ON A COMPARISON BETWEEN DWORK AND RIGID COHOMOLOGIES OF PROJECTIVE COMPLEMENTS

Abstract For homogeneous polynomials 
$G_1,\ldots ,G_k$
 over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork’s theory. In this article, we will construct an explicit cochain map from the Dwork complex of 
$G_1,\ldots ,G_k$
 to the Monsky–Washnitzer complex associated with some affine bundle over the complement 
$\mathbb {P}^n\setminus X_G$
 of the common zero 
$X_G$
 of 
$G_1,\ldots ,G_k$
 , which computes the rigid cohomology of 
$\mathbb {P}^n\setminus X_G$
 . We verify that this cochain map realizes the rigid cohomology of 
$\mathbb {P}^n\setminus X_G$
 as a direct summand of the Dwork cohomology of 
$G_1,\ldots ,G_k$
 . We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz’s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.


§1. Introduction
Let X be an algebraic variety over a finite field F q of characteristic p > 0. The zeta function of X is defined to be the following exponential sum: where N s is the number of F q s -rational points of X.This function is known to be a rational function in t with coefficients in Z by Dwork [13].For a projective hypersurface X, Dwork expressed the zeta function of X as an alternating product of characteristic polynomials of a suitably chosen representative of a Frobenius action in a series of articles [14]- [17], following his proof of the rationality of zeta functions.Based on Dwork's theory, Adolphson and Sperber developed a cohomology theory and got an estimate for the zeta function when X is a closed subvariety of A r × G s m in [1], and when X is a smooth projective complete intersection in [4], [5].
On the other hand, Monsky and Washnitzer developed rather an intrinsic cohomology theory in [29] when X is a smooth affine variety admitting a nice p-adic lift.Then Monsky proved the Lefschetz fixed-point theorem in [25], [27] to express the zeta function of X as an alternating product of characteristic polynomials of a Frobenius action on Monsky-Washnitzer cohomology.Later, van der Put [32] removed the technical condition on X assumed by Monsky and Washnitzer to make the theory work for every smooth affine variety X over F q .Berthelot [9] extended this theory to not necessarily affine varieties, and the resulting theory is called rigid cohomology theory.
Since Dwork cohomology and rigid cohomology compute the same important invariant, one may ask whether there is a connection between the two theories.For smooth hypersurfaces in projective spaces, Katz answered this question in [19].It is strongly believed that the corresponding comparison results hold for more general cases, but up to the best of author's knowledge, there is no written proof so the author hope that this article provides a detailed proof for general cases with several equations.
Let us briefly explain the contents of [19].Let k/Q p be a finite extension with the valuation ring O k .Given a homogeneous polynomial G ∈ O k [x 0 , . . ., x n ] of degree d ≥ 1, consider a k-linear span of some monomials (cf.[19, p. 77]): For a fixed constant γ ∈ k, there are differential operators on L 0,+ : On the other hand, suppose that the hypersurface X G ⊆ P n k defined by G is smooth.If H i ⊆ P n k is the hyperplane defined by x i = 0 for i = 0, . . ., n, and [19,Th. 1.16], there is an exact sequence 0 / / D y L 0,+ + n i=0 D x i L 0,+ / / L 0,+ Θ / / H n dR (X ∅ G ) / / 0, where the local description of Θ is given in [19,Th. I].Here, H • dR denotes algebraic de Rham cohomology.One way of getting a global description of Θ is using the complement of X G .Namely, denote T n k := P n k \ (H 0 ∪ • • • ∪ H n ) with local coordinates t i := x i /x 0 for i = 1, . . ., n.Then there is a k-linear map given by (cf.[19, p. 78]) inducing an exact sequence (cf.[19, p. 79]) Here, the map is defined via the inhomogeneous coordinates of P n k \ H 0 .One gets a description in the homogeneous coordinates using the relation: To relate R and Θ, we use the canonical exact sequence: where Res G is the residue map uniquely characterized by the property Then, by [19,Th. 1.18], Res G • R = Θ.The remaining part of [19] is dedicated to compute representatives of Frobenius actions.To achieve this, we need to develop a p-adic analytic theory.Then Θ and R extend by continuity, and they are compatible with the Frobenius actions in a suitable sense.Since we discuss the corresponding version of the p-adic analytic theory in this article, we do not explain the remaining part of [19].
Monsky's lecture note [26] gave a more detailed discussion of the algebraic version of the Dwork complex in p-adic setting together with its relations with algebraic de Rham cohomology and Monsky-Washnitzer cohomology.Then the complex algebraic analog of Dwork theory together with the connection of de Rham cohomology has been studied.Adolphson and Sperber dealt with the smooth complete intersections in affine varieties in [3].Dimca, Maaref, Sabbah, and Saito studied the singular subvarieties embedded in smooth varieties in [12] using the theory of algebraic D-modules.These results were again implemented in the rigid setting by Baldassarri and Berthelot for singular projective hypersurfaces in [7] using the theory of arithmetic D-modules.On the other hand, Bourgeois [11] directly constructed a quasi-isomorphism between the Dwork complex used by Adolphson and Sperber in [1] and the complex of Monsky and Washnitzer in the smooth affine setting.
The goal of this article is to construct an explicit comparison between the Dwork cohomology of given homogeneous polynomials and the rigid cohomology of the complement of their common zero in a projective space, together with Frobenius actions defined on both sides.This generalizes the complement comparison result in [19] described above, but with a different choice of cochain complexes.Note that if the given homogeneous polynomials define a smooth complete intersection, then we can recover the essential information of the rigid cohomology of the common zero.The more detailed exposition will be given in the following two subsections.
As mentioned before, Adolphson and Sperber studied Dwork complexes in various settings, and it seems that the Dwork complex which appears in this article resembles the one in [5].Our academic contribution is to find a correct version of the p-adic Dwork complex which is appropriate to construct the desired comparison map, and give a systematic treatment of getting a connection between the two theories via the Cayley trick 1 as the author did in [31,22] to study the period integrals in the complex geometric setting.

The idea and motivation
One remarkable observation so far is that the comparison becomes more transparent when we consider the complement of the hypersurface X in the ambient projective space P n F q .Moreover, we may extract information of , where H • rig denotes rigid cohomology.Indeed, if X ⊆ Y is a codimension k closed embedding of smooth varieties 1 The Cayley trick gives an isomorphism between the cohomology of the open complement in the projective space and the cohomology of the hypersurface complement in a larger space.For the detail, see §2. over F q , then there is a commutative diagram with exact rows: where the top row is a special case of the excision exact sequence [10, Prop.2.5], and the isomorphisms in the columns come from the Gysin isomorphism [23, §9.3].Therefore, if X G ⊆ P n F q is a smooth projective complete intersection given by homogeneous polynomials G 1 , . . ., G k ∈ F q [x 0 , . . ., x n ], then there is a long exact sequence, called the Gysin exact sequence: which is a rigid cohomology analog of the excision exact sequence of algebraic de Rham cohomology.As in the case of algebraic de Rham cohomology, this sequence induces an isomorphism where Using the interpretation of the zeta function as the characteristic polynomial of the Frobenius action on the cohomology (see, e.g., [18]), one can deduce that the zeta function of X G can be written as and P (t) is completely determined by the Frobenius action on the primitive part.Hence, the computation of the cohomology of the projective complement has its own importance.Once we decide to focus on the cohomology of the complement, we may forget about the regularity of X G ⊆ P n F q because P n F q \ X G is always smooth, being an open subset of the smooth space P n F q .On the other hand, the Dwork complex can be defined for any homogeneous polynomials G 1 , . . ., G k ∈ F q [x 0 , . . ., x n ], regardless of the regularity of their common zero X G ⊆ P n F q .Namely, taking the Teichmüller lifts of the coefficients of each G i , we get homogeneous polynomials G i defined over some finite extension k/Q p with deg G i = degG i such that the reduction of each G i becomes G i .Then, we define the Dwork complex associated with G 1 , . . ., G k to be the twisted de Rham complex of the form where k{x, y} is the Tate algebra over k (see Definition 4.8), and , γ ∈ k × are some parameters.
Although the Dwork complex is defined for homogeneous polynomials, its cohomology would depend only on their common zero locus.For example, when we are working with one homogeneous polynomial G ∈ F q [x 0 , . . ., x n ], there are comparison theorems between the Dwork cohomology of G and the rigid cohomology of P n F q \ X G .In the existing results, [19] and [7], they remove the hyperplane divisors in P n F q defined by x 0 , . . ., x n to get an affine open subset, where one can write down a comparison map, and then use the log de Rham complex to recover the original situation.Consequently, their Dwork complexes are exactly the ones defined by Adolphson and Sperber in [2, §2].
Instead of removing hyperplane divisors in P n F q , we use the Cayley trick to convert the computation involving polynomials to the computation involving a hypersurface contained in a larger space.With the above notation, the hypersurface is cut out by k for a suitably chosen locally free O P n k -module E of finite rank, where y 1 , . . ., y k play the role of fiber coordinates.Consequently, we get the Dwork complex as in (1.2) which resembles Adolphson and Sperber's Dwork complex defined in [5,  §2].The difference of our Dwork complex and the one in [5] comes from the different choice of Dwork's splitting functions (for a definition, see §5), which causes the different choice of the lift of x n ] over the p-adic field.Since the lift of Adolphson and Sperber, denoted by H in [5, eq.(2.10)], is a power series in y 1 , . . ., y k , it does not define a hyperplane in P(E).Hence, we cannot get the desired geometric object.However, our lift . ., y k so it indeed define a hypersurface in P(E).Although the two Dwork complexes are different, their reductions on the finite field are exactly the same so one may expect that both Dwork complexes have the same cohomology.This is true when G 1 , . . ., G k defines a smooth projective complete intersection in P n F q (see Remark 4.13).Hence, the two Dwork complexes may be regarded as equivalent at least for this case.

The main results
Let k/Q p be a finite extension with the valuation ring (O k , m k ).Denote val p the p-adic valuation such that val p (p) = 1.Given homogeneous polynomials G 1 , . . ., G k ∈ O k [x 0 , . . ., x n ] of positive degrees d 1 , . . ., d k not divisible by the uniformizer of O k , we introduce formal variables y 1 , . . ., y k corresponding to G 1 , . . ., G k so that the polynomial defines a hypersurface in an affine space.Consider the twisted de Rham complex where , γ ∈ O k are regarded as parameters.Introduce gradings so that S and dS become homogeneous of bidegree (deg c , deg w ) = (0, 1) and the twisted de Rham complex is graded with respect to deg c .Then the Dwork complex associated with G 1 , . . ., G k will be defined by the module of m k -separated differentials.Then the above gradings extend to defined by the formula together with the k-linearity.Here, u, v, α, β are multi-indexes with (1.4) We will see later that the inclusion induces a surjection of cohomology spaces with one-dimensional kernel generated by the class [dS].By Definition 4.6, where F q is the residue field of O k , and is the reduction of S. Since Monsky-Washnitzer cohomology is canonically isomorphic to rigid cohomology for smooth affine schemes, the ρ S in Theorem 1.1 is a comparison map from Dwork cohomology to rigid cohomology.
On the other hand, if X G ⊆ P n k is the common zero of G 1 , . . ., G k , then we will see in §2 that there is a canonical map Spec k[x, y, S −1 inducing a quasi-isomorphism on rigid cohomology spaces.Moreover, by Corollary 4.7, the Monsky-Washnitzer cohomology associated with the bidegree (0, 0)-subalgebra above is computed via the complex of m k -adically separated forms of C † S .The corresponding statement for algebraic de Rham cohomology is Proposition 2.4.This is a direct generalization of [26, Th. 9.2] which covers the case of projective hypersurface complement.With the notations so far, we can say more about the comparison map ρ S .
Theorem 1.2.ρ S induces an isomorphism for every i ≥ 2. On the other hand, The q-power map induces an endomorphism, called the Frobenius endomorphism on F q [x, y].This map lifts to endomorphisms both act on the zero forms by sending x i and y j to its qth power x q i and y q j , respectively.These endomorphisms admit retractions, that is, endomorphisms such that Ψ q,S • Φ q,S and ψ • Fr are the identity maps, respectively.The detailed expositions will be given in §5.Now, we have the following comparison of the endomorphisms above.
Theorem 1.3.ρ S is compatible with the Frobenius and the Dwork operators defined on the source and the target, respectively.More precisely, the diagrams Remark 1.4.In Theorem 1.3, we dropped the subscript (deg c = 0, deg w = 0) in the target of ρ S because we are not sure that an arbitrary lift Fr of the q-power map preserves the bidegree (0, 0)-subcomplex.However, the particular choice such that and the ψ coming from this choice are compatible with the bidegrees so we can recover the subscript (deg c = 0, deg w = 0) in Theorem 1.3.See § §5.1 and 5.2 for the details.We will see in Theorem 4.5 that any lifts of the q-power map define homotopic cochain maps so we can always make such choices.
We have the following remark concerning formal deformation theory of the Dwork operator, which is not covered in the rest of this article.
Remark 1.5.Using the twisted de Rham complex in Theorem 1.1, we may directly construct a DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra with the isomorphic underlying complex, as the authors of [20] did on the complex geometry setting, and we may develop the formal deformation theory as in [31].Theorem 1.3 enables us to apply this type of formal deformation theory to the Dwork operator.For the detailed discussion in the DGBV aspects of Dwork theory, see [21].

Outline of the article
In §2, we explain the Cayley trick.In particular, Proposition 2.4 gives a direct sum decomposition of the algebraic de Rham complex of the affine cone and the corresponding decomposition of the algebraic de Rham cohomology.This identification is used in the rest of the article.
In §3, we explicitly write down a comparison map ρ S (Definition 3.1) in a corresponding algebraic setting.The comparison for this algebraic ρ S will be given in Propositions 3.10 and 3.16.
After establishing the algebraic theory, we will define the required p-adic analytic complexes in §4 and give a proof of Theorems 1.1 and 1.2.In §4.1, we recall the basics on Monsky-Washnitzer cohomology.In particular, Proposition 4.7 is the Monsky-Washnitzer version of Proposition 2.4.This gives the target complex of the ρ S in Theorem 1.1.In §4.2, we recall the basics on Dwork complexes and introduce the source complex of the ρ S in Theorem 1.1.Now, the main results of §3 yield Theorem 4.11 which is the combination of Theorems 1.1 and 1.2.
The appendix A is an explanation of the computation of algebraic de Rham cohomology via the cosimplicial algebra coming from Čech covering of affine open subsets which is in the proof of Proposition 2.4.Up to the best of author's knowledge, the suitable reference for this simplest case is not available in the literature so the appendix is added for this article to be more self-contained.§2.The Cayley trick In this section, we give a detailed explanation of the Cayley trick and its consequences.We begin with motivation.Let k be a field and X ⊆ P n k a smooth projective complete intersection of codimension k.For a "reasonable" cohomology theory H • defined for quasiprojective schemes over k, one may obtain the Gysin exact sequence of the following form: The particular cases we consider are: (1) k is a field of characteristic zero, and (2) k = F q is a finite field, and Case ( 2) is mentioned in the introduction (1.1) briefly.For (1), see [30, §3.1] for example.
In particular, in the cases ( 1) and ( 2) above, by the weak Lefschetz property and Poincaré duality.Denote in this situation Then Res X induces an isomorphism: Therefore, we may focus on the cohomology of the complement P n k \ X.Because we decided to consider the complements, we may assume that X = X G ⊆ P n k is defined by any finite set of homogeneous polynomials G 1 , . . ., G k ∈ k[x 0 , . . ., x n ] of positive degrees d 1 , . . ., d k , respectively.The Cayley trick is a method of translating the computation of H • (P n k \ X G ) to a computation of the cohomology of the complement of a hypersurface living in a larger space.This larger space is simply given by the projective bundle Another way of describing P(E) comes from the toric geometry, via the geometric quotient: where the G m × G m -action is given by Here, the new variables y 1 , . . ., y k correspond to

and the action above explains the gradings (1.3). Moreover, S := y
-bundle.In this setting, if an abstract cohomology theory H • satisfies the Künneth formula, and which is true for the cases (1) and ( 2) above.In this section, we focus on In what follows, we denote In the rest of this section, we will describe the algebraic de Rham cohomology of A using the algebraic de Rham cohomology of C S .Note that SpecA is smooth over k, being an open subset of a smooth k-scheme P(E), and Spec B is smooth over k, being an open subset of We will see in Proposition 2.4 that these induce split injections of cohomology spaces.Since the de Rham differential preserves the bidegree (deg c , deg w ), the inclusion from Ω , where the bidegree (0, 0) part of Ω • C S /k is the k-linear span of differential forms , where u, v, α, β are multi-indexes following convention (1.4) such that This explains the gradings (1.3) in the introduction.Then each grading has the corresponding Euler vector field: respectively.Denote the contraction with each Euler vector field.
Lemma 2.2.θ c and θ w above have the following properties.

Proof. The results follow from direct computations.
There are several basic but important consequences of Lemma 2.2.Lemma 2.3.With the notations above, the following hold.
(1) All inclusions in the following commutative square are quasi-isomorphisms: (2) There are cochain maps induced from θ c and θ w , respectively: (3) We can identify and there is a cochain map Moreover, the following relations hold: .
(2) The above relations also show that each θ becomes a cochain map on the subcomplex of homogeneous elements of degree zero.
Proposition 2.4.With the notations above, there is a decomposition of complexes and for every i ∈ Z, an isomorphism Consequently, there is an isomorphism for every i ∈ Z: then we may rewrite ξ as Since θ w ξ = 0 and θ w ω = 0, Therefore, (2.4) becomes a direct sum decomposition: Since the restriction of θ w on the subspace of deg w = 0 induces a cochain map by Lemma 2.3, we get a direct sum as a complex: To compute the direct summand, consider the open subsets for j = 1, . . ., k where X y j G j is the zero locus of y j G j .These open subsets form an affine open covering of P(E) \ X S .On each U j , there is a section of θ c given by .
They combine to give a section of the associated Čech-de Rham complex by Proposition A.6 and Example A.7. Since Ω . Now, the rest part of the proposition follows from combining the two observations so far.

§3. Cayley trick and twisted de Rham complexes
In this section, we develop the algebraic de Rham version of Theorems 1.1 and 1.2.This section is a generalization of Monsky's lecture note [26,Ch. 9].We continue with the notation of §2.For fixed , γ ∈ k × which we regard as formal parameters, consider the twisted de Rham complex over k[x, y]: equipped with the gradings as in (1.3).Then the corresponding comparison map is given as follows.
Definition 3.1.Define the map by the formula together with the k-linearity.
Under this map, we will obtain comparison results Propositions 3.10 and 3.16.These will be properly completed to give Theorems 1.1 and 1.2.We saw in Lemma 2.3 that the target complex of ρ S in Definition 3.1 computes the algebraic de Rham cohomology of C S = k[x, y, S −1 ].On the other hand, the following lemma explains the effect of taking the subcomplex.

Lemma 3.2. The inclusion
is a quasi-isomorphism.On the other hand, the inclusion as a Z-graded k-vector space so that .
Also, we will often abbreviate the subscripts (1) ρ S is a k-linear cochain map.
(2) ρ S commutes with θ c and θ w .Here, θ w is regarded as a degree −1 map of Z-graded k-vector spaces.
(3) If ξ 1 and ξ 2 are deg w -homogeneous of positive degree, then In particular, if ξ is deg w -homogeneous of positive degree, then for every integer i > 0.
Proof.The results follow from direct computation.
To describe the kernel of ρ S , we introduce the following auxiliary map: Definition 3.6.Define the map By Lemma 2.2, if ξ is homogeneous with respect to deg w , then w,S (ξ) = ( deg w ξ + γS)ξ.
Remark 3.7.This map corresponds to the one in [26, Lem.9.1] which is defined via the congruence condition on the degree of a defining hypersurface.However, we are working reversely via the Cayley trick.In our context, [26, Lem.9.1] becomes Definition 3.6, and Monsky's definition follows from Lemma 2.2.(1) w,S is a k-linear cochain map.
(3) w,S restricts to the bidegree (deg c = 0, deg w > 0)-subcomplex: Proof.(3) We proceed by induction on the power i.The case i = 1 follows immediately from construction.If i > 1, then where the third line follows from induction hypothesis.
Proposition 3.10.With notation 3.4, there is an exact sequence of cochain complexes Consequently, there is an exact sequence for every i ∈ Z, and in particular, Proof.Since the target of ρ S admits a k-basis which is in the image of ρ S , the surjectivity of ρ S follows.The injectivity of w,S follows from Lemma 3.9. .
Define the map of graded k-vector spaces we see that σ is well-defined.Note that this forces σ(1) = γS.By construction, ρ S σ is the identity.Since coker w,S is spanned over k by S i x u y v dx α ∧ dy β with i ≥ 0 and |v| + |β| > 0, and is in the image of σ, we conclude that σ is surjective.Hence, ρ S and σ are mutually inverses.Therefore, we achieve the desired exactness.
For the second part, take the cohomology long exact sequence.Since w,S is homotopic to zero by definition of w,S and Lemma 3.9, we get the desired exact sequences.In particular, the long exact sequence begins with Hence, we get the desired vanishing and the δ becomes an isomorphism Since Spec C S = Speck[x, y, S −1 ] is connected, the right-hand side is one-dimensional with a basis [dS] coming from 3.2.
Corollary 3.11.The cohomology groups of the twisted de Rham complex are finite-dimensional k-vector spaces for every i ∈ Z.In particular, Proof.By Lemma 3.2, we may compute the cohomology of the twisted de Rham complex by using L • (0,+) , D ,γS .Hence, the results follow from Proposition 3.10 and the finiteness of the algebraic de Rham cohomology of smooth k-algebras [28, Th. 3.1].
To describe the image of ρ S on the cohomology spaces, we introduce the following auxiliary map.Definition 3.12.Define the k-linear map as follows: If ξ is a deg w ξ-homogeneous of positive degree, then denote and define χ(ξ) := χ ξ • ξ on deg w ξ-homogeneous elements.
Remark 3.13.This map corresponds to the one in [26, p. 110].The additional grading deg w coming from the Cayley trick replaces the role of the congruence condition on the degree of a defining hypersurface in Monsky's definition.Lemma 3.14.As a cochain map, the following hold.
Proof.If ξ is a deg w -homogeneous element, then so the lemma follows.
Lemma 3.15.With Notations 2.1 and 3.4, the square commutes up to homotopy where we use the identification coming from the decomposition in Proposition 2.4.
Proposition 3.16.In the exact sequence as in Proposition 3.10 for i ∈ Z: Here, we use the identification of Proposition 2.4.Consequently, ρ S induces for every i ≥ 2.
Proof.Suppose that ξ ∈ Ω i+1 k[x,y]/k is a D ,γS -closed form.If we take Since ω is in the image of ρ S , it represents a class in H i (Ω • B/k , d).Now, θ w ξ is a lift of ω along ρ S and, since D ,γS ξ = 0, we have Therefore, by the construction of connecting map δ, by the surjectivity of δ observed above.Hence, but this implies ξ = 0 by Lemma 3.15.Hence, ρ S is surjective as well, that is, it is an isomorphism.By the identification of Proposition 2.4, this implies that δ is an isomorphism as well.The last assertion follows from Proposition 2.4 together with Lemma 3.

§4. p-adic cohomology and Cayley trick
In this section, we will prove Theorems 1.1 and 1.2, by constructing p-adic models of the complexes studied in §2 and §3, respectively.From now on, k will be a finite extension of Q p with the valuation ring (O k , m k ) and the residue field F q .Also, we keep the notation in §2 and §3, but we assume that G 1 , . . ., G k belong to O k [x 0 , . . ., x n ] and their reductions G 1 , . . ., G k are nonzero in F q [x 0 , . . ., x n ].

Monsky-Washnitzer cohomology
In this subsection, we briefly review the theory of Monsky-Washnitzer cohomology, which gives a p-adic model of algebraic de Rham cohomology studied in §2.Using this, we translate Proposition 2.4 into Monsky-Washnitzer setting and get the corresponding results in Proposition 4.7.
Then a weakly complete finitely generated algebra over O k is a homomorphic image of some overconvergent power series ring.
Theorem 4.5.Given a smooth F q -algebra A, there is always a lift A of A. Moreover, the following hold.
(1) Every lift of A is isomorphic to A as an O k -algebra.
(2) Let B be a smooth F q -algebra with a lift B. If ϕ : A → B is an F q -algebra map, then there is an O k -algebra map ϕ : A → B such that Proof.This is [32,Th. 2.4.4].
Definition 4.6.Let A be a smooth F q -algebra.Define where A is any lift of A given by Theorem 4.5.
Return to the situation of §2, but with the assumption that G 1 , . . ., G k belong to O k [x 0 , . . ., x n ] and their reductions G 1 , . . ., G k in F q [x 0 , . . ., x n ] are nonzero.As we observed in §2, there is an isomorphism coming from (2.3).Following Notations 2.1 and 3.4, we denote Then the w.c.f.g.O k -algebra satisfies Moreover, its subalgebras Proposition 4.7.With the notations above, there is a decomposition of complexes and for every i ∈ Z, an isomorphism . Consequently, there is an isomorphism for every i ∈ Z: [29, Th. 4.5]).Since θ c and θ w in §2 acts only on dx and dy, the proof of Proposition 2.4 works for overconvergent algebras to give the desired decomposition: To get the second assertion, consider the affine weak formal scheme (P(E) \ X S ) † in the sense of [24,Def. 15], that is, the topological space P(E) \ X S endowed with the structure sheaf associated with A † .Then the open subsets for j = 1, . . ., k give a covering {U † j } j=1,...,k of (P(E) \ X S ) † by principal open subsets associated with the w.c.f.g.O k -algebras (B † j ) deg c =0 where B † j := O † B j .From the vanishing of higher cohomology [24,Th. 14] of finitely generated modules on affine weak formal schemes, we deduce that the Čech-de Rham complexes of m k -separated differentials compute the Monsky-Washnitzer cohomology of the corresponding reduction.On the other hand, the section of θ c on each U † j , as in the proof of Proposition 2.4 still works.Since restriction to a principal open subset is given by tensoring with weakly completed principal localizations (cf.[24, p. 4]), the Čech-de Rham cosimplicial algebra for m k -separated differentials is 0-coskeletal as in algebraic de Rham case (Example A.7). Hence, we obtain Now, the rest part of the proposition follows from combining the two observations so far.

Dwork cohomology
In this subsection, we introduce the Dwork complex associated with G 1 , . . ., G k , which gives a p-adic model of twisted de Rham complexes studied in §3.Then, we extend the ρ S in Definition 3.1 to the Dwork complex in Proposition 4.11, which proves Theorems 1.1 and 1.2.
the ring of restricted power series over O k (in N variables), and the Tate algebra over k (in N variables).
Remark 4.9.Tate algebra can be written as Hence, given an N -tuple = ( 1 , . . ., N ) of positive real numbers, we denote We sometimes use notation k{ −1 z}.In terms of rigid geometry, this algebra corresponds to the closed polydisk of radius .If = |c| for some c ∈ k ⊕N , then where k{c −1 z} is the Tate algebra with respect to the variables c −1 Denote N := n + k + 1, and denote for ∈ k with val p > 0 so that C( ) ∼ = k{x, y}.Then the twisted de Rham complex of the form 3) is valid on our Dwork complex.
Notation 4.10.We will often denote as a Z-graded k-vector space so that Now, Theorems 1.1 and 1.2 follow from the following theorem.
Theorem 4.11.If val p γ ≤ 1 p − 1 and val p > 0, then the ρ S in Definition 3.1 extends continuously to p-adic analytic complexes, that is, there is a commutative square . Moreover, the extended ρ S induces an isomorphism for every i ≥ 2. On the other hand, Proof.To extend ρ S , we need to check the overconvergence of the expression.To see this, it suffices to show that there is some r > 1 such that lim From the degree condition, we have Since |α| ≤ n + 1 and |β| ≤ k are bounded by constants, we may ignore them so roughly |u| ∼ |v|d max for large |u|.On the other hand, we have Consequently, it suffices to take r such that Next, since w,S acts only on dx 0 , . . ., dx n , dy 1 , . . ., dy k , it extends to Dwork complexes.Then, we get a commutative diagram where the top row is exact by Proposition 3.10.Since the polynomial complexes are dense and the maps are all m k -adically continuous, the bottom row is exact as well.Since the relation in Lemma 3.9 holds on L • ,(0,+) by continuity, w,S becomes the zero map on the Dwork cohomology.Therefore, we get an exact sequence Moreover, since Lemma 3.2 applies to the inclusions we get the desired vanishing of H 1 (Ω • , D ,γS ).On the other hand, we have for every positive integer m so ρ S χ, ρD ,γS χ, and ρ S θ w χ in Lemmas 3.14 and 3.15, all converge as maps from L Therefore, the argument as in Proposition 3.16 works in p-adic setting to give isomorphisms Here, we use the identification of Proposition 4.7.Then ρ S induces an isomorphism for every i ≥ 2. Note that here we use the canonical isomorphism which exists because P(E) \ X S is smooth affine.Finally, there is an isomorphism as we have observed in (2.3).Therefore, the proposition follows.
The following corollary is a generalization of Monsky's remark in [26, p. 115].
Corollary 4.12.With the assumptions in Proposition 4.11, if X G ⊆ P n F q is a smooth complete intersection, then the inclusion Proof.By Lemma 3.2, it suffices to show that the inclusion is a quasi-isomorphism.By Propositions 3.16 and 4.11 together with its proof, this follows if we show that the inclusion and these isomorphisms are compatible with the Gysin sequences for H • dR and H • rig , we conclude that the above inclusion of algebraic de Rham complexes coming from Arguing as in the proof of Proposition 4.11, we see that all inclusions are quasi-isomorphisms.On the other hand, C( γ) admits a filtration which induces a ring isomorphism The filtration on C( γ) extends to given as follows: Then the above ring isomorphism extends to the isomorphism where we denote S ∈ F q [x, y] the reduction of S. By this observation, we may apply [5,Prop. A.2] to lift a basis for the cohomology of Ω • F q [x,y]/F q , dS ∧ − to get a basis for the cohomology of (Ω • , D ,γS ) whenever the cohomology over the residue field is finitedimensional.For the detailed computation over the residue field when G 1 , . . ., G k define a smooth projective complete intersection, see [4].

§5. Operators on p-adic analytic cohomologies
In this section, we will give more precise statement of Theorem 1.3 together with its detailed proof.This section is a generalization of [19,§III].We begin with reviewing some necessary constructions.For each i ≥ 1, the equation For each choice of γ i , the corresponding Dwork's splitting functions is defined to be Each θ i has integral coefficients and converges for In this section, we will take γ = γ 1 so that γ p−1 = −p and T are canonically identified.Then the commutativity follows from direct computation:

The Frobenius operator
Denote Fr the endomorphism on C † S lifting the qth power endomorphism over the residue field such that Fr : C † S / / C † S (x i , y j ) / / (x q i , y q j ) .
This map is injective and extends to a cochain map , d .Note that the above Fr sends the bidegree (c, w)-subspace to the bidegree (qc, qw)-subspace.Hence, our Fr restricts to the bidegree (0, 0)-subcomplex: On the other hand, denote Using this, we may extend Φ q to the cochain map as follows: with the k-linearity.We may write where we extend convention (1.4) to monomials: y β := y β 1 . . .y β j , dy β := dy β 1 ∧ . . .∧ dy β j .
For S := by the formal identity Φ q,S := exp − γ S(x, y) which converges because we can rewrite Φ q,S = exp(γFr(S) − γS) Φ q,S is still a cochain map.Now, we may compare Fr and Φ q,S via ρ S .
Proposition 5.2.There is a commutative diagram Proof.We will follow Katz's computation in the proof of [19, Th. 2.14] and [19,Th. 2.8].Since the Frobenius on F q [x, y] can be decomposed into where the superscript (q) on each ring means that F q acts by qth power.Denote the lifting of each factor by Fr We abuse notation to denote Fr G S := y where in the second and the third rows, we set deg c y i := −qd i .The top square is commutative because The middle square is commutative by Lemma 5.1.For the bottom square, we first compute Hence, the commutativity follows if we show that To do this, consider Note that g satisfies Moreover, g is by definition a formal solution of the differential equation is a constant multiple of exp(−t) which converges only for val p t > 1 p − 1 , g is the unique power series solution.On the other hand, there is a solution of the form h(t q ) exp Fr y Fr G S • t q γ q−1 S q − t .
where the isomorphism comes from the inclusion Ω ) is the usual trace map for fields extends of finite degree.This gives a description of ψ: where, following (1.4), we denote By our choice of Fr in §5.1, Fr −1 sends the bidegree (qc, qw)-subspace to the bidegree (c, w)-subspace.Hence, the corresponding Tr restricts to the bidegree (0, 0)-subcomplex; and hence, our ψ restricts to the bidegree (0, 0)-subcomplex: .
On the other hand, denote Using this, we may extend Ψ q to the cochain map analogously to Φ q,P n k : by the formal identity Ψ q,S := exp − γ S(x, y) • Ψ q,P n k • exp γ S(x, y) , which converges because we can rewrite Ψ q,S = Ψ q,P n k • exp(γS − γFr(S)) and the final expression converges.Hence, Ψ q,S is still a cochain map by the same reason as in §5.1.Now, we may compare ψ and Ψ q,S via ρ S .
In other words, Ω • X/S is a bounded below complex of Γ(X, −)-acyclic objects so the canonical map is an isomorphism.
Remark A.3.To show the second assertion of Lemma A.2, one may argue with the Čech spectral sequence for a chosen covering U of X : where H q (Ω • X/S ) is the presheaf associate with U, a complex of abelian groups H q (U, Ω 0 X/S ) / / H q (U, Ω 1 X/S ) / / • • • and Tot takes the total complex of a double complex.For this, one may even use the covering {Id X : X → X} to get the desired vanishing because X is affine.
In computing algebraic de Rham cohomology of affine schemes, one may rely on cosimplicial de Rham algebras.For this, we introduce some terminologies on (co-)simplicial objects.Let Δ be the simplex category and C a finitely bicomplete category, that is, C has finite limits and finite colimits.For n ∈ N, denote Δ ≤n the full subcategory of Δ consisting of [0], . . ., [n] and the obvious inclusion y]/k , D ,γS deg c =0 induces a surjection on cohomology spaces with one-dimensional kernel spanned by the class of dS.Proof.Since deg c dS = 0, the differential D ,γS = d + d(γS) ∧ − is compatible with deg c so the subcomplex is well-defined.Moreover, by Lemma 2.2(3), each ξ ∈ Ω • k[x,y]/k homogeneous with respect to deg c satisfies the relation (D ,γS θ c + θ c D ,γS )ξ = ( deg c ξ) ξ so if D ,γS ξ = 0, then ξ is in the image of D ,γS unless deg c ξ = 0. Hence, the first inclusion is a quasi-isomorphism.Note that 1 ∈ Ω • k[x,y]/k is the only bidegree (deg c , deg w ) = (0, 0) element up to scalar multiplication by k.Since D ,γS (1) = γdS, D ,γS (dS) = 0, 1 ∈ Ω • k[x,y]/k does not contribute to the cohomology and kills the class [dS].On the other hand, if deg w f > 0, then the equation D ,γS (f ) = γdS ⇐⇒ df = (1 − f )γdS has no solutions in k[x, y].Hence, [dS] defines a nontrivial class in the subcomplex with (deg c = 0, deg w > 0).Remark 3.3.Since (−1)! is not a well-defined number, in order to extend ρ S to the deg c = 0 complex, we have to choose the value manually.Since 1 is the only bidegree (deg c , deg w ) = (0, 0) element up to scalar multiplication by k, it suffices to consider ρ S (1) only.For ρ S to be a cochain map, ρ S (1) must satisfy dρ S (1) = ρ S (D ,γS (1)) = ρ S ( γdS) = dS S so ρ S (1) = log S.However, this is impossible in the polynomial ring, and even in the corresponding overconvergent power series ring (Definition 4.1).Notation 3.4.In what follows, we will often denote

Definition 4 . 1 .
Denote the ring of overconvergent power series over O k by Hence, A † , B † , and C † S compute the Monsky-Washnitzer of O A , O B , and O C S , respectively: for R = A, B, or C S ,

k
and the final expression converges.Since we may formally write D ,γS = exp − γ S(x, y) • D ,0 • exp γ S(x, y)

b
yG) m y β y v t |m|+|v|+|β| qm−v−e β y m G m−v−e β t |m| = 1 q m∈Z ⊕k ≥0 b qm−v−e β (−1) |m|−1 (|m| − 1)! y m G m−v−e β γ |m| Fr G S |m| t,that is, (5.2) holds, and the proof is completed.§A Remarks on algebraic de Rham cohomology Definition A.1.Given a map of schemes X → S, define its (relative) de Rham cohomology to beH n dR (X/S) := H n RΓ(X, Ω • X/S ) ,where Ω • X/S is the algebraic de Rham complex.Lemma A.2.Given a map of affine schemes X → S, if A = Γ(S, O S ), X = Γ(X, O X ), and A → B the corresponding ring map, thenRΓ(X, Ω • X/S ) ∼ = Ω • B/A in D(A),the derived category of A. Consequently, H n dR (X/S) ∼ = H n (Ω • B/A , d), where d is the usual de Rham differential.
D ,γS , where k{x, y} will be a version of the Tate algebra (see Remark 4.9).
d , where d is the de Rham differential.With the valuation conditions on γ and for the convergence (see Theorem 4.11), we have the following comparison map.