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The Monsky-Washnitzer and the overconvergent realizations

We construct the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classic \'etale site provides a new direct definition of the overconvergent de Rham cohomology and rigid cohomology and shows that their finite dimensionality follows formally from the one of Betti cohomology for smooth projective complex varieties.


INTRODUCTION
The problem of the definition of a well-behaved cohomology theory "à la de Rham" for analytic varieties over a non-archimedean field lies on the pathological properties of the de Rham complex in this context. Even though it behaves as expected when applied to proper varieties, or analytifications of algebraic varieties, its (hyper)cohomology computed on very basic affinoid smooth rigid varieties (such as the closed disc B 1 ) is oddly infinite dimensional. This is related to the impossibility to integrate general holomorphic rigid forms while preserving their radius of convergence.
This classical problem has been studied and positively resolved by different authors (see [22], [25], [42]). The common strategy is to consider the (hyper)cohomology of an alteration of the de Rham complex, namely the overconvergent complex Ω † that is, the subcomplex of those forms which can be extended on a "strict neighborhood" of the variety. In order to give sense to this definition one needs first to consider an affinoid variety and endow it with an overconvergent structure, which amounts to embedding it in the interior of a bigger one. Secondly, one has to prove than two different choices would induce two canonically equivalent cohomology groups, and that this definition can be extended functorially to arbitrary varieties.
The technical tool which is behind these facts is the rigid version of Artin's approximation theorem proved by Bosch [14] stating that any map of varieties can be approximated with a new one preserving two given overconvergent structures, combined with a homotopic argument. In this article, we follow the approach of Große-Klönne [23] and [25] and we give a motivic version of this procedure which can be stated as follows (see Theorem 4.23): Theorem. For any ring of coefficients Λ the canonical functor l : RigSm † /K → RigSm /K from smooth rigid varieties with an overconvergent structure to rigid varieties induces a monoidal, triangulated equivalence of the associated categories of motives: We will actually prove a relative statement, where the base dagger variety is not necessarily the spectrum of the field K. We remark that, in particular, it is possible to define the spectrum of the overconvergent de Rham cohomology as the motive Ll * Ω † .
The main technical ingredient for the proof is Proposition 3.10 which provides a way to approximate maps of rigid analytic varieties having an overconvergent structure, with maps that preserve such structures. It is reminiscent of (a cubical version of) Artin's approximation lemma, but completely independent of it.
We can apply the previous theorem to give a new definition of rigid cohomology, which is a good cohomological theory "à la de Rham" for algebraic varieties X over a discrete field k of positive characteristic. The idea, due to Monsky and Washnitzer [42] is to find (whenever possible) a smooth formal model X of X over the ring of integers K • of a mixed-characteristic valued field K with residue k, and then consider the overconvergent de Rham cohomology of the associated generic fiber X η . Also in this case, the major task which is solved in the literature consists in proving that this definition does not depend on the various choices made at each step, as well as in generalizing it to arbitrary varieties. Classically, the tools which are used include the convergent site of Ogus [46] and the crystalline site of Berthelot [11] for proper varieties, or the overconvergent site developed by Le Stum [37].
Using the motivic language, these problems are alternatively solved by the following remark: a theorem of Ayoub [2,Corollary 1.4.24] states that the special-fiber functor (·) σ : FormSm /K • → Sm /k from the category of smooth formal schemes over K • to the category of smooth varieties over the residue field k induces an equivalence of motives. In particular, by letting (·) η : FormSm /K • → RigSm /K be the generic-fiber functor, there is a motivic triangulated monoidal functor: As a whole, by considering the functor MW * and the complex Ω † we therefore obtain automatically a functorial cohomology theory on algebraic varieties over k satisfyingétale descent and homotopy invariance, which coincides with the one of Monsky-Washnitzer whenever this one is defined. It is formal to show that MW * has a right adjoint MW * and that the motive MW * Ω † represents the "classic" rigid cohomology, providing an alternative to its usual definition and to the rigid spectrum considered by Deglise-Mazzari [20] and Milne-Ramachandran [41] following Besser [13]. Our construction only uses canonical, explicit functors, the classić etale sites on algebraic and analytic varieties and no hypothesis on the valuation of K.
Another crucial fact which is proved in the literature concerns the finite dimensionality of the cohomological theories mentioned above (the most general statements are in [35]), as well as their compatibility with base extensions. The classic proofs rely on several reduction procedures, involving resolutions of singularities, localizations and homotopy. They decompose the general statement into direct, computable checks on varieties of a special kind, such as the ones which are projective and smooth. We remark that these ad hoc constructions are encapsulated in a fundamental theorem of Ayoub [2,Theorem 2.5.35]. When combined with the results of [52], it states that the category of rigid analytic motives with rational coefficients over a base field RigDA ef et (K, Q) is generated, in a suitable sense, by the motives M(X) associated with smooth projective algebraic varieties X over K. Admittedly, the proof of the theorem consists in an elaborated composition of the standard reduction procedures, enhanced with the triangulated language allowed by the motivic setting.
As shown above, the main outcome of this article is proving that the overconvergent de Rham cohomology and rigid cohomology factor over the triangulated category RigDA ef et (K, Q). In particular, using the theorem of Ayoub, we deduce (see Corollary 5.24) their finitedimensionality as well as their compatibility with base change by reducing to the motives M(X) of the aforementioned form, and hence to well-known facts related to the classic de Rham cohomology of complex smooth projective varieties X(C). Our proof makes no distinction between the discrete-valuation case and the general case, and is independent on the classic proofs (see [10], and partial results in [12], [24], [40]).
In the Appendix, we prove that an overconvergent structure of a variety corresponds to a presentation of its (adic) compactification as an inverse limit (in a weak sense defined by Huber) of strict inclusions of rigid varieties. This connects the theory of dagger spaces of Große-Klönne [23] to the theory of adic spaces of Huber [32] and strengthens the parallel between the techniques used in this paper and the ones of [53] where smooth perfectoid spaces arise as inverse limits of finite maps of rigid varieties.

OVERCONVERGENT RIGID VARIETIES
From now on we fix a complete valued field K endowed with a non-archimedean valuation of rank 1 and residue characteristic p > 0. We denote by π a pseudo-uniformizer of K that is, an invertible, topologically nilpotent element. We also denote by K • the ring of integers and by k the residue field. We consider rigid analytic varieties as adic spaces, using the language of Huber [32]. In particular, when we consider a point x ∈ X for a variety X we mean a point in the sense of Huber (or, equivalently, a point of the G-topos of X). We only consider rigid analytic varieties over K which are separated and taut (that is, the closure of a quasi-compact subset is quasi-compact, see [33,Definition 5.1.2]). If R is a Tate algebra, we sometimes denote by Spa R the associated affinoid space Spa(R, R • ).
The starting point to define overconvergent, or dagger varieties are the so-called dagger algebras. For the sake of completeness, we report here their definition and some basic properties, proved in [23] and [42]. We refer to the Appendix for a link between these definitions and the language of adic spaces of Huber. [23]). For c ∈ K and m, d ∈ N >0 we denote by K c m/d τ 1 , . . . , c m/d τ n or simply by K c m/d τ the subring of K τ 1 , . . . , τ n of those power series a α τ α such that lim |a α |λ |α| = 0 for λ = |c| m/d . We denote by K τ † = K τ 1 , . . . , τ n † the following topological subring of K τ 1 . . . , τ n that is, the ring of those power series a α τ α such that lim |a α |λ |α| = 0 for some λ ∈ R >1 . A dagger algebra is a topological K-algebra R isomorphic to a quotient K τ † /I of K τ † . Its completionR is the Tate algebra K τ /I. A morphism of dagger algebras R → S is a K-linear (hence continuous) morphism. The category Aff † of affinoid dagger spaces is the opposite category of dagger algebras. We denote by Spa † R the object on Aff † associated with R. We say that its limit is SpaR whereR is the completion of R and vice-versa we say that Spa † R is a dagger structure of SpaR. We say that Spa † R [resp. a morphism Spa † R ′ → Spa † R] has the property P if SpaR [resp. the induced morphism of affinoid spaces SpaR ′ → SpaR] has the property P. The category of smooth morphisms of affinoid dagger spaces X → S to a fixed affinoid dagger space S = Spa † R is denoted by AffSm † /S. Similarly, a collection of We now recall some basic facts about the category of dagger spaces. In particular, we isolate in the following proposition the fundamental Artin's approximation lemma. It will not be used under this general form, but rather in a smooth "cubical" fashion (see 3.10).
Proposition 2.8 ([22, Corollary 7.5.10]). Suppose char K = 0 or char K = p > 0 and [K : K p ] < ∞. Let X and Y be two affinoid dagger spaces with limitX andŶ respectively. We fix a Banach norm || · || on O(X) and O(Ŷ ). For any [iso-]morphism φ :X →Ŷ and any is an open rational subspace of Spa † R canonically independent on the choice of f i , g. Moreover O † is a sheaf of topological K-algebras on SpaR.
Definition 2.13. Let X be an affinoid dagger space with limitX. We denote by O † X or simply by O † the sheaf of topological algebras on the rational site of X as well as the sheaf of topological algebras onX introduced in Proposition 2.12.
In the category of affinoid rigid analytic spaces, the functor Spa R → R • is represented by B 1 . The next proposition shows the role of the dagger disc B 1 † introduced above.
Proof. Let X = Spa † R be an affinoid dagger space. A continuous map K τ →R amounts to the choice of an element s ∈R • and it preserves the dagger structures if and only if for any n the induced map K π 1/n τ →R factors over a map K π 1/n τ →R h for some h, that is, if and only if πs n ∈ lim − →R • h for all n. Since R is a K-algebra, integrally closed inR (see [15,Theorem 2]) we deduce that such an s lies inR • ∩ R.
Vice-versa, we claim that any element s ∈R • ∩ R † satisfies the condition. If f ∈R 1 ∩ R • then we deduce from [2, Proposition 2.1.16] the following inclusions in X 1 = SpaR 1 : Since {X h } is coinitial among the rational subvarieties W of X 1 such thatX ⋐ X 1 W we deduce in particular that X h ⊂ U(πf /1) for some h that is, πf ∈ lim − →R  (1) The functor U →Û defines an equivalence between the categories of inclusions of rational subspaces in X and inX. (2) The functor U →Û defines an equivalence between the categories of finiteétale affinoid spaces over X and overX.
Proof. The first claim follows from Proposition 2.12. For the second claim, by [48,Lemma 7.5] we know that, up to shifting indices, any mapÛ →V of finiteétale affinoid spaces over X is induced by a map U 1 → V 1 of finiteétale spaces over is a dagger algebra. We now use the equivalence between dagger affinoid spaces and their presentations, proved in the Appendix (see Proposition A.22). In particular, we can alternatively prove that the sequence U h is a presentation ofÛ . It suffices to show thatÛ lies in Int(U 1 ) (see the notations of Definition A.7). Since f : U 1 → X 1 is finite, by [9, Corollary 2.5.13(i)] and Corollary A.8 we deduce that Int(U 1 /X 1 ) = U 1 and hence by Corollary A.11 we get Int(U 1 ) = f −1 Int(X 1 ). We then need to prove that the image ofÛ lies in the interior of X 1 and this is clear as it factors overX lying in Int(X 1 ). Remark 2.16. If Y → X is a finiteétale morphism of affinoid dagger spaces, the dagger algebra for some finiteétale algebraŜ 1 overR 1 (up to shifting indices). In particular, it is finiteétale over O † (X).
Proposition 2.21. Let X be an affinoid dagger space and F be a coherent O † -module over it. ( where we denote by S the dagger algebra associated with Y (see Proposition 2.15). By Remark 2.16 the map R → S is finiteétale, and in particular the dagger tensor product coincides with the usual one (see [23,Lemma 1.10]). The claim then follows from [27,Section I.3].
We now recall the definition of Große-Klönne of "global" dagger spaces  Proposition 2.12). A morphism of dagger spaces is a morphism of the underlying locally ringed spaces over K (see [23,Definition 2.12]) and the category they form is denoted by Rig † . We say that the rigid spaceX is the limit of X and vice-versa we say that X is a dagger structure ofX. We say that X [resp. a morphism X → X ′ ] has the property P ifX [resp. the induced morphism of rigid spacesX →X ′ ] has the property P. Whenever S is a dagger space, we denote by Rig † the category of rigid spaces over it. The category of smooth morphisms of dagger spaces X → S to a fixed dagger space S is denoted by RigSm † /S and its full subcategory of affinoid objects by AffSm † /S. A collection 6 of morphisms {X i → X} in RigSm /S is a cover if the induced collection {X i →X} is a topological cover, that is, if X i →X is surjective.
Example 2.23. Any rigid variety without boundary has a dagger structure by [23,Theorem 2.27]. We denote by P 1 † a dagger variety having as limit the projective line P 1 .
The following proposition is straightforward.
Proposition 2.24. Let S be a dagger space. The functor is fully faithful and induces an equivalence of the associated open analytic and theétale topoi.
From now on, we will use the term affinoid dagger space also to indicate the objects in the essential image of the functor above.
We easily obtain also the following version.
Corollary 2.25. Let S be a dagger space. The functor is fully faithful and induces an equivalence of the associated open analytic and theétale topoi.

APPROXIMATION RESULTS
From now on, we fix a dagger space S with limitŜ. In this section, we recall the analytic version of the inverse function theorem and we use it as an alternative to Artin's approximation theorem for smooth dagger algebras [14]. As a matter of fact, it induces a weaker form of this theorem (see Corollary 3.4) but also a cubical version of it that we will need in what follows (see Propositions 3.9 and 3.10).
The reader who believes in Proposition 3.10 can safely skip this technical section.
Proposition 3.1. Let R be a dagger algebra with completionR. If an element ξ ofR is algebraic over Frac R then it lies in R.
Proof. We denote by X the space Spa † R. Since R is algebraically closed inR (see [15, Theorem 2]) we conclude that Frac R is algebraically closed in (R \ {0}) −1R and therefore ξ ∈ Frac R. We can also assume ξ ∈ FracR 1 up to shifting indices. Let I ξ = (d 1 , . . . , d n ) be the ideal of denominators ofR 1 associated with the meromorphic function ξ (see [22,Lemma 4.6.5]) and let V (I ξ ) be the induced (Zariski) closed subvariety of X 1 = SpaR 1 . From now on, we denote by T c the closure of a subset T in X 1 . We recall that ξ is analytic around a point x if and only if ( We obtain in particular the following fact.

Corollary 3.2. Let R be a dagger algebra with completionR. If f ∈ R is invertible inR then it is invertible in R.
We recall the following version of the inverse mapping theorem in the analytic context.
Corollary 3.4. Suppose that S is affinoid. Let X and Y be two affinoid dagger algebras smooth over S such thatŶ isétale over a poly-disc B m ×Ŝ. For any [iso-]morphism φ :X →Ŷ over S and any ε > 0 there exists a [iso-]morphism ψ : X → Y over S such that ||φ −ψ|| ≤ ǫ.
Proof. We suppose S = Spa † A, X = Spa † R and we denote their limits byŜ = SpaÂ, We first assume that The map φ is uniquely determined by the association (σ, τ ) → (s, t) fromÂ σ, τ /(P ) tô R for an m-tuple s and an n-tuple t inR. By Corollary 3.3 there exists power series defines a new map ψ fromX toŶ for any choice ofs ∈R • ∩R such thats is in the convergence radius of F and F (s) is inR • . The field Frac(R)(F (s)) is finiteétale over Frac(R). By Proposition 3.1 we deduce that F (s) is in R and therefore ψ is a map from X to Y . By the density of R inR and the continuity of F we can also assume that the m-tuples induces a map ψ such that ||φ − ψ|| ≤ ǫ.
If we takeX =Ŷ we can find an endomorphism ψ ofX inducing a map X → Y such that || id −ψ|| ≤ ǫ. If we take ǫ sufficiently small, we deduce that ψ is an automorphism ofX and hence any two dagger structures on it are isomorphic. In particular, we obtain the general case of the proposition.
In the previous proof, we also showed the following structure theorem. Corollary 3.5. Suppose S affinoid. LetŶ be an affinoid rigid space which isétale over the poly-disc B m ×Ŝ. Then it admits a dagger structure Y isomorphic to Spa R with Proof. The claim is local on S so we can assume it is affinoid. Let X be a smooth dagger variety over it. By Proposition 2.15 and [2] we can find a rational covering {U i → X} such 8 that each limitÛ i isétale over some poly-disc B n V i whereV i is rational insideŜ. The claim then follows from Corollary 3.5.
We recall (see [16, Definition 1.1.9/1]) that a morphism of normed groups φ : G → H is strict if the homomorphism G/ ker φ → φ(G) is a homeomorphism, where the former group is endowed with the quotient topology and the latter with the topology inherited from H. In particular, we say that a sequence of normed K-vector spaces is strict and exact at R ′ if it exact at R ′ and if f is strict that is, the quotient norm and the norm induced by R ′ on R/ ker(f ) ∼ = ker(g) are equivalent.
For any finite set Σ of such maps and any dagger algebra R with limitR the following diagram of topological K-algebras has vertical inclusions and strict and exact lines Moreover, the ideal σ∈Σ I σ is generated by a finite set of polynomials with coefficients in Z.
Proof. The fact that the first line is strict and exact as well as the description of the generators of I σ is proved in [53]. The same statement applied to the ringsR h π 1/h θ and a direct limit argument show that also the second line is exact. We now prove that the vertical maps are inclusions. This can be proved only for the last two columns, where the statement is clear. As the second line is isometrically contained in the first, we also deduce that it is strict as well.
for all σ then the element f can be chosen so that |f − g| < Cε. Moreover, iff σ ∈ R θ † /I σ for all σ then the element f can be chosen inside R θ † .
Proof. The first claim and the first part of the second are simply a restatement of Lemma 3.7, where C = C(Σ) is the constant defining the compatibility || · || 1 ≤ C|| · || 2 between the norm || · || 1 onR θ / I σ induced by the quotient and the norm || · || 2 induced by the embedding in R θ /I σ . We now turn to the last sentence of the second claim. By Lemma 3.7 and what proved above there exist two lifts of {f σ }: an element f 1 of R θ † and an element f 2 ofR θ such that |f 2 − g| < Cε and their difference lies in I σ . Hence, we can find elements γ i ∈R θ such that Proposition 3.9. Let R be a dagger algebra with completionR. Let s 1 , . . . , s N be elements of R θ 1 , . . . , θ n • . For any ε > 0 there exist elementss 1 , . . . ,s N ofR θ • ∩ R θ † satisfying the following conditions.
(2) For any α, β ∈ {1, . . . , N} and any k ∈ {1, . . . , n} such that Proof. We will actually prove a stronger statement, namely that we can reinforce the previous conditions with the following: (5) For any α, β ∈ {1, . . . , N} any subset T of {1, . . . , n} and any map σ : Above we denote by s| σ the image of s via the substitution (θ t = σ(t)) t∈T . We proceed by induction on N, the case N = 0 being trivial. We remark that if ǫ is sufficiently small, any element a such that |a − s k | < ε lie inR θ • as this ring is open. We are left to prove that we can pick elementss k in R θ † .
Consider the conditions we want to preserve that involve the index N. They are of the form and are indexed by some pairs (σ, i) where i is an index and σ varies in a set of maps Σ. Our procedure consists in determining by induction the elementss 1 , . . . ,s N −1 first, and then deduce the existence ofs N by means of Lemma 3.8 by lifting the elements {s i | σ } (σ,i) . Therefore, we first define ε ′ := 1 C ε where C = C(Σ) is the constant introduced in Lemma 3.8 and then apply the induction hypothesis to the first N − 1 elements with respect to ε ′ .
By the induction hypothesis, the elementss i | σ satisfy the compatibility condition of Lemma 3.8 and lie in R θ † . By Lemma 3.8 we can find an elements N of R θ † lifting them such that |s N − s N | < Cε ′ = ε as wanted.
We now restate Proposition 3.9 in more geometric terms. Proposition 3.10. Suppose that S is affinoid. Let X be an affinoid dagger space smooth over S and let Y be an affinoid dagger algebra such thatŶ isétale over B m ×Ŝ. For a given finite set Proof. We let S be Spa A with limitŜ = SpaÂ. By Corollary 3.5 we can assume that Y = Spa A σ 1 , . . . , σ n , τ 1 , . . . , τ m † For any h ∈ Z we denote by R the dagger algebra O † (X) θ † and by R χ † the dagger algebra associated with X × B n † × B 1 † . Each f k is induced by maps (σ, τ ) → (s k , t k ) from K σ, τ /(P ) toR for some m-tuples s k and n-tuples t k in R. By Corollary 3.3 there exists a sequence of power series F k = (F k1 , . . . , F km ) associated with each f k such that We prove that any such map satisfies the first claim. By Proposition 2.14 this amounts to prove that the elementst k := F k (s) lie inR • ∩ R. By our choice ofs k we already know that they lie inR • . We remark that the field (Frac R)(t k ) is finiteétale over Frac R. By Lemma 3.1 we deduce that eacht k lies in R as wanted.
Let now ε be a positive real number, smaller than all radii of convergence of the series F kj and such that F (a) ∈ R for all |a − s| < ε. Denote bys ki the elements associated with s ki by applying Proposition 3.9 with respect to the chosen ε. In particular, they induce a well-defined map H k and the elementss ki lie inR • ∩ R. We show that the maps H k induced by this choice also satisfy the second, third and fourth claims of the proposition. Suppose This implies that both F k | θr=ǫ and F k ′ | θr=ǫ are two m-tuples of formal power seriesF with coefficients in O(X ×B n−1 ) converging around s and such that P (σ,F (σ)) = 0,F (s) =t. By the uniqueness of such power series stated in Corollary 3.3, we conclude that they coincide.
Moreover, by our choice of the elementss k it follows thats :=s k | θr=ǫ =s k ′ | θr=ǫ . In particular one has and therefore H k • d r,ǫ = H k ′ • d r,ǫ proving the second claim.
The third claim follows from the fact that the elementss ki satisfy the condition (4) of Proposition 3.9.

DAGGER RIGID MOTIVES
Motives (here, a short form for mixed, derived, effective motives with coefficients in a ring Λ) can be defined out of an arbitrary site with products (C, τ ) and a choice of a "contractible object" I inside it. The underlying idea is to construct the universal category with respect to derived Λ-functors defined on C, which satisfy τ -descent and invariance with respect to the contractible object I. Typically, such functors are related to (co-)homology theories. Not surprisingly, the category of motives is simply a Verdier quotient of the derived category of τ -sheaves with values in Λ-modules D(Sh τ (C, Λ)) obtained by imposing the condition that all projections X × I → X are invertible. For the reader familiar with the theory of Voevodsky's motives, we remark that we make no (explicit) use of correspondences, and hence our categories of motives are without transfers (but see Remark 5.22).
Hereunder, we introduce the category of motives RigDA † ef et (S, Λ) associated with thé etale site on smooth dagger varieties over S, inspired by the construction of the motives RigDA ef et (Ŝ, Λ) associated with theétale site on smooth rigid analytic varieties (see [2]). In the dagger case, the "contractible object" is the dagger disc B 1 † while in the rigid analytic setting it is the closed disc B 1 .
For computations, and more crucially to invert the Tate twist, the description of motives as Verdier quotients is sometimes unsatisfactory, and specific "models" of these triangulated categories are needed to have control on Hom-sets. This is why the language of model categories is used, applied to the categories of complexes of presheaves. We borrow all the notations from Ayoub (see [2] and [4]) and we refer to his survey [1] for a gentle explanation of these constructions in the algebraic context. We also refer to [53] for a brief collection of some standard results about the change of models, which are based on the results of [4] and inspired by the classic papers [34], [39] and [43].
The reader who is not interested in these formal constructions can skip the technical statements of the section, focusing only on its main result which is Theorem 4.23. It is formally obtained from Proposition 3.10, proved in the previous section.
From now on, we fix a commutative ring Λ and work with Λ-enriched categories. In particular, the term "presheaf" should be understood as "presheaf of Λ-modules" and similarly for the tem "sheaf". The presheaf Λ(X) represented by an object X of a category C sends an object Y of C to the free Λ-module Λ Hom C (Y, X).
The category Ch(Psh(C)) of complexes of presheaves over a category C can be endowed with the projective model structure for which weak equivalences are quasi-isomorphisms and fibrations are maps F → F ′ such that F (X) → F ′ (X) is a surjection for all X in C (cfr [29, Section 2.3] and [4,Proposition 4.4.16]). Its homotopy category is the usual (unbounded) derived category D(Psh(C)). The homotopy category of Ché t,B 1 † Psh(RigSm † /S) will be denoted by RigDA † eff (S, Λ) and its element Λ(X) will be called the motive of X for any dagger space X in RigSm /S. In case S = Spa K then we simply write RigDA † eff (K, Λ).
It is a monoidal category such that the tensor product Λ(X) ⊗ Λ(Y ) of two motives associated with varieties X and Y coincides with Λ(X ×Ŝ Y ) and the unit object is Λ(Ŝ). The same is true for the dagger counterpart. Motives have been introduced as quotients of the derived categories of presheaves. On the other hand, the canonical quotient functor admits a fully faithful right adjoint. Therefore, motives can equally be defined as triangulated subcategories of the derived categories of presheaves, which is particularly useful for computing their Hom-sets. We now investigate better this point of view (see [8]).
We say that an object F of the derived category D = D(Psh(RigSm † /S)) is η-local if the functor Hom D (·, F ) sends maps in S η to isomorphisms. This amounts to say that F is quasiisomorphic to a η-fibrant object. We use the same terminology for the model categories on Ch Psh(C) for any full subcategory C generating the sameétale topos.
Remark 4.6. The existence of these localizations at the level of model categories is granted by the results of Hirschhorn [28] used in the references above. At the level of the homotopy categories, using the language of [8], these localizations induce endofunctors C η of D(Psh(RigSm † /S)) such that C η F is η-local for all F and there is a natural transformation C η → id which is a pointwise η-weak equivalence. The functor C η restricts to a triangulated equivalence on the objects F that are η-local and one can compute the Hom set Hom(F , F ′ ) in the homotopy category of the η-localization as D(F , C η F ′ ). The same is true for the category D(Psh(C)) for any subcategory C generating the sameétale topos. There is also a characterization of the B 1 -localization, as described in [2] and [53]. Such descriptions admit a natural dagger analogue, that we now describe. This is based on the fact that B 1 † is an interval object (see [47]).

Remark 4.8. Let B 1
h be the open U(πχ k /1) = U((πχ) k /π k−1 ) in Spa K πχ . The dagger variety B 1 † is an interval object with respect to the maps i 0 and i 1 induced by the points χ → 0 and χ → 1 respectively, and the multiplication induced by the multiplication on the limit B 1 . Indeed, the map B 1 × B 1 → B 1 → B 1 h factors over B 1 2h × B 1 2h . In the following part, we have opted for the cubical rather than the simplicial approach in order to have clearer computations, and a stronger parallel with the perfectoid case [53]. Definition 4.9. We denote by † the Σ-enriched cocubical object (see [6, Appendix A]) defined by putting †n = B n † × S = Spa K τ 1 , . . . , τ n † × S and considering the morphisms d r,ǫ induced by the maps B n † → B (n+1) † corresponding to the substitution τ r = ǫ for ǫ ∈ {0, 1} and the morphisms p r induced by the projections B n † → B (n−1) † . For any dagger variety X and any presheaf F of dagger varieties [resp. rigid varieties] with values in an abelian category, we can therefore consider the Σ-enriched cubical object F (X × S † ) (see [6, Appendix A]). Associated to any Σ-enriched cubical object F there are the following complexes: the complex C ♯ • F defined as C ♯ n F = F n and with differential (−1) r (d * r,1 − d * r,0 ); the simple complex C • F defined as C n F = n r=1 ker d * r,0 and with differential (−1) r d * r,1 ; the normalized complex N • F defined as N n F = C n ∩ F n r=2 ker d * r,1 and with differential −d * 1,1 . By [7, Lemma A.3, Proposition A.8, Proposition A.11], the inclusion N • F ֒→ C • F is a quasi-isomorphism and both inclusions C • F ֒→ C ♯ • F and N • F ֒→ C • F split. For any complex of presheaves F of 13 dagger varieties we let Sing B 1 † F be the total complex of the simple complex associated with Hom(Λ( † ), F ). It sends the object X to the total complex of the simple complex associated with F (X × S † ).
We now show that the complex Sing B 1 † F defined above gives rise to the "universal" homotopy-invariant cohomology theory attached to F .
We are now interested in finding a convenient set of compact objects which generate the categories above, as triangulated categories with small sums. This will simplify many definitions and proofs in what follows. We first briefly recall the notion of compactness in triangulated categories.    Proof. Since any smooth dagger variety is locallyétale over a dagger poly-disc over a rational open of S by Proposition 3.6, the set of functors H i Hom • (Λ(X), ·) detect quasi-isomorphisms betweenétale local objects, by letting X vary among spaces of the prescribed form and i vary in Z. We are left to prove that the motive Λ(X) of any affinoid dagger smooth variety X is compact. Since Λ(X) is compact in D(Psh(RigSm † /S)) and Sing B 1 † commutes with direct sums, it suffices to prove that if {F i } i∈I is a family ofét-local complexes, then also i F i isét-local. If I is finite, the claim follows from the isomorphisms H −n Hom A coproduct over an arbitrary family is a filtered colimit of finite coproducts, hence the claim follows from [4,Proposition 4.5.62].
The previous proof can be generalized to the following result.  Proof. This follows from the previous proposition and Corollary 3.5.
The categories of motives have been introduced by imposingétale descent and homotopy invariance on (co-)homological theories. If one wants to impose the extra condition that the Tate twist is invertible, they are not yet enough. Nonetheless, there is a canonical way to do so (by means of the language of model categories) via the introduction of spectra. We refer to [30] and [4,Section 4.3] for the details of this construction, that we simply apply to our dagger context. The aim of the following part is to compare the categories of motives RigDA † and RigDA introduced above. We start by defining the canonical adjunction pair between them. The completion functor Spa R → SpaR defined on affinoid dagger spaces can be extended (by glueing) to a functor l from dagger spaces to rigid analytic spaces (see [23,Theorem 2.19]). Proof. The existence of the Quillen adjunction follows formally from the continuity of l and the formula l(B 1 † ) ∼ = B 1 . Also, the fact that Ll * is monoidal follows from the formula l * (X × S Y ) ∼ = l * (X) ×Ŝ l * (Y ). We are left to prove that Rl * ∼ = l * .
By its very definition, l * preserves quasi-isomorphisms of complexes of presheaves. We claim that l * preserves alsoét-weak equivalences. Fix any affinoid dagger space X and any presheaf F on AffSm. Let R(X) resp. R(X) be the class of covering families of X resp.X. By Corollary 2.17 we know that any covering ofX can be refined into one coming from a covering of X. This proves that the canonical map from (l * F ) By the explicit construction of the sheafification functor as F → F ++ (see e.g. [49, Tag 00W1]) we then deduce that l * commutes with thé etale sheafification functor, and hence it also preservesét-weak equivalences as claimed.
We also remark that for any complex F and any dagger variety X we obtain so that l * Sing B 1 ∼ = Sing B 1 † l * . We then deduce by Proposition 4.10 that l * maps B 1 -weak equivalences to B 1 † -weak equivalences, as wanted. We will see that the following technical proposition implies the the fully faithfulness of the functor Ll * . It relies heavily on the approximation result (Proposition 3.10) obtained in the previous section.
Proposition 4.22. Suppose that S is affinoid. Let X be an affinoid dagger algebra smooth over S with limitX and Y be a dagger algebra,étale over a poly-disc B m † . The canonical map Proof. We need to prove that the natural map defines bijections on homology groups.
We start by proving surjectivity. Suppose that β ∈ Λ Hom(X × n ,Ŷ ) defines a cycle in N n that is, β • d r,ǫ = 0 for 1 ≤ r ≤ n and ǫ ∈ {0, 1}. This means that β = λ k f k with λ k ∈ Λ, f k ∈ Hom(X × n ,Ŷ ) and λ k f k • d r,ǫ = 0. This amounts to say that for every k, r, ǫ the sum λ k ′ over the indices k ′ such that f k ′ • d r,ǫ = f k • d r,ǫ is zero. By Proposition 3.10, we can find maps H k ∈ Hom(X × n × B 1 ,Ŷ ) such that We denote by H the cycle λ k H k ∈ Λ Hom(X × n × B 1 ,Ŷ ). By [53, Lemma 3.14] we conclude that i * 1 H and i * 0 H define the same homology class, and therefore β defines the same class as i * 1 H which is the image of a class in Λ Hom(X × n † , Y ) as wanted.
We are now ready to prove the main result of this section. By Proposition 4.20 the functor Rl * is l * . Fix now a dagger space Y smooth over S. We claim that the motive l * Ll * Λ(Y ) is isomorphic to Λ(Y ). By Proposition we can find a cover . By the dagger, effective version of [5,Lemma 3.4] we obtain that the collection of functors {Lj * i } i is conservative. In particular, we can prove that The functor Lj * i obviously commutes with Ll * . We claim that it also commutes with l * . We can alternatively prove that j i♯ commutes with l * (see Proposition 4.11) and this is also straightforward. We are then left to prove that the canonical map is invertible. Therefore, we assume from now on that the base dagger variety S is affinoid, and that Y isétale over a poly-disc. We will equivalently show that l * Sing B 1 Λ(Ŷ ) ∼ = Sing B 1 † (Y ). It is enough to show that for any affinoid dagger space X smooth over S and with limitX, the canonical map (Sing B 1 † Λ(Y ))(X) → (Sing B 1 Λ(Ŷ ))(X) is a quasi-isomorphism, and this is true by means of Proposition 4.22.
We finally conclude that the triangulated subcategory of RigDA † ef et (S, Λ) of those objects F such that the canonical map F → l * Ll * F is invertible contains Λ(Y )[i] for any Y smooth over S. By Proposition 4.15 we deduce that this subcategory coincides with the whole RigDA † ef et (S, Λ) and therefore l * Ll * ∼ = id as wanted. We now turn to the assertion for the stable categories. From what we proved above we conclude that the Quillen adjunction of Proposition 4.20 is a Quillen equivalences. By means of [30,Theorem 5.5] the same is true also for the adjunction of Corollary 4.21, hence the claim.

THE MONSKY-WASHNITZER REALIZATION FUNCTOR
It is possible to use the equivalence obtained above to define some realization functors for algebraic motives, as well to give an explicit description of the motives representing rigid cohomology or the overconvergent de Rham cohomology. In this respect, motives will be used as a convenient formalism allowing choices modulo homotopy, localizations and reductions to special cases. This language will provide extremely concise proofs of the functoriality and the finite-dimensionality for both the overconvergent de Rham cohomology and rigid cohomology.
Assumption 5.1. From now on, we suppose that K is a complete valued filed of mixed characteristic (0, p) with a valuation of rank 1 (we do not suppose that the valuation is discrete). We recall that we denote by K • its ring of integers and by k its residue field of characteristic p.
Fix a formal scheme X of topological finite type over K • . We can adapt Definition 4 to the setting of algebraic varieties and define the category DA ef et (X σ , Λ) ofétale motives over X σ (see e.g. [5,Page 20]). Analogously, we can define the category FormDA ef et (X , Λ) of etale motives of formal varieties over X . If we let M be the model category Ch(Λ -Mod), these constructions are denoted by SH eff M (X σ ) and FSH eff M (X ) in [2, Definition 1.4.12] respectively, with the only difference that we are considering the (finer)étale topology rather than the Nisnevich site. They are both monoidal categories with respect to the tensor product inherited by the direct product of varieties, and they also admit stable versions DAé t (X σ , Λ) and FormDAé t (X , Λ) obtained by inverting the Tate twists. By [2, Corollaries 1.4.24 and 1.4.29] the special fiber functor induces (triangulated, monoidal) equivalences: : R(·) σ * On the other hand, the generic fiber functor induces Quillen adjunctions: The functor (·) η * as well as theétale and B 1 -localizations commute with direct sums (see the proof of [53,Proposition 4.18]). Therefore, the functor R(·) η * also commutes with direct sums. It has a right adjoint MW eff * induced by R(·) η * : RigDA(K, Λ) ef et → FormDA ef et (K • , Λ). This adjoint pair has also a stable version (MW * , MW * ) whose left adjoint is called the stable Monsky-Washnitzer realization functor.
Remark 5.4. Fix a formal scheme X of topological finite type over K • with special fiberX = X σ and such that its generic rigid fiberX = X η has a dagger structure X. The previous definition can be generalized to the following functor (also admitting a right adjoint and a stable version): We now remark that our construction overlaps with the classic definition of Monsky-Washnitzer.
Proposition 5.5. Let X be a smooth formal scheme of topological finite type over K • with special fiberX and generic fiberX admitting a dagger structure X. Then MW eff * Λ(X ) ∼ = Λ(X).
It is immediate to see that these functors can be used to define a cohomology theory for varietiesX over k satisfyingétale descent and homotopy invariance, and such that it coincides with the de Rham cohomology of the generic fiber X η wheneverX admits a smooth formal model X over K • . We now describe this construction and we will later show (Proposition 5.12) that this definition overlaps with the classical definition of rigid cohomology.
Definition 5.6. We denote by Ω 1 † the presheaf on smooth dagger spaces Ω 1 O † /K and by Ω q † its q-th exterior power. It associates to a smooth dagger space X the space of overconvergent differential d-forms Ω d † X/K (X). When restricted to the analytic site of a dagger space X it defines a coherent O † -module. For any fixed map Λ → K, the overconvergent de Rham complex is the object Ω † in Ch Psh(RigSm † /K) concentrated in negative degrees with Ω † −q = Ω q † for q ≥ 0 and with the usual differential maps. It associates to a smooth dagger variety X [22,Definition 7.5.11]). We also denote with Ω † the associated motive in RigDA Proof. Fix anétale morphism f : Y → X. We want to prove that Ω q † is isomorphic to the sheaf induced by Ω q † X . Up to considering a rational cover, we can also assume that f is a composition of rational embeddings and finiteétale maps between affinoid dagger spaces. We are then left to consider the case in which f is finiteétale.
We recall that the analytification functor X → X an from algebraic varieties over K to rigid analytic varieties over K induces the following adjunction: L Rig * : DA ef et (K, Λ) ⇄ RigDA ef et (K, Λ) :R Rig * . which also admits a stable version (see [2,Page 54]).
We will use the term "representing a cohomological theory" in the following sense.  Remark 5.10. As soon as the functor H * as an extra structure or satisfies extra axioms, the previous proposition can be reinforced. For example, one may impose an isomorphism between the two δ-functor structures (if H * has one).
Remark 5.11. Any cohomology theory on smooth algebraic varieties represented by M can be canonically generalized to arbitrary (not necessarily smooth) quasi-projective varieties Π : is defined using the extraordinary direct and inverse image functors (Π ! , Π ! ) which are part of the six-operation formalism of algebraic motives (see [3] and [4]).
We compare the cohomology theories which naturally arise by considering Ω † and the functors MW * , L Rig * and Rl * with some classical ones: the overconvergent de Rham cohomology, which we will denote by H * odR , algebraic de Rham cohomology, which we will denote by H * dR , and rigid cohomology, which we will denote by H * rig . For their definitions, we refer to the survey in [22,Chapter 7].
(3) The motive MW eff * Ω † represents rigid cohomology H * rig on smooth varieties over k. (4) The motive R Rig * Ll * Ω † represents the de Rham cohomology H * dR on smooth varieties over K.
Proof. Fix an affinoid smooth dagger space X with limitX. Each Ω q † is a sheaf of coherent O †modules, hence it is acyclic for theétale topology by Proposition 2.21 and Lemma 5.7. From a spectral sequence argument, we conclude that H i (Γ(X, Ω † )) = H í et (X, Ω † ) which shows that Ω † isét-local and that H i Γ(X, Ω † ) = H i odR (X). We now prove that Ω † is B 1 † -local. From the computations above, this amounts to prove that H i odR (X × B 1 † ) ∼ = H i odR (X) which is classical (see [42,Theorem 5.4] or [23,Theorem 4.12] and [25, Example 1.8]) in case K has a discrete valuation, and follows from [10, Corollary 5.5.2] in the general case, since any smooth affinoid space has a model over a discrete-valuation complete sub-field of K.
We observe that for any smooth rigid affinoid spaceX with a chosen associated dagger structure X by Theorem 4.23 we see that: with the composite isomorphism canonical onX. This is enough to show that Ll * Ω † represents the overconvergent de Rham cohomology on rigid varieties over K (see Remark 5.9). Fix now a smooth schemeX over k having a smooth formal model X . We have by Theorem 4.23 and the composite isomorphism is canonical onX. Since rigid cohomology satisfiesétale descent [17], this implies that it is represented by MW eff * Ω † (see Remark 5.9).
Let now X be a smooth algebraic variety over K. By [25, Theorem 2.3] we obtain a canonical sequence of isomorphisms Remark 5.13. The previous proof can easily be made independent on [10, Corollary 5.5.2]. Indeed, the proof of [42,Theorem 5.4] applied to K • and to π = p does not use the fact that the p-adic norm is discrete. Remark 5.14. As a whole, we have then provided new formulas computing rigid and overconvergent de Rham cohomologies by using canonical functors on the categories of motives. The usual formulas, involving colimits over the possible choices of lifting and dagger structures (see [13]) are encoded in the description of the functors R(·) σ * and Rl * respectively.
Remark 5.15. Various authors have already given examples of complexes of presheaves representing rigid cohomology (see [13], [20] and [41]). By means of the previous proposition, one can show that such complexes are isomorphic to MW eff * Ω † in DA ef et (k, Λ). By what proved above, the following definition is well posed.
Definition 5.16. Let Λ be K and i be in Z.
(1) For any M in RigDAé t (K, Λ) we denote ). Remark 5.17. The above definition of rigid cohomology is obviously functorial. Whenever k is a finite field, we then obtain automatically an action of Frobenius on each cohomology group H i rig (N) since relative Frobenius maps are invertible in DAé t (k, Q) (this category is equivalent to the one with transfers, see e.g. [19,Section 16.2]).
We conclude by pointing out a straightforward consequence of our constructions, the comparison theorems of Große-Klönne [23], [25] and a theorem of Ayoub [2,Theorem 2.5.35]. It can be summarized by saying that the finite dimensionality of the overconvergent de Rham, Monsky-Washnitzer and rigid cohomology can be formally deduced by the finitedimensionality of the "classic" de Rham cohomology for projective algebraic varieties in characteristic zero.
Remark 5.19. The category DAé t,gm (K, Λ) is denoted by DA ct et (K, Λ) in [3] and [4] where ct stands for "constructible" and gm for "geometric". The equivalence of all these notions follows from [ [39] and [2]). We can state it also for motives without transfers by means of the equivalences proved in [52] (see also [5,Appendix B]). The proof of the statement above is highly non-trivial, and uses the whole equipment of resolution of singularities, induction on dimension and localization. Proof. From the previous theorem, we deduce that the motives Λ(X)(d) with d ∈ Z and with X [analytification of] a smooth projective variety are generators of the stable category DAé t (K, Λ) [resp. RigDAé t (K, Λ)]. The first statement follows then from [3, Proposition 2.1.24] (sometimes referred to as the theorem of Neeman and Ravenel, see [45]). Since L Rig * is monoidal and the motive Λ(X)(d) is strongly dualizable in DAé t,gm (K, Λ) for any smooth projective variety X (see [2,Lemma 1.3.29]), we also deduce the final claim. . By what showed above, it suffices to show that for any smooth projective variety X the complex Hom • (Λ(X an ), Ll * Ω † ) is compact. But this complex is, by Theorem 4.23 and [23, Theorem 2.26] canonically isomorphic to Ω(X an ) which is in turn quasi-isomorphic, by [25,Theorem 2.3] to the complex Ω(X) which is manifestly compact.
For the last assertion, by Definition 5.16 it suffices to remark that both the functor L(·) * η and the functor R(·) σ * preserve compact objects: the former has a sum-preserving right adjoint (see Remark 5.2) and the latter is a triangulated equivalence.
Remark 5.25. When we apply the previous corollary to the motives M = Λ(X) for a quasiprojective variety X over k of finite type [resp. a smooth quasi-compact rigid analytic variety X over K] (see Example 5.20) we obtain the finite dimensionality and the boundedness of the rigid [resp. overconvergent de Rham] cohomology groups of X.
Remark 5.26. The proof consists in a big "formal" step, which reduces the assertion to the study of Ω(X) for a smooth projective variety X over K. We can therefore conclude that the finite dimensionality of the overconvergent and rigid cohomologies follows formally from the finite dimensionality of the cohomology of Ω(X) aka the Betti cohomology of X(C) (fixing a map of abstract fields K → C to an algebraically closed complete archimedean field C).
Remark 5.27. We point out that we do not require that the valuation of K is discrete, and that the previous proof easily generalizes to arbitrary cohomology theories represented in RigDA(K, Q) which are finite dimensional on analytifications of projective smooth algebraic varieties over K. 22 Similarly, we obtain an alternative proof of a result of Gabber [10,Corollary 5.5.2]. We now indicate with Ω † K the complex Ω † to emphasize its dependence on the base field K. Corollary 5.28. Let L/K be an extension of complete valued fields and suppose Λ = L. Let f be the induced map Spa L → Spa K. The canonical map in RigDA † ef et (K, Λ) Proof. We can prove that the cone of the map C is zero, which by Theorem 5.21 amounts to prove that Hom(Rl * Λ(X an )[−i], C) = 0 for all smooth projective varieties X/K and all i. This follows from the following isomorphisms In the same spirit, inspired by the work of Cisinski-Déglise [18], we can also obtain the Künneth formula, generalizing [24,Section 9.4] where the discrete-valuation case is considered. and we can prove it is invertible by checking this on the set of generators of RigDAé t,gm (K, Λ) formed by analytifications of smooth projective algebraic varieties. The result then follows from the usual Künneth formula for algebraic de Rham cohomology [36].

APPENDIX A. DAGGER SPACES AND INVERSE LIMITS OF ADIC SPACES
In this section, we study the connection between the language of dagger algebras with the theory of adic spaces developed by Huber [32]. Our aim is to show that a dagger algebra is a presentation of the compactification of Spa(A, A • ) as an inverse limit of strict inclusions. We do not claim much originality here, as the results are straightforward reformulations of [9], [23] and [33].
We start by recalling the notion of compactification of Huber. In order to state it, it is crucial to use the language of adic spaces, which allows more flexibility for the choice of the ring of integral functions associated with a Banach algebra.
In this Appendix, K is any complete non-archimedean field with respect to a valuation of rank 1 having a pseudo-uniformizer π and ring of valuation K • . We recall that all our rigid analytic varieties over the base field K (and hence morphisms between them) are separated and taut, and that we denote by Spa R the rigid analytic space Spa(R, R • ) whenever R is a Banach K-algebra. Example A.2. The space B 1 is not partially proper over K. Consider the total order in R >0 × δ Z induced by putting 1 < δ < R >1 and let ||·|| be the Gauss norm on K[τ ]. Consider the valuation ring K(τ ) + in K(τ ) induced by the following valuation, taking values in R >0 × δ Z ∪ {0}: | · | ∞ : f = a i τ i → ||f || · δ max{i : |a i |=||f ||} .
The associated valuation of rank 1 is the Gauss norm on K(τ ) which defines a point in B 1 . Nonetheless, there is no lift of Spa(K(τ ), K(τ ) + ) → Spa K to B 1 since |τ | ∞ = δ > 1.