Documenta Math. 147 De Rham-Witt Cohomology and Displays

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).


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: 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 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 ιi → P i → . . .→ P 1 ι0 → P 0 .
2) For each i ≥ 0 a W (R)-linear map 3) For each i ≥ 0 a Frobenius linear map The composition of ι and α is the multiplication I R ⊗ P i → P i .Moreover we have the equation: 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: 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: 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 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: 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 −1 0 P 0 ⊂ P 0 ⊗ Q.By the theory of elementary divisors we find a decomposition by W (R)-modules 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 m crys (X/W (R)) has a display structure.Therefore we consider the following assumptions: There is a compatible system of smooth liftings Xn /W n (R) for n ∈ N of X/R such that the following properties hold: (*) The cohomology groups H j ( Xn , Ω i Xn/Wn(R) ) 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 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 m crys (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 m crys (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 m crys (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 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/Wn(R) .Let us denote by J X/Wn(R) ⊂ O X/Wn(R) the sheaf of pd-ideals.We denote by J 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 : 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: The filtered comparison Theorem 4.6 says that for m < p we have an isomorphism in the derived category We would like to have a similar comparison theorem for the truncated Nygaard complex The advantage of the Nygaard complex is that the restriction of the Frobenius from 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 X/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 • X/Wn(R) : Assume that we have for varying n a compatible system of smooth liftings Xn /W n (R).We obtain a formal scheme X = lim −→ Xn .We set: 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 : Moreover we can show in support of our conjecture, that the complexes 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: This is what strong divisibility asserts.Documenta Mathematica 12 (2007) 147-191 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 Φ: Definition 2.1 A predisplay over R consists of the following data: 1) A chain of morphisms of W (R)-modules 2) For each i ≥ 0 a W (R)-linear map 3) For each i ≥ 0 a Frobenius linear map 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.
For i = 0 the following map is the multiplication: We will denote a predisplay as follows: 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: 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 : ) be the sheaf of ideals.We define three sets of maps: (5) These maps are given in this order by the maps between the following vertically written procomplexes (the index n is omitted): The first unlabeled arrows on the left hand side denote the maps V ξ ⊗ω → ξω, where the product is taken in W Ω i X/S (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: The predisplay structure on the P i is easily obtained by taking the cohomology of the maps (5).
Here we write N 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 αi and ιi are in this case the maps F 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 ) 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: Indeed, let y ∈ P i+1 .Then we obtain from (D1) that If we apply F i+1 to the last equation and use (D2), we obtain: 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 sep 0 = P 0 and P sep i+1 is the image of P i+1 in the fibre product of: 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: Here the maps α pick up the last factor of I ⊗ .The following map is called the "Verjüngung": For a separated display the iteration (7) factors through the Verjüngung: The last arrow will be called α i .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: 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: 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 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 be a predisplay.Then we define its Tate-twist as follows: For i ≥ 1 we set 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: The chain of the maps ι is as follows: where the last map ι 0 is the natural inclusion.
The maps F i : 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: We obtain maps 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 of Frobenius linear maps for i = 0, . . .d: We require that the map From these data we obtain a predisplay in the following manner: We set: 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: The restriction of F i to I ⊗ L k for k < i is Φk , 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: The map α i : I ⊗ P i → P i+1 is given by the following diagram: 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": 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": The morphisms F i for i ≥ 0 induce Frobenius linear morphisms: These are Frobenius linear isomorphisms of R-modules.Indeed, if we choose a normal decomposition {L i } we obtain identification: 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 : On the summand (I ⊗ L i−k ) the map τ i is the composite: On the summand L i+j the map τ i is the composite: 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: 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 • F i ) may be constructed as follows.We choose normal decompositions More precisely this means, that we fix isomorphisms of P resp.P ′ with standard displays.We obtain: 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 Then ⊕ l K l = P 0 ⊗ P ′ 0 , and we define Frobenius linear maps From the standard data K l , • Φ l we obtain a display 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 consists of the following data.λ is a sequence of maps of W (R)-modules We require that the following diagrams are commutative: Remark: We will consider also the maps 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: We will denote the set of bilinear forms of displays in this sense by 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: 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 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 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 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 Thus we are in the case Z 3 with factor p n−i = 1.Again the submodule is in the image of 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 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 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 But the diagram is commutative by the definition of a bilinear form.Now the commutativity of ( 19) gives a map: 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) does not depend up to canonical isomorphism on the normal decompositions of P and P ′ .We write 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: Composing this with the iteration of ι, Q i+j → Q 0 , we obtain a bilinear form Let us assume that the following diagrams are commutative: Then the β ij extend uniquely to a bilinear form 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 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 such that the following diagrams are commutative 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 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: 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 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 Assume we have already constructed W (S)-module homomorphisms which are compatible with F, ι and α for j ≤ i.

Consider the diagram
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.

Documenta Mathematica 12 (2007) 147-191
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 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 : ).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: 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: 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 splits into short exact sequences: 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 are injective for each q and m, as one may see easily from the exact cohomology sequence.This injectivity may be restated as follows: 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 Then the spectral sequence 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:

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: If we apply the induction assumption to I • we obtain an exact sequence for each q and the given m.
The morphism of complexes ( 25) induces a commutative diagram: 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.
We have seen this for n = m.For n > m we have to consider the maps.
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 is surjective.But this map factors as: 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 Then for each integer m the corresponding spectral sequence of hypercohomology associated to the complex

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

Documenta Mathematica 12 (2007) 147-191
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 It is shown in [LZ] 3.2 and 3.3, that even if X 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: The aim of this section is to prove a filtered version of this isomorphism.
For typographical reasons we use the abbreviations: Let us denote by 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): 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 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 is for any m ≥ 0 a quasi-isomorphism.
Proof: We use the W n (R)-basis of Ω l Ã/Wn(R) given by p-basic differential forms.For each weight function k : [1, d] → Z ≥0 we fix an order on the set For any ascending partition of Supp k into disjoint intervals 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 , 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 The map F looks as follows V ξ ⊗ ẽ(k, P) −→ e n−1 (ξ, pk, P).

Documenta Mathematica 12 (2007) 147-191
For each weight k : , 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): -submodule generated by the p-basic differentials ẽ(k, P) of fixed integral weight k.Then F m Ω• is the direct sum of the following subcomplexes F m (k): 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 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 We denote δ n : 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 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: 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 Bn/Wn(R) is zero.Since B m+n /A m+n is étale, the extension dψ m is zero too.Consider the commutative diagram It induces a morphism of algebras which are étale over A n : 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 ⊗ An A/pA.But then we obtain the well-known isomorphism From the isomorphism (36) we deduce an isomorphism 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 We consider it as a complex of W n+m (A)-modules via . 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)⊗ Wn+m(A),F m .Because of the obvious isomorphism ( [LZ] (3.2)) the result is the same if we tensorize (38) by 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.
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 l X/R and define I m W n Ω X/R starting in degree 0.
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 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 which makes the following diagram commutative Proof of the lemma: Let n > k ≥ i.For ξ ∈ W n (R) we denote by ξ its restriction to W n−k (R).Then we have a well-defined map Indeed, write ξ = (x 0 , . . ., x n−1 ).Then Then we obtain a morphism of W n+1 (R)-modules because of the following commutative diagram 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 l A/R may be expressed, in terms of basic Witt-differentials: 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 Using loc.cit.we find: 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: 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 coincides with p i V W n Ω B/R .This follows from the following commutative diagram 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 The composition of the last map with p i V : is just the restriction.This proves the lemma.
Q.E.D. The proposition follows immediately because we obtain an inverse to the map (40): The first m vertical maps defined by the lemma are equivariant with respect to 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/Wn(R) ⊂ O X/Wn(R) the sheaf of pd-ideals.We denote by J [m] X/Wn(R) its m-th divided power.Let 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: 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 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 ΩD/Wn(R) has the following subcomplex F il m ΩD/Wn(R) : Then the left hand side of ( 45) is isomorphic to the hypercohomology of ( 46).The Witt-lift defines a morphism It maps the ideal sheaf of X ⊂ Ỹ to the ideal sheaf . Since I X is endowed with divided powers, we obtain mapping I D to I X .The homomorphism (47) induces a map of the pd-de Rham complexes Taking into account that I [h] X = p h−1 I X for h < p, we obtain the desired morphism from (46) to the complex 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 ⊗ An+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 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 m B/R 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 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 • Xn/Wn(R) } n∈N .Passing to the projective limit we obtain: A window is called standard if it arises in the following way.Let L 0 , . . ., L d be finitely generated projective A-modules.Let 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 It is clear that (P i , F i ) form a window.Each window is isomorphic to a standard window.Indeed, let E 0 = ⊕ Lj be a splitting of the filtration (52) in the definition: Let L i be a finitely generated projective A-module which lifts Li .We find homomorphisms L i → P i which make the following diagrams commutative: It follows from the lemma of Nakayama that ⊕L i → P 0 is an isomorphism, since it is modulo a.By induction we obtain 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 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 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 αn → R. The kernel a n a pd-structure.We consider the crystalline topos (X/A) crys .Let J X/An ⊂ O X/An be the pd-ideal sheaf.We are interested in the cohomology groups: Remark: It would be more accurate to consider the cohomology groups of R lim X/An ).But under the Assumptions 5.2 and 5.3 we are going to make these groups will coincide.
X/An ) are the hypercohomology groups of the following complex F il We will make the following assumptions: Assumption 5.2 The cohomology groups H q (Y n , Ω p Yn/An ) are for each n locally free A n -modules of finite type.
Assumption 5.3 The de Rham spectral sequence degenerates at E 1 Since Y n is quasicompact and separated by assumption the cohomology sheafs R m f n * Ω • Yn/An are quasicoherent.From the assumption we see that these sheaves are locally free of finite type.Hence the complex Rf n * Ω • Yn/An is quasiisomorphic to the direct sum of its cohomology groups.This implies that the cohomology groups R m f n * Ω • Yn/An commute with arbitrary base change.The same applies to the cohomology groups R q f n * Ω p Yn/An .By Proposition 3.2 and the projection formula (Proposition 3.1) we obtain a degenerating spectral sequence The image of this map lies in p [m] • Ω• D/An = p [m] A n ⊗ L An ΩD/An .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 But the cohomology of the last complex is p [m] H i (Y n , Ω • Yn/An ), 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/An ) = p [m] H i (Y n , Ω • Yn/An ).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 are injective.The A-modules P m = H n crys (X/A, J [m] X/A ) for m ≤ n together with the maps 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 m X/A = J [m] X/A , a m = a [m] .We write F il [m] Yn/An .Then we find a canonical isomorphism 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

[m]
Y/A will be the complex F il m Consider the following morphism: 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 This equality holds for m = 0 by the projection formula.Indeed, consider the canonical map: F il m Ω Y/A → a m Ω 0 Y/A → 0. The kernel is the following complex C: 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 0 Y/A ) = a m+1 H n (X, Ω 0 Y/A ).
The assertion (64) follows from the diagram 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 Therefore we obtain an exact sequence 0 → H n (X, aF 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 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 with the convention that a k = A if k ≤ 0.
Concretely we can find the liftings L i as follows.We consider the maps: Then L m is obtained by splitting the last surjection.This construction gives isomorphisms: L m ∼ = H (n−m) (X, Ω m Y/A ) 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 The assertion that {P m } is a window is then equivalent with the condition that ⊕ n i=0 Φ i : ⊕ n i=0 L i → ⊕ u i=0 L i is a σ-linear isomorphism, or in other words that det(⊕ n i=0 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 is a nonzero.The compositum map A → W (k) respects the Frobenius and induces a map on crystalline cohomology which respects the Frobenius.It is induced by the base change map for de Rham cohomology.