Independence of `

In this section we use the results on weight monodromy from the previous section to prove independence of`-results for the cohomology of varieties over k((t)) which include the case` = p.

Definition 2.22. • Let X be a smooth proper variety over k((t)), so that H_crisⁱ (X /E_K^†) is defined as a (ϕ,∇)-module over E_K^†. Then we define

Hⁱ_p(X ) := WD(H_crisⁱ (X /RK)), this is a K^un-valued Weil–Delgine representation over k((t)).

• Let X be a smooth curve, so that Hⁱ

rig(X /E_K^†) is defined as a (ϕ,∇)-module over E_K^†. Then we define

Hⁱ_p(X ) := WD(H_rigⁱ (X /RK)), this is a K^un-valued Weil–Delgine representation over k((t)).

• Let X be a smooth curve, so that Hⁱ_c,rig(X /E_K^†) is defined as a (ϕ,∇)-module over E_K^†. Then we define

Hⁱ_c,p(X ) := WD(Hⁱ_c,rig(X /RK)), this is a K^un-valued Weil–Delgine representation over k((t)).

Then following Deligne, these are objects that can be compared with the families of

`-adic representations {H<sub>ét</sub><sup>i</sup>(X<sub>k((t))</sub><sup>sep</sup>,Q`)}<sub>`6=p</sub>, {H<sup>i</sup><sub>c,ét</sub>(X<sub>k((t))</sub><sup>sep</sup>,Q`)}_{`6=p</sub>, or rather with the family of associated Weil–Deligne representations. {H<sub>`}ⁱ(X )}_{`6=p</sub>, {H<sup>i</sup><sub>c,`}(X )}_`6=p.

Definition 2.23 ( [2], §8). i) Let F/E be an extension of characteristic 0 fields, and V an F-valued Weil–Deligne representation of k((t)). Then we say that F is defied over E if for any algebraically closed fieldΩcontaining F, V ⊗Ωis isomorphic to all of its Aut(Ω/E)-conjugates.

ii) Let {E_i}_i∈Ibe a family of field extensions of some fixed characteristic 0 field E, and {V_i}_i∈I a family ofΩi-valued Weil–Deligne representations of k((t)). Then we say that {V_i}_i∈I is compatible if each V_iis defined over E, and for any i, j and any al-gebraically closed fieldΩcontaining E_iand E_j, the Weil–Deligne representations V_i⊗Ω^{and V}j⊗Ωare isomorphic.

Then we have the following, more easily checkable, characterisation of compatibility.

Lemma 2.24 ( [2], §8). A family {V_i}_i∈Ia family ofΩi-valued Weil–Deligne representa-tions ofk((t)) as above is compatible if and only if for all k the character

Tr(−|Gr^M_kVi) : W_k((t))→ Ei

of thek-th graded piece of the monodromy filtration has values in E and is independent ofi.

As with the weight-monodromy conjecture, the main ingredient of the restricted ver-sion of`-indepdence we wish to explain is essentially a result on the p-divisible groups of abelian varieties (or more generally, 1-motives). If A is an abelian variety over k((t)), we abuse notation slightly and write V<sub>`</sub>(A) for the Weil–Deligne representation asso-ciated to the`-adic Tate module of A, and Vp(A) for the Weil–Deligne representation associated to D_R(A).

Proposition 2.25. Let A be an abelian variety over k((t)). Then the family of Weil-Delgine representations

{V_{`</sub>(A)}<sub>`}

as` ranges over all primes (including ` = p) is compatible.

Proof. First suppose that A has semistable reduction. In this case we know exactly what the graded piece of the monodromy filtration look like - if we have an exact se-quence

0 → T0→ A₀^◦→ B0→ 0

expressing the connected component of the special fibre of the Néron model as an ex-tension of an abelian variety by a torus, then for all`, including ` = p we have

Gr₋₁^MV<sub>`</sub>(A) ∼= V`(T0), Gr^M₀V<sub>`</sub>(A) ∼= V`(B0), Gr₁^MV<sub>`</sub>(A) ∼= V`(T⁰₀)^∨(1)

where T⁰₀is the torus occurring in the corresponding exact sequence for the dual abelian variety A⁰. The claim then follows from independence of` for tori and abelian varieties over finite fields.

In general, we use the theory of 1-motives from [17], which gives the graded parts of the monodromy filtration Gr_k^MV_`(A) as:

• the Tate module (`-adic or p-adic) of a torus T⁰over k((t));

• the Tate module (`-adic or p-adic) of an abelian variety A⁰over k((t)) with poten-tially good reduction;

• the Weil–Deligne representation (`-adic or p-adic) associated to some continuous homomorphism

ρ : Gal(k((t))^sep/k((t))) → GLn(Z) (necessarily with finite image).

Since`-independence for the third of these is clear, and the first is of the same form but with a Tate twist, we can reduce to the case where A has potentially good reduction, and the monodromy filtration is trivial.

Then (since k is finite) there exists some finite, separable, totally ramified extension F/k((t)) such that A_F has good reduction, let A⁰_k denote the corresponding abelian va-riety over k. Then Gal(F/k((t))) acts on A⁰_kvia k-automorphisms, and exactly as in the

`-adic case, we can describe the p-adic Tate module Vp(A) as follows: the monodromy operator N is trivial, and the Galois representation is given by the induced action of Gal(F/k((t))) on H<sub>cris</sub><sup>1</sup> (A<sup>0</sup><sub>k</sub>/K )<sup>∨</sup>⊗KK<sup>un</sup>. Hence it suffices to show that for any subgroup G ⊂ Autk(A<sup>0</sup><sub>k</sub>), the character of the induced action on H<sub>cris</sub><sup>1</sup> (A<sup>0</sup><sub>k</sub>/K ) is equal to that char-acter of the induced action on H<sup>1</sup><sub>ét</sub>(A<sup>0</sup><sub>¯k</sub>,Q`) for all` 6= p. Since crystalline cohomology of abelian varieties coincides with the Dieudonne module of the associated p-divisible group, this is exactly the statement of (for example) the Corollary in Chapter V, §2 of [3].

Corollary 2.26. Let X be a smooth and proper variety over k((t)) of dimension n, and i ∈ {0,1,2n − 1,2n}. Then the family of Weil–Deligne representations

n H_`ⁱ(X )o

`

as` ranges over all primes (including ` = p) is compatible.

Proof. As before, the only real case of interest is i = 1, which follows from the previous proposition.

Corollary 2.27. Let X /k((t)) be a smooth curve, and i ≥ 0. Then the two families of Weil–Deligne representations

n H_`ⁱ(X )o

`, n

Hⁱ_c,`(X )o

`

as` ranges over all primes (including ` = p) are both compatible.

Proof. It suffices to treat the case of cohomology without supports, since the supported version then follows from Poincaré duality. Let X be a smooth compactification for X , and D = X \ X , we may assume that D 6= ;. The only case of interest again is i = 1, and for all` we have the excision exact sequence

0 → H_{`</sub><sup>1</sup>(X ) → H<sup>1</sup><sub>`}(X ) → H⁰_{`</sub>(D)(−1) → H<sub>`}²(X ) → 0.

Them some straightforward linear algebra shows that Gr^M₋₁H¹_{`</sub>(X ) = Gr<sup>M</sup><sub>−1</sub>H<sub>`}¹(X )

Gr^M₁ H¹_{`</sub>(X ) = Gr<sup>M</sup><sub>1</sub> H<sup>1</sup><sub>`}(X ) and that we have an exact sequence

0 → Gr₀^MH_{`</sub><sup>1</sup>(X ) → Gr<sup>M</sup><sub>0</sub> H<sup>1</sup><sub>`}(X ) → W`→ 0 where

W_`:= ker³

H⁰_{`</sub>(D, K )(−1) → H<sub>`}²(X )´ .

Since we know `-independence for both H<sub>`</sub>⁰(D, K )(−1) and H²_`(X ), we therefore know

`-independence for W`. Since we know`-independence for Gr<sup>M</sup><sub>k</sub> H<sup>1</sup><sub>`</sub>(X ),`-independence for Gr<sup>M</sup><sub>k</sub> H<sup>1</sup><sub>`</sub>(X ) then follows.

A Non-embeddable varieties and descent

In this appendix, for completeness as much as any other reason, we show how to ex-tend the construction of rigid cohomology Hⁱ

rig(X /E_K^†,E ), which applied to embeddable varieties X /k((t)), to those which are not necessarily embeddable. The method, using Zariski descent, is completely standard, and we will allow k to be an arbitrary field of characteristic p.

Definition A.1. Let X be a k((t))-variety. An embedding system for X is a simplicial triple (X_•, Y_•,P_•) where X_•→ X is a Zariski hyper-cover of X , each (Xn, Yn,Pn) is a smooth and proper frame overVJ^tKand all the face mapsPn→P_n−1are smooth around Xn. A Frobeniusϕ on an embedding system (X_•, Y_•,P_•) is an endomorphismϕ :P_•→ P_•lifting the absolute q-power Frobenius on P_•=P_•⊗V

J^tKkJ^tK^.

Proposition A.2. Every k((t))-variety X admits an embedding system (X_•, Y_•,P_•) with a Frobenius liftϕ.

Proof. Let {Ui} be a finite open affine covering for X . Then there exists an embedding Ui→ Pⁿ_k((t))ⁱ for some ni, and we let Uibe the closure of UiinPⁿ_kⁱ

J^tK

. We can thus consider the frame (U,U,U) where U =`

iU_i, U =`

iU_i and U₌^`_iPbⁿ_Vⁱ J^tK

. Now define U_n= U ×X. . . ×XU, with n copies of U, and similarly define U_n= U ×k. . . ×kU and U_n= U_×V

J^tK. . . ×_V

J^tKU, fibre product in the category of formalV -schemes. Then we have a simplicial triple (U_•,U_•,U_•), and we get a framing system (X_•, Y_•,P_•) for X by taking X_n= Un, Y_nto be the closure of X_nin U_nandP_n=U_n. Since eachP_nis a disjoint union of products of projective space, the simplicial formal scheme P_• admits a Frobenius lift.

Let X be k((t))-variety, and (X_•, Y_•,P_•) an embedding system for X with a Frobenius liftϕ. Then for someE ∈ (F-)Isoc^†(X /E_K^†) or (F-)Isoc^†(X /K ) we can realise eachE |Xn on (X_n, Y_n,P_n) to give a compatible collection of modules with connection on the simplicial rigid space ]Y_•[_P•.

Definition A.3. Define H_rigⁱ ((X_•, Y_•,P_•)/E_K^†,E ) := Hⁱ(]Y_•[_P•,E ⊗Ω^∗_]Y

•[_P•/SK), these are vector spaces overE_K^†.

Of course, we expect that H_rigⁱ ((X_•, Y_•,P_•)/E_K^†,E ) only depends on the pair (X.E ), and when X is embeddable coincides with the rigid cohomology defined in previous sections.

Happily this is the case.

Lemma A.4. Suppose that X is embeddable, and (X_•, Y_•,P_•) is as above. Then there is a natural isomorphism

H_rigⁱ ((X_•, Y_•,P_•)/E_K^†,E ) ∼= Hrigⁱ (X /E_K^†,E )

which is compatible with Frobenius whenE ∈ F-Isoc^†(X /E_K^†) and the connection when E ∈ Isoc^†(X /K ).

Proof. Let (X , Y ,P) be a smooth and proper frame containing X , we form a new em-bedding system for X as follows. Embed each connected component of X_n into Y via the canonical map X_n→ X → Y , so we get an smooth and proper frame (Xn, Y_n⁰ :=

`k<sub>n</sub>Y ,P<sup>0</sup><sub>n:=</sub><sup>`</sup>_knP) for all n. These fit together to form an embedding system (X_•, Y_•⁰,P⁰_•).

Then we defineP⁰⁰_n=P_n×V

J^tKP⁰_nand Y_n⁰⁰to be the closure of X_ninsideP⁰⁰_nfor the diag-onal embedding, so that we have a diagram of embedding systems

(X_•, Y_•⁰⁰,P⁰⁰_•) then follows from the proof of Theorem 4.5 of [11] that

Rp_1∗(E ⊗Ω^∗_]Y00 re-peated application of Lemma 4.4 from [11] gives a resolution

j^†_XE ' Tot( j^†_X•E)

for any sheaf E on ]Y [_P, where Tot denotes the un-normalised cochain complex associ-ated to a cosimplicial sheaf on ]Y [_P. Hence

H_rigⁱ (]Y [_P,E ⊗Ω]Y [_P/S_K) ∼= Hrigⁱ (]Y_•[_P•,E ⊗Ω]Y_•[_P•/S_K) as required.

The isomorphism H_rigⁱ ((X_•, Y_•,P_•)/E_K^†,E ) ∼= H_rigⁱ (X /E_K^†,E ) is easily checked to be com-patible with ground field extensions and functoriality in X , and hence is comcom-patible with Frobenius whenE ∈ F-Isoc^†(X /E_K^†). It is also easily seen to be compatible with the sim-plicial object is the category of smooth and proper frames overVJ^tK. Then for each fixed n, (X_n,•, Y_n,•,P_n,•) is an embedding system for the embeddable variety X_n, and if fact we get a whole augmented simplicial triple (X_n,•, Y_n,•,P_{n,•) → (Xn}, Y_n,P_n). Examining the proof of Lemma A.4 tells us that

Rπ_n∗(E ⊗Ω^∗_]Yn,•[

Pn,•/SK) ∼= E ⊗Ω^∗_]Yn[

Pn/SK

where πn:]Y_n,•[_Pn,•→]Yn[_Pn is the natural morphism. Hence if we letπ :]Y•,•[_P•,•→ ]Y_•[_P• denote the natural morphism, we get an isomorphism

Rπ_∗(E ⊗Ω^∗_]Y•,•[

P•,•/SK) ∼= E ⊗Ω^∗_]Y•[

P•/SK

and hence the natural morphism Hⁱ(]Y_•[_P•,E ⊗Ω^∗_]Y•[

P•/SK) → Hⁱ(]Y_•,•[_P•,•,E ⊗Ω^∗_]Y•,•[

P•,•/SK)

is an isomorphism. Of course, the same is true replacing (X_•, Y_•,P_•) by (X_•⁰, Y_•⁰,P⁰_•) and so we get a canonical isomorphism

Hⁱ(]Y_•[_P•,E ⊗Ω^∗_]Y•[

P•/SK) → Hⁱ(]Y_•⁰[_P0

•,E ⊗Ω^∗_]Y0

•[ P0•/SK) by composing this isomorphism with the inverse of

Hⁱ(]Y_•⁰[_P0

•,E ⊗Ω^∗_]Y0

•[

P0•/S_K) → Hⁱ(]Y_•,•[_P•,•,E ⊗Ω^∗_]Y•,•[

P•,•/S_K).

These are easily checked to compatible with the isomorphisms to the cohomology of any third embedding system (X⁰⁰_•, Y_•⁰⁰,P⁰⁰_•).

Hence we get well-defined and functorial cohomology groups H_rigⁱ (X /E_K^†,E ) which ad-mit semi-linear Frobenii whenE ∈ F-Isoc^†(X /E_K^†), and connections whenE ∈ Isoc^†(X /K ).

These structures are compatible whenE ∈ F-Isoc^†(X /K ). Of course, entirely similar con-siderations apply to constructingE_K^†-valued rigid cohomology with compact supports for non-embeddable varieties. The only new ingredient needed is (a repeated application of) the following analogue of Lemma 4.4 from [11].

Lemma A.6. Let (X, Y ,P) be aVJ^tK-frame, and X = U1∪ U2an open cover ofX . Then for any sheafE on ]Y [_Pthere is an exact triangle

RΓ]U1∩U2[_PE → RΓ]U1[_PE ⊕ RΓ]U2[_PE → RΓ]X [_PE⁺¹→ of complexes of sheaves on ]Y [_P.

Proof. Choose complements Z and H_j for X and U_j in Y respectively, so that H₁∪ H2

is a complement for U₁∩U2. Let

i :]Z[_P→]Y [P, i_j:]H_j[_P→]Y [P, i₁₂:]H₁∪ H2[_P→]Y [P

denote the corresponding open immersions. Note that for any sheaf F on ]Y [_Pwe have a diagram

F //



F ⊕ F //



F



+1 //

Ri_12∗i⁻¹₁₂F //_Ri1∗i⁻¹

1 F ⊕ Ri2∗i⁻¹₂ F //_Ri∗i⁻¹_F ⁺¹ //

in which both rows are exact (since ]H₁∪ H2[_P=]H1[_P∪]H2[_Pand ]H₁∩ H2[_P=]Z[P).

Hence we get an exact triangle

¡F → Ri12∗i⁻¹₁₂F¢ [1] → ¡F → Ri1∗i⁻¹₁ F¢ [1] ⊕ ¡F → Ri2∗i⁻¹₂ F¢ [1] → ¡F → Ri∗i⁻¹F¢ [1]⁺¹→ . Now we let Y⁰= Y ⊗kJ^tKk((t)) and denote by k :]Y⁰[_P→]Y [Pthe inclusion, applying this to F = k_∗k⁻¹E gives the result.

References

[1] Aise Johan de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. I.H.E.S. 82 (1995), 5–96.

[2] Pierre Deligne, Les constantes des équations fonctionnelles des fonctions L, Mod-ular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, pp. 501–597. Lecture Notes in Math., Vol.

349.

[3] Michel Demazure, Lectures on p-divisible groups, Lecture Notes in Mathematics, vol. 302, Springer-Verlag, Berlin-New York, 1972.

[4] Kazuhiro Fujiwara and Fumihara Kato, Foundations of rigid geometry I, preprint (2013), arXiv:math/1308.4734v1, to appear in EMS Monographs in Mathematics.

[5] Alexander Grothendieck, SGA 7 I: Groupes de monodromie en géométrie al-gébrique, Expose IX: Modèles de Néron et monodromie, Lecture Notes in Mathe-matics, no. 288, Springer-Verlag, 1973.

[6] Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, As-pects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996.

[7] Kiran Kedlaya, Descent theorems for Overconvergent F-crystals, Ph.D. thesis, Mas-sachusetts Institute of Technology, June 2000.

[8] , Full faithfulness for overconvergent F-isocrystals, Geometric Aspects of Dwork Theory, vol. II, de Gruyter, 2004, pp. 819–835.

[9] , A p-adic local monodomy theorem, Ann. of Math. 160 (2004), no. 1, 93–

184.

[10] , Finiteness of rigid cohomology with coefficients, Duke Math. J. 134 (2006), no. 1, 15–97.

[11] Christopher Lazda and Ambrus Pál, Rigid cohomology over Laurent series fields I:

First definitions and basic properties, preprint (2014), arXiv:math/1411.7000.

[12] , Rigid cohomology over Laurent series fields II: Finiteness for smooth curves, preprint (2014), arXiv:math/1412.5300.

[13] Bernard Le Stum, Rigid cohomology, Cambridge Tracts in Mathematics, vol. 172, Cambirdge University Press, 2007.

[14] Adriano Marmora, Facteurs epsilon p-adiques, Comp. Math. 144 (2008), no. 2, 439–483.

[15] Arthur Ogus, F-isocrystals and de Rham cohomology II - Convergent isocrystals, Duke Math. J. 51 (1984), no. 4, 765–850.

[16] Ambrus Pál, The p-adic monodromy group of abelian varieties over global function fields in charcateristicp, preprint.

[17] Michel Raynaud, 1-motifs et monodromie géométrique, Astérisque (1994), no. 223, 295–319, Périodes p-adiques (Bures-sur-Yvette, 1988).

[18] Nobuo Tsuzuki, Slope filtration of quasi-unipotent overconvergent F-isocrystals, Ann. Inst. Fourier, Grenoble 48 (1998), no. 2, 379–412.

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