≥ 0 and n > 1, and that the de Rham spectral sequence

E_{1}^{p,q} = H^{q}(X, Ω_{X/R}^{p}) ⇒ ℍ^{p+q}(X, Ω_{X/R}^{·})

degenerates. Then the canonical predisplay structure on P_{m} = ℍ^{n}(X, N^{m}W Ω_{X/R}) is a display structure.

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Andreas Langer Mathematics University of Exeter Exeter, EX4 4QE Devon, UK a.langer@ex.ac.uk Thomas Zink Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld zink@math.uni-bielefeld.de 192

Documenta Mathematica 12 (2007)