INTEGRAL p-ADIC HODGE THEORY OF FORMAL SCHEMES IN LOW RAMIFICATION

LOW RAMIFICATION

YU MIN

Abstract. We prove that for any proper smooth formal scheme Xover OK, where OK is the ring of integers in a complete discretely valued nonarchimedean extension K ofQ^p with perfect residue fieldk and ramification degree e, thei-th Breuil-Kisin cohomology group and its Hodge-Tate specialization admit nice decompositions when ie < p−1. Thanks to the comparison theorems in the recent works of Bhatt, Morrow and Scholze [BMS18], [BMS19], we can then get an integral comparison theorem for formal schemes, which generalizes the case of schemes proven by Fontaine and Messing in [FM87] and Caruso in [Car08].

Contents

0. Introduction 1

Acknowledgements 4

1. Recollections onAinf-cohomology 5

2. A lemma in commutative algebra 7

3. Hodge-Tate cohomology 8

4. The unramified case: comparison theorem 11

4.1. Decomposition of Hodge-Tate cohomology groups 11

4.2. Hodge-to-de Rham spectral sequence 12

5. The ramified case: comparison theorem 15

6. Breuil-Kisin modules 19

References 22

0. Introduction

In this paper, we study theA_inf-cohomology theory and the Breuil-Kisin cohomology theory constructed respectively in [BMS18], [BMS19], now unified as prismatic cohomol- ogy in [BS19]. Our main result concerns their structure in low ramification.

Theorem 0.1 (Theorem 3.8, Theorem 5.7). Let X be a proper smooth formal scheme overO_K, whereO_K is the ring of integers in a complete discretely valued nonarchimedean extensionK of Qp with perfect residue field k and ramification degree e. Let O_C be the ring of integers in a completed algebraic closureC of K and X be the adic generic fibre ofX¯ :=X⊗_OKO_C. Assumeie < p−1. Then there is an isomorphism ofS:=W(k)[[u]]- modules

H_Sⁱ(X)∼=H_´etⁱ (X,Zp)⊗_ZpS

whereH_Sⁱ(X)is the Breuil-Kisin cohomology ofX. Similarly under the same assumption, there is an isomorphism ofO_K-modules

H_HTⁱ (X)∼=H_´etⁱ (X,Zp)⊗_ZpO_K.

1

where H_HTⁱ (X) is the Hodge-Tate cohomology of X.

Remark 0.2. Note that the definition of Breuil-Kisin modules (see Definition 1.9) in [BMS18], [BMS19] is slightly more general than the original definition given by Kisin in [Kis06]. The difference lies in the existence ofu-torsion (note thatS=W(k)[[u]] is a two dimensional regular local ring). However, the theorem above shows that the Breuil- Kisin cohomology theory constructed by Bhatt, Morrow and Schloze does take values in the category of Breuil-Kisin modules in a traditional sense, at least when ie < p−1.

Unfortunately, we can not give any canonical isomorphisms between these modules.

Our method only enables us to compare the module structure. The proof of this theorem relies essentially on the existence of Breuil-Kisin cohomology and the construction of Ainf-cohomology in [BMS18] by using the Lη-functor and the pro-´etale site. In fact, this construction presents a close relation between A_inf-cohomology and p-adic ´etale cohomology. The Lη-functor provides us with two morphisms between H_Aⁱ

inf( ¯X) (resp.

H_HTⁱ ( ¯X)) andH_´etⁱ (X,Zp)⊗_ZpA_inf (resp. H_´etⁱ (X,Zp)⊗_ZpO_C), whose composition in both direction is µⁱ (resp. (ζ_p−1)ⁱ). For the definitions ofµand ζ_p, see Section 1.

Note that H_HTⁱ ( ¯X) is just the base change of H_HTⁱ (X) along the canonical injection O_K → O_C. We can then directly obtain the statement about the Hodge-Tate cohomology groups in Theorem 0.1 by studying the two morphisms provided by theLη-functor.

For the part concerning the Breuil-Kisin cohomology groups, we need to prove some torsion-free result. Namely, whenie < p−1, the Breuil-Kisin cohomology groupH_Sⁱ (X) isE(u)-torsion free (equivalently,u-torsion free), whereE(u)∈Sis the Eisenstein poly- nomial for a fixed uniformizer π in O_K. Moreover for any positive integer n, we have H_Sⁱ(X)/pⁿ is alsoE(u)-torsion free.

As a consequence of Theorem 0.1, we can get an integral comparison theorem about p-adic ´etale cohomology and crystalline cohomology both in the unramified case and ramified case, which generalizes the case of schemes studied by Fontaine and Messing in [FM87] and Caruso in [Car08]. This is actually the main motivation of this work. So next we give some background about integral comparison theorems.

Integralp-adic Hodge theory. For a proper smooth (formal) schemeXoverO_Kwhere O_K is the ring of integers in a complete discretely valued nonarchimedean extension K ofQp with perfect residue fieldk, we can consider the de Rham cohomologyH_dRⁱ (X/O_K) ofX, thep-adic ´etale cohomologyH_´etⁱ (X,Zp) of the geometric (adic) generic fiberX and the crystalline cohomologyH_crysⁱ (Xk/W(k)) of the special fiberX_k, wherekis the residue field ofO_K. Integralp-adic Hodge theory then studies the relations of these cohomology theories.

The first result concerning integral comparison was given by Fontaine and Messing in [FM87].

Theorem 0.3 ([FM87]). Let X be a proper smooth scheme over W(k) and Xn = X×_Spec(W(k)) Spec(W(k)), where k is a perfect field of characteristic p. Let G_K0 de- note the absolute Galois group of K0 = W(k)[¹_p]. Then for any integer r such that 0≤r≤p−2, there exists a natural isomorphism ofG_K0-modules

T_crys(H_dR^r (X_n))'H_´et^r(XK¯0,Zp/pⁿ)

where Tcrys is a functor from the category of torsion Fontaine-Laffaille modules to the category of Zp[G_K0]-modules, which preserves invariant factors.

Note thatH_dR^r (X_n)∼=H_crys^r (X_k/W_n(k)). Here we have used implicitly thatH_dR^r (X_n) is in the category of torsion Fontaine-Laffaille modules, which is actually one of the main difficulties. The proof of Fontaine-Messing’s theorem relies on syntomic cohomol- ogy which acts as a bridge connecting integral p-adic ´etale cohomology and crystalline cohomology.

Recall that rationalp-adic Hodge theory provides an equivalence between the category of crystalline representations and the category of (weakly) admissible filteredϕ-modules.

The idea of Fontaine-Laffaille’s theory is to try to classify GK0-stable Zp-lattices in a crystalline representation V by ϕ-stable W(k)-lattices in D satisfying some conditions, whereD is the corresponding admissible filteredϕ-module.

To generalize Fontaine-Laffaille’s theory to the semi-stable case, Breuil introduced the ring S and related categories of S-modules in order to add a monodromy operator. He has also obtained an integral comparison result in the unramified case when r < p−1 in [Bre98a]. Later, this result was generalized to the case thate(r+ 1)< p−1 by Caruso in [Car08].

Theorem 0.4 ([Bre98a] [Car08]). Let X be a proper and semi-stable scheme over O_K, whereO_Kis the ring of integers in a complete discretely valued nonarchimedean extension K of Qp with perfect residue field k and ramification degree e. Let X_n be X×_SpecOK Spec(O_K/pⁿ). Fix a non-negative integer r such that er < p−1. Then there exists a canonical isomorphism of Galois modules

H_´etⁱ(XK¯,Z/pⁿZ)(r)∼=Tst∗(H_log−crysⁱ (Xn/(S/pⁿS))) for anyi < r.

Tst∗ is a functor from the category Mod^r,ϕ_/S

∞ (see Definition 6.9) to the category of Zp[G_K]-modules, which preserves invariant factors. The proof also relies on the use of syntomic cohomology. One of the main difficulties in their proof is to show H_log−crysⁱ (X_n/(S/pⁿS)) is in the category Mod^r,ϕ_/S

∞, in particular, to showH_log−crysⁱ (X_n/(S/pS)) is finite free over S/pS.

Remark 0.5. One crucial point of Breuil’s theory is that it highly depends on the restriction r ≤ p−1 which is rooted in the fact that the inclusion ϕ(Fil^rS) ⊂ p^rS is true only whenr ≤p−1. One way to remove this restriction is to consider Breuil-Kisin modules. In fact, one of the main motivations of A_inf-cohomology theory is to give a cohomological construction of Breuil-Kisin modules. The techniques in [BMS18] can not directly give the desired Breuil-Kisin cohomology. However, this goal is achieved in [BMS19] by using topological cyclic homology and in [BS19] by defining prismatic site in a more general setting.

Recently, Bhatt, Morrow and Scholze have obtained a more general result about the relation between p-adic ´etale cohomology and crystalline cohomology in [BMS18] by using A_inf-cohomology. Their result does not impose any restriction on the ramification degree, roughly saying that the torsion in the crystalline cohomology gives an upper bound for the torsion in thep-adic ´etale cohomology.

As we have said, by studyingAinf-cohomology and its descent Breuil-Kisin cohomology, we can generalize the results of Fontaine-Messing, Breuil and Caruso to the case of formal schemes, at least in the good reduction case.

Theorem 0.6 (Theorem 4.9, Theorem 5.9). Let X be a proper smooth formal scheme overO_K, whereO_K is the ring of integers in a complete discretely valued nonarchimedean extension K of Qp with perfect residue field k and ramification degree e. Let C be a completed algebraic closure of K andX¯ =X×_OKO_C. WriteX for the adic generic fiber ofX. Then when¯ e= 1, there is an isomorphism ofW(k)-modulesH_´etⁿ(X,Zp)⊗_ZpW(k)∼= H_crysⁿ (X_k/W(k)) for any n < p−1. When e >1, the same isomorphism exists for any n such that(n+ 1)e < p−1.

For the proof in the unramified case, we need the following theorem:

Theorem 0.7 (Theorem 4.8). With the same assumptions as the theorem above, when e= 1, then for any n < p−1, we have

length_Zp(H_´etⁿ(X,Zp)tor/p^m))≥length_W(k)(H_crysⁿ (Xk/W(k))tor/p^m) for all positive integer m.

In fact, the (truncated) Hodge-Tate complex of sheaves is formal in this case, which enables us to compare Hodge-Tate cohomology and Hodge cohomology. Then we need to study the Hodge-to-de Rham spectral sequence. By Theorem 0.1, we can finally relate de Rham (or crystalline) cohomology top-adic ´etale cohomology. Note that the theorem above gives a converse to Theorem 1.8 in [BMS18], which implies that H_´etⁿ(X,Zp) and H_crysⁿ (X_k/W(k)) have the same invariant factors.

In the ramified case, the integral comparison theorem follows directly from Theorem 0.1 and Theorem 1.6 in [BMS18].

Remark 0.8. The A_inf-cohomology theory in the semistable case has been studied in [CK19]. The Breuil-Kisin cohomology might be also generalized to the semistable case by using the prismatic site. Then one could also hope to generalize Theorem 0.1 and Theorem 0.6 to the semistable case.

Acknowledgements. I would like to express my great gratitude to my advisor Matthew Morrow for suggesting this problem to me and many helpful discussions during the prepa- ration of this work. I also want to thank Takeshi Tsuji and Shizhang Li for their valuable comments.

Notations. In the whole paper, letK be a complete discretely valued nonarchimedean extension of Qp with the ring of integers O_K, perfect residue field k and ramification degree e = [K : W(k)[¹_p]]. Fix a uniformizer π of O_K and write E = E(u) for an Eisenstein polynomial for π. Define S:= W(k)[[u]] and S_n := S/pⁿ. Then there is a natural W(k)-linear surjective morphism β :S → O_K sending u to π, whose kernel is generated byE.

Let X be a proper smooth formal scheme over O_K, and ¯X= X⊗_OK O_C, where O_C is the ring of integers of a complete algebraic closure C of K. We useX to denote the adic generic fibre of ¯X and X_k to denote the special fibre of X. Fix an algebraic closure

¯kof kand define X_k¯ :=X_k⊗_kk.¯

LetA be a commutative ring andM be anA-module. For any a∈A, we writeM[a]

for the submodule ofM consisting ofa-torsion elements.

1. Recollections on Ainf-cohomology

In this section, we simply recall the necessary ingredients for defining theAinf-cohomology theory in [BMS18]. In fact, we will stick to the method using the pro-´etale site and the d´ecalage functor Lη, which will provide us with some useful morphisms between A_inf- cohomology groups and ´etale cohomology groups.

We first recall some basic definitions in p-adic Hodge theory.

Definition 1.1. (i) Define O_C^[ := lim←−x→x^pO_C/p and Ainf := W(O_C^[), the Witt vector ring of O^[_C. Note that Ainf is equipped with a natural Frobenius map ϕ, which is an isomorphism of rings.

(ii) Fix a compatible system of primitive p-power roots of unity {ζ_pⁿ}_n∈N such that ζ_p^pn+1 =ζpⁿ. Under the isomorphism of multiplicative monoidsO^[_C ∼= lim←−x→x^pO_C, we define := (1, ζp, ζ_p²,· · · , ζpⁿ,· · ·)∈ O^[_C and µ:= []−1∈Ainf.

(iii) There is a map θ : A_inf → O_C defined by Fontaine. The map θ is surjective and ker(θ) is generated by ξ = µ/ϕ⁻¹(µ). There is also another projection θe:=θ◦ϕ⁻¹ :A_inf → O_C, whose kernel is generated by ξe:=ϕ(ξ) =ϕ(µ)/µ.

Next we define some sheaves on the pro-´etale siteX_pro´et. Recall that there is a natural projection map of sites

ω :X_pro´et→X_´et

Definition 1.2 ( [BMS18] Section 5). Consider the following sheaves on X_pro´et. (i) The integral structure sheaf O⁺_X :=ω^∗O_X⁺

´ et. (ii) The structure sheaf O_X :=ω^∗O_X´et.

(iii) The completed integral structure sheaf Ob⁺_X := lim←−nO_X⁺/pⁿ. (iv) The completed structure sheaf Ob_X :=Ob_X⁺[¹_p].

(v) The tilted completed integral structure sheaf Ob⁺

X^[ := lim←−ϕO_X⁺/p.

(vi) Fontaine’s period sheafAinf,X, which is the derivedp-adic completion ofW(Ob⁺

X^[).

The other ingredient for definingA_inf-cohomology is the d´ecalage functor.

Definition 1.3(TheLη-functor, [BMS18] Section 6). Let (T,O_T) be a ringed topos and I ⊂ O_T be an invertible ideal sheaf. For any I-torsion-free complex C^• ∈ K(O_T), we can define a new complexηIC^•= (ηIC)^• ∈K(O_T) with terms

(ηIC)ⁱ :={x∈Cⁱ|dx∈ I ·Cⁱ⁺¹} ⊗_OT I^⊗i

For every complexD^•∈K(O_T), there exists a stronglyK-flat complexC^• ∈K(O_T)and a quasi-isomorphismC^• →D^•. By saying stronglyK-flat, we mean that eachCⁱ is a flat O_T-module and for every acyclic complexP^• ∈K(O_T), the total complex Tot(C^•⊗P^•) is acyclic. In particular,C^• is I-torsion free. Then we can define

LηI :D(O_T)→D(O_T) LηI(D^•) :=ηI(C^•)

We consider the natural projectionν:Xpro´et→X¯_zar, which is actually the composition X_pro´et→X_´et→X¯_´et→X¯_zar. Then we are ready to define theA_inf-cohomology theory.

Definition 1.4 ( [BMS18] Definition 9.1). Define AΩX¯ := LηµRν∗(Ainf,X) and ΩeX¯ :=

Lη_ζp−1Rν∗(Ob_X⁺). The A_inf-cohomology is defined to be the Zariski hypercohomology of

the complex of sheaves AΩX¯, i.e. RΓAinf( ¯X) := RΓzar( ¯X, AΩX¯). We can also define Hodge-Tate cohomology RΓHT( ¯X) :=RΓzar( ¯X,ΩeX¯).

Now we introduce the category of Breuil-Kisin-Fargues modules in which the Ainf- cohomology takes values.

Definition 1.5 ( [BMS18] Definition 4.22). A Breuil-Kisin-Fargues module is a finitely presentedA_inf-moduleM which becomes free overA_inf[¹_p]after invertingpand is equipped with an isomorphism

ϕM :M⊗_Ainf,ϕAinf[1

ξe]−→^' M[1 ξe].

The main theorem about theAinf-cohomology theory is the following:

Theorem 1.6 ([BMS18] Theorem 14.3). The complexRΓAinf( ¯X)is a perfect complex of A_inf-modules with a ϕ-linear mapϕ:RΓ_Ainf( ¯X)→RΓ_Ainf( ¯X) which becomes an isomor- phism after inverting ξ resp. ξ. The cohomology groupse H_Aⁱ

inf( ¯X) := Hⁱ(RΓ_Ainf( ¯X)) are Breuil-Kisin-Fargues modules. Moreover, there are several comparison results:

(i) ´Etale specialization: RΓ_Ainf( ¯X)⊗_Ainf W(C^[) ' RΓ_´et(X,Zp)⊗_ZpW(C^[), where C^[= Frac(O_C^[).

(ii) Crystalline specialization: RΓA_inf( ¯X)⊗^L_A

inf W(¯k)'RΓcrys(X¯k/W(¯k)).

(iii) De Rham specialization: RΓ_Ainf( ¯X)⊗^L_A

inf,θO_C 'RΓ_dR( ¯X/O_C).

(iv) Hodge-Tate specialization: ΩeX¯ ' AΩX¯ ⊗^L

Ainf,eθO_C and RΓ_Ainf( ¯X)⊗^L

Ainf,eθO_C ' RΓ_HT( ¯X).

Corollary 1.7. For all i≥0, we have (i) H_Aⁱ

inf( ¯X)⊗_AinfW(C^[)∼=H_´etⁱ (X,Zp)⊗_ZpW(C^[).

(ii) 0→H_Aⁱ

inf( ¯X)⊗_Ainf W(¯k)→H_crysⁱ (X¯k/W(¯k))→T or^A₁^inf(H_Aⁱ⁺¹

inf( ¯X), W(¯k))→0.

(iii) 0→H_Aⁱ

inf( ¯X)⊗_Ainf,θ O_C →H_dRⁱ ( ¯X/O_C)→H_Aⁱ⁺¹

inf( ¯X)[ξ]→0.

(iv) 0→H_Aⁱ

inf( ¯X)⊗_A

inf,eθO_C →H_HTⁱ ( ¯X)→H_Aⁱ⁺¹

inf( ¯X)[eξ]→0.

One of the most important applications of theA_inf-cohomology theory is to show that the torsion in the crystalline cohomology gives an upper bound for the torsion in the

´etale cohomology.

Theorem 1.8 ([BMS18] Theorem 14.5). For any n, i≥0, we have the inequality length_W(k)(H_crysⁱ (X_k/W(k))_tor/pⁿ)≥length_Zp(H_´etⁱ (X,Zp)_tor/pⁿ).

As we have mentioned, there is a refinement of theA_inf-cohomology, i.e. Breuil-Kisin cohomology, which recovers A_inf-cohomology after base change along a faithfully flat mapα :S→A_inf. In particular, we have (α(E)) = (eξ). The first construction of Breuil- Kisin cohomology is given in [BMS19] by using topological cyclic homology. Another construction is given in [BS19] by using the prismatic site. We will not say anything about the construction of Breuil-Kisin cohomology theory here but choose to state a similar comparison theorem as in theAinf case.

Definition 1.9. A Breuil-Kisin module is a finitely generatedS-moduleM together with an isomorphism

ϕM :M ⊗_S,ϕS[1

E]→M[1 E].

Theorem 1.10 ([BMS19] Theorem 1.2). For any proper smooth formal scheme X/O_K, there is a S-linear cohomology theory RΓ_S(X) which is a perfect complex of S-modules.

Moreover, it is equipped with a ϕ-linear map ϕ:RΓ_S(X) →RΓ_S(X) which induces an isomorphism

RΓS(X)⊗_S,ϕS[1

E]'RΓS(X)[1 E]

The cohomology groups H_Sⁱ(X) := Hⁱ(RΓ_S(X)) are Breuil-Kisin modules. There are several specializations that recover other p-adic cohomology theories:

(i) After base change alongα :S→Ainf, it recoversAinf-cohomology : RΓS(X)⊗_S,α A_inf 'RΓ_Ainf( ¯X).

(ii) Etale specialization:´ RΓS(X)⊗_S,eαW(C^[) ' RΓ´et(X,Zp)⊗_ZpW(C^[), where αe is the composition S−→^α A_inf ,→W(C^[).

(iii) De Rham specialization: RΓS(X)⊗^L

S,eβO_K 'RΓdR(X/O_K), where βe:=β◦ϕ: S→ O_K.

(iv) Crystalline specialization: after base change along the map S→W(k) which is the Frobenius on W(k) and sends u to 0, it can recover crystalline cohomology of the special fiber: RΓ_S(X)⊗^L_SW(k)'RΓ_crys(X_k/W(k)).

2. A lemma in commutative algebra

In this section, we want to prove the following key lemma which will be used in the comparison of Hodge-Tate cohomology andp-adic ´etale cohomology.

Lemma 2.1. Let M = Lm

i=1O_C/β^mi and N = Ln

j=1O_C/β^nj, where β 6= 0 is in the maximal ideal m of O_C and all m_i, n_j are positive integers. Suppose there are two O_C- linear morphisms f :M →N and g:N →M such that g◦f =α and f◦g=α, where β =αx in O_C and x is not a unit. Then m=n and the multi-sets {m_i} and {n_j} are the same, i.e.,M ∼=N.

In order to prove this lemma, we consider all finitely presented torsion modules over O_C. Any such module looks like Ln

i=1O_C/π_i for some non-zeros π_i ∈ m. We call trk(N) :=nthe torsion-rank ofN. We will also use the normalized lengthlOC for finitely presented torsion O_C-modules, as used in [CK19], see also [Bha17]. In particular, this length behaves additively under short exact sequences andlO_C(O_C/p) = 1.

Now we prove a lemma concerning the torsion-rank:

Lemma 2.2. LetN ,→M be an injection of finitely presented torsionO_C-modules, then trk(N)≤trk(M). Dually ifN M, then trk(N)≥trk(M).

Proof. Write N =Ln

i=1O_C/π_i and M =Lm

i=1O_C/$_i. Letπ be the smallest of the π_i (i.e., the one with the smallest valuation), and let$ be the largest of$i. Then

(O_C/π)ⁿ⊂N ,→M ⊂(O_C/$)^m,

which shows that$∈πO_C; write $=πx. The composition of these maps lands in the π-torsion of (O_C/$)^m, which is isomorphic to (xO_C/$O_C)^m ∼= (O_C/πO_C)^m. So now we have an injection (O_C/π)ⁿ,→(O_C/π)^m. Taking length shows that n≤m.

IfN M, we can consider the injectionHom(M,O_C/t),→Hom(N,O_C/t) wheretis any element inm. Then we have trk(M) = trk(Hom(M,O_C/t))≤trk(Hom(N,O_C/t)) =

trk(N).

Lemma 2.3. Let g:N → M be a morphism of finitely presented torsion O_C modules;

Write N =Ln

i=1O_C/πi and M =Lm

i=1O_C/$i. Assume that Ker(g) is killed by some elementα ∈ O_C which is strictly smaller that all of theπ_i. Then trk(N)≤trk(M).

Proof. By assumption Ker(g) is contained in theα-torsionN[α] ofN, which is given by N[α]∼=Ln

i=1π

αO_C/πO_C. So

N N/Ker(g)N/N[α]∼=

n

M

i=1

π

αO_C/πO_C.

Taking torsion-ranks, Lemma 2.2 for surjections shows that trk(N/Ker(g)) = trk(N).

ButN/Ker(g),→M, so Lemma 2.2 also shows that trk(N/Ker(g))≤trk(M).

Now we are ready to prove Lemma 2.1.

Proof of Lemma 2.1. Note that the number of invariant factor β^k in M is equal to trk(β^k−1M) −trk(β^kM). By Lemma 2.3, we have trk(β^kM) = trk(β^kN) for any k.

This means that the number of invariant factorβ^k inM and that inN are equal for any

k. So we must have M ∼=N.

Lemma 2.4. Let M =O^r_C⊕(Lm

i=1O_C/β^mi) andN =O_C^s ⊕(Ln

j=1O_C/β^nj). Suppose there are twoO_C-linear morphismsf :M →N and g:N →M such thatg◦f =α and f◦g=α, where β =αx in O_C and x is not a unit. Then M ∼=N.

Proof. According to Lemma 2.1, M/β^k and N/β^k are isomorphic for all k. Then the

lemma follows.

3. Hodge-Tate cohomology

In this section, we study the Hodge-Tate specialization of Breuil-Kisin cohomology of a proper smooth formal scheme X over O_K and prove the isomorphism concerning Hodge-Tate cohomology groups in Theorem 0.1 under the restriction: ie < p−1.

Our strategy is to first study the Hodge-Tate specialization ofA_inf-cohomology of ¯X.

We can take advantage of the Lη-construction of Ainf-cohomology and its Hodge-Tate specilizaton. This will provide us with two morphisms which enable us to use Lemma 2.4. In order to make this more precise, we need to introduce the framework of almost mathematics (derived category version) following [Bha18].

Definition 3.1 (The pair (O_C,m)). For any complete and algebraically closed nonar- chimedean field C of characteristic 0, with ring of integers O_C, we say an O_C-module M is almost zero if m·M = 0 where mis the maximal ideal of O_C. A mapf :K →L in D(O_C) is an almost isomorphism if the cohomology groups of the mapping cone of f are almost zero.

Now we consider the almost derived category ofO_C-modules. Precisely, there are two functors:

D(O_C) ⁽⁾

a

−−→D(O_C)^a :=D(O_C)/D(k), K7→K^a D(O_C)^a−⁽⁾−→^∗ D(O_C), K^a7→(K^a)∗:= RHomO_C(m, K)

where the Verdier quotient D(O_C)/D(k) is actually the localization of D(O_C) with respect to almost isomorphisms. The functor ()∗ is right adjoint to the quotient functor ()^a.

Lemma 3.2. Let C be spherically complete, i.e. any decreaing sequence of discs in C has nonempty intersection. For any perfect complexK ∈D(O_C), we have K'(K^a)∗.

Proof. See [Bha18, Lemma 3.4].

There are similar constructions and results in the setting ofAinf-modules.

Definition 3.3 (The pair (Ainf, W(m^[))). An Ainf-module M is called almost zero if W(m^[)·M = 0, where W(m^[) = Ker(A_inf → W(k)). A map f : K → L in D(A_inf) is called an almost isomorphism if the cohomology groups of the mapping cone of f are almost zero.

Similarly, we consider the almost derived category ofA_inf-modules. LetDcomp(A_inf)⊂ D(A_inf) be the full subcategory of all derivedp-adically complete complexes. There are two functors:

D_comp(A_inf) ⁽⁾

a

−−→D_comp(A_inf)^a:=D_comp(A_inf)/D_comp(W(k)), K 7→K^a Dcomp(Ainf)^a−⁽⁾−→^∗ Dcomp(Ainf), K^a→(K^a)∗:= RHomAinf(W(m^[), K)

where the Verdier quotient Dcomp(A_inf)^a := Dcomp(A_inf)/Dcomp(W(k)) is actually the localization ofD_comp(A_inf) with respect to almost isomorphisms. The functor ()∗ is also right adjoint to ()^a.

Lemma 3.4. Let C be spherically complete. If K ∈ Dcomp(Ainf) is perfect, then K ' (K^a)∗.

Proof. See [Bha18, Lemma 3.10].

Next, we state a lemma concerning theLη-functor, which gives us two morpshims.

Lemma 3.5. Let A be a commutative ring and a ∈ A be a non-zero divisor. Assume K∈D^[0,d](A) withH⁰(K) being a-torsion free. Then there are natural maps Lη_a(K)→ K and K→Lηa(K) whose composition in either direction is a^d.

Proof. See [Bha18, Lemma 5.20].

We may apply this lemma toA =O_C, a=ζ_p−1 andK =τ^≤iRν∗Ob⁺_X. Then there are two natural maps which we denote by f, g,

f :Lηζp−1(τ^≤iRν∗Ob⁺_X)'τ^≤iΩeX¯ →τ^≤iRν∗Ob⁺_X g:τ^≤iRν∗Ob_X⁺ →τ^≤iΩeX¯

The quasi-isomorphismLη_ζp−1(τ^≤iRν∗Ob⁺_X)'τ^≤iΩeX¯ is due to the commutativity of the Lηfunctor and the canonical truncation functorτⁱ (see [BMS18, Corollary 6.5]). Passing to sheaf cohomology, we get two natural maps

f :τ^≤iRΓzar( ¯X, τ^≤iΩeX¯)→τ^≤iRΓzar( ¯X, τ^≤iRν∗Ob⁺_X) g:τ^≤iRΓ_zar( ¯X, τ^≤iRν∗Ob_X⁺)→τ^≤iRΓ_zar( ¯X, τ^≤iΩeX¯)

whose composition in either direction is (ζ_p−1)ⁱ. Since there is a quasi-isomorphism τ^≤iRΓzar( ¯X, τ^≤iΩeX¯) ' τ^≤iRΓzar( ¯X,ΩeX) which is induced by the canonical morphism τ^≤iΩeX¯ →ΩeX¯, we get two maps

f :τ^≤iRΓzar( ¯X,ΩeX¯)→τ^≤iRΓzar( ¯X, Rν∗Ob⁺_X) g:τ^≤iRΓ_zar( ¯X, Rν∗Ob_X⁺)→τ^≤iRΓ_zar( ¯X,ΩeX¯)

whose composition in either direction is (ζp−1)ⁱ.

Note that there is a quasi-isomorphism RΓzar( ¯X, Rν∗Ob⁺_X) ' RΓ_pro´et(X,Ob⁺_X). What we want to study at the end is thep-adic ´etale cohomology but not pro-´etale cohomology.

But actually we get almost what we want. Recall the primitive comparison theorem due to Scholze.

Theorem 3.6 ([Sch13, Theorem 8.4]). For any proper smooth adic space X over C, there are natural almost isomorphisms

RΓ_´et(X,Zp)⊗^L

ZpO_C 'RΓ_pro´et(X,Ob_X⁺) and

RΓ_´et(X,Zp)⊗^L

ZpA_inf 'RΓ_pro´et(X,Ainf,X)

Then by passing to the world of almost mathematics, we get two natural maps in D(O_C)^a:

f^a: (τ^≤iRΓ_zar( ¯X,ΩeX¯))^a→(τ^≤iRΓ_zar( ¯X, Rν∗Ob_X⁺))^a'(τ^≤iRΓ_´et(X,Zp)⊗_ZpO_C)^a g^a: (τ^≤iRΓzar( ¯X, Rν∗Ob⁺_X))^a'(τ^≤iRΓ´et(X,Zp)⊗_ZpO_C)^a→(τ^≤iRΓzar( ¯X, τ^≤iΩeX¯))^a Lemma 3.7. The complex τ^≤iRΓ_HT( ¯X) = τ^≤iRΓ_zar( ¯X,ΩeX¯) (resp. τ^≤iRΓ_Ainf( ¯X)) is a perfect complex of O_C-modules (resp. Ainf-modules).

Proof. Note that we have RΓHT(X)⊗_OK O_C ' RΓHT( ¯X). Since RΓS(X) is a perfect complex of S-modules and RΓ_S(X)⊗_SO_K ' RΓ_HT(X), the Hodge-Tate cohomology RΓ_HT(X) is a perfect complex of O_K-modules by [Sta19, Lemma 066W]. Moreover as O_K is a Notherian local ring, the cohomology groups H_HTⁱ (X) are finitely generated O_K-modules and so finitely presented O_K-modules. So we see that every Hodge-Tate cohomology groupH_HTⁱ ( ¯X) is also finitely presented overO_C. By [Sta19, Lemma 0ASP], this meansH_HTⁱ ( ¯X)∼=Oⁿ_Cⁱ⊕(L

j∈JO_C/xj) whereni is a non-negative integer,xj ∈ O_C andJ is a finite set of positive integers. SoH_HTⁱ ( ¯X) is perfect. The lemma hence follows from [Sta19, Lemma 066U]. For τ^≤iRΓA_inf( ¯X), this follows from [BMS18, Lemma 4.9]

stating that each H_Aⁱ

inf( ¯X) is perfect.

Asτ^≤iRΓ_´et(X,Zp) andτ^≤iRΓ_zar( ¯X,ΩeX¯) are perfect complexes, then Lemma 3.2 tells

us that ifCis spherically complete, then (τ^≤iRΓ( ¯X,ΩeX¯))^a_∗ 'τ^≤iRΓ(X,ΩeX) and (τ^≤iRΓ´et(X,Zp)⊗_Zp O_C)^a_∗ 'τ^≤iRΓ_´et(X,Zp)⊗_ZpO_C. By moving back to the real world, we have two maps

(f^a)∗ :τ^≤iRΓzar( ¯X,ΩeX¯)→τ^≤iRΓ_´et(X,Zp)⊗_ZpO_C (g^a)∗:τ^≤iRΓ_´et(X,Zp)⊗_ZpO_C →τ^≤iRΓ_zar( ¯X,ΩeX¯)

whose composition in either direction is (ζ_p−1)ⁱ. These two maps induce maps between cohomology groups for anyn≤i.

f :Hⁿ( ¯X,ΩeX¯)→H_´etⁿ(X,Zp)⊗_ZpO_C g:H_´etⁿ(X,Zp)⊗_ZpO_C →Hⁿ( ¯X,ΩeX¯) whose composition in either direction is (ζ_p−1)ⁱ.

Now we come to the following key theorem:

Theorem 3.8. Let X be a proper smooth formal scheme over O_K, where O_K is the ring of integers in a complete discretely valued nonarchimedean extensionK of Qp with perfect residue field k and ramification degree e. Let O_C be the ring of integers in a completed algebraic closure C of K andX be the adic generic fibre of X¯ :=X⊗_OK O_C. Assume ie < p−1, then there is an isomorphism of O_C-modules between Hodge-Tate cohomology group and p-adic etale cohomology group,´

H_HTⁱ ( ¯X) :=Hⁱ( ¯X,Ωe_X)∼=H_´etⁱ (X,Zp)⊗_ZpO_C.

Proof. Note that replacing C by its spherical completion will not make any difference to this theorem. The spherical completion always exists (see [Rob13, Chapter 3]) and p-adic ´etale cohomology is insensitive to such extensions in the rigid-analytic setting (see [Hub13, 0.3.2]). So we could assumeC is spherically complete. Using the flat base change along α : S → A_inf from Breuil-Kisin cohomology to A_inf-cohomology, we see thatHⁱ( ¯X,ΩeX¯) has a decomposition asO_Cⁿ ⊕(Lm

j=0O_C/π^mj). By requiringie < p−1, we havev((ζp−1)ⁱ)< v(π) inO_C asv((ζp−1)^p−1) =v(p) andv(p) =v(π^e). Now the theorem follows from Lemma 2.4 and the existence of maps f, g.

4. The unramified case: comparison theorem

In this section, we study the relation betweenp-adic ´etale cohomology groupH_´etⁱ (X,Zp) and crystalline cohomology groupH_crysⁱ (Xk/W(k)) in the unramifed case, i.e. the rami- fication degree e=1 andi < p−1. Note that in the unramified case, crystalline cohomol- ogy RΓ_crys(X_k/W(k)) is canonically isomorphic to de Rham cohomologyRΓ_dR(X/O_K) (O_K =W(k) when e= 1).

In order to prove the integral comparison theorem, we first relate Hodge-Tate coho- mology to Hodge cohomology. And then we can use Theorem 3.8 to get a link between Hodge cohomology andp-adic ´etale cohomology. The last step is to study the Hodge-to- de Rham spectral sequence and we can prove the converse to [BMS18, Theorem 14.5], which results in the final comparison theorem.

4.1. Decomposition of Hodge-Tate cohomology groups. In this subsection, we explain how to relate Hodge-Tate cohomology to Hodge cohomology. In fact, we can show that the complex of sheaves τ^≤p−1ΩeX¯ is formal in the unramified case.

Theorem 4.1. For any proper smooth formal scheme X over W(k) and X¯ =X⊗_W(k) O_C, the complex of sheaves τ^≤p−1ΩeX¯ is formal, i.e. there is a quasi-isomorphism γ : Lp−1

i=0 Ωⁱ_X¯{−i}[−i] ' τ^≤p−1ΩeX¯. Here Ωⁱ_X¯ := lim←−Ωⁱ_{( ¯X/p}n)/(O_C/pⁿ) is the OX¯-module of continuous differentials andΩⁱ_X¯{−i}is the Breuil-Kisin-Fargues twist ofΩⁱ_X¯ (cf. [BMS18, Definition 8.2].

Proof. We proceed by first showing that τ^≤1ΩeX¯ is formal and then constructing the general quasi-isomorphism in the statement.

By [BMS18, Proposition 8.15], there is a quasi-isomorphismτ^≤1ΩeX¯ 'L\X/Z¯ p{−1}[−1], where L\X/¯ Zp is the derived p-adic completion of the cotangent complex of ¯X over Zp. Considering the sequenceZp →W(k)→OX¯, we have a distinguished triangle

LW(k)/Zp⊗^L_W(k)OX¯ →LX/Z¯ p →LX/W¯ (k)

Since LW\(k)/Zp vanishes, we have

L\X/¯ Zp{−1}[−1]'L\X/W¯ (k)'LX⊗_W\_(k)O_C/W(k){−1}[−1]

By the K¨unneth property of cotangent complex (cf. [Ill06]), we then have LX⊗_W\_(k)O_C/W(k)'LO\_C/W(k)⊗OX¯ ⊕L\X/W(k)⊗ O_C 'OX¯{1}[1]⊕Ω¹_X¯ So we get a decompositionτ^≤1ΩeX¯ 'OX¯⊕Ω¹_¯

X{−1}[−1]. In particular, we have a mapγ₁ : Ω¹_X¯{−1}[−1]→ ΩeX¯ which gives the Hodge-Tate isomorphism C⁻¹ : Ω¹_X¯{−1} → H¹(ΩeX¯) (cf. [BMS18, Theorem 8.3]).

Now we consider the map for anyi≤p−1 given by

(Ω¹_X¯)^⊗i →Ωⁱ_X¯, ω₁⊗ · · · ⊗ω_i 7→ω₁∧ · · · ∧ω_i It has an anti-symmetrization sectionaas shown in [DI87], given by

a(ω₁∧ · · · ∧ω_i) = (1/i!) X

s∈Sym_i

sgn(s)ω_s(1)⊗ · · · ⊗ω_s(i) Then we defineγ_i as the composition

Ωⁱ_X¯{−i}[−i]−→^a (Ω¹_X¯{−1})^⊗i[−i]'(Ω¹_X¯{−1}[−1])^{⊗Li γ}

⊗Li

−−−→1 (eΩX¯)^⊗Li −−−→^multi ΩeX¯

By applyingHⁱ, we have

ΩⁱX¯{−i}−→ H^{a i}((Ω¹X¯{−1}[−1])^⊗Li)∼= (H¹(Ω¹X¯{−1}[−1]))^{⊗i γ}

⊗Li

−−−→1 (H¹(ΩeX¯))^⊗i→ Hⁱ((ΩeX¯)^⊗Li)−−−→ H^{multi i}(ΩeX¯) Since the Hodge-Tate isomorphism is compatible with multiplication, this composition

is exactly the Hodge-Tate isomorphismC⁻¹ : Ωⁱ_X¯{−i} ' Hⁱ(ΩeX¯). So we get the desired quasi-isomorphismγ =Lp−1

i=0γi:Lp−1

i=0 Ωⁱ_X¯{−i}[−i]'τ^≤p−1ΩeX¯. Remark 4.2. Note that the key input in the proof above is the Hodge-Tate isomorphism C⁻¹ : Ωⁱ_X¯{−i} → Hⁱ(ΩeX¯). In fact, there is a Hodge-Tate isomorphism for any bounded prism (A, I) (cf. [BS19, Theorem 4.10]) and also a generalization of the quasi-isomorphism τ^≤1ΩeX¯ ' L\X/Z¯ p{−1}[−1]. This quasi-isomorphism shows that τ^≤1ΩeX¯ is formal if and only if ¯Xcan be lifted to A_inf/eξ². In the ramified case, this seems to be hardly satisfied due to the non-vanishing of the cotangent complexLOK/W(k).

Corollary 4.3. There is a natural decomposition for any non-negative integern≤p−1, H_HTⁿ ( ¯X) =Hⁿ( ¯X,ΩeX¯)∼=

n

M

i=0

Hⁿ⁻ⁱ( ¯X,Ωⁱ_X¯{−i}).

4.2. Hodge-to-de Rham spectral sequence. In this subsection, we study the Hodge- to-de Rham spectral sequence and finish the proof of the integral comparison theorem in the unramified case. More precisely, we will prove the converse to Theorem 1.8 by analyzing the length of the torsion part of de Rham cohomology groups andp-adic ´etale cohomology groups.

Note that we have the Hodge-to-de Rham spectral sequence E₁^i,j =H^j( ¯X,ΩⁱX¯) =⇒H^i+j( ¯X,Ω^•X¯) =H_dR^i+j( ¯X/O_C)

As ¯X = X⊗_W(k) O_C, this spectral sequence can be seen as the flat base change to O_C of the Hodge-to-de Rham spectral sequence of X overW(k). This tells us E∞^i,j has

a decomposition E∞^i,j ∼= O_C^mL (Laij

r=1O_C/p^nr) (note that E∞^i,j is also a subquotient of H^j( ¯X,Ωⁱ_X¯)).

Fix a non-negative integern such thatn < p−1, we have the abutment filtration 0 =Fⁿ⁺¹⊂Fⁿ⊂ · · · ⊂F⁰ =H_dRⁿ ( ¯X/O_C)

and the short exact sequences

0→Fⁱ⁺¹ →Fⁱ→E_∞^i,n−i→0

Consider the normalized length lO_C for finitely presented torsion O_C-modules. Recall that this length behaves additively under short exact sequences andlOC(O_C/p) = 1. We have the following result.

Lemma 4.4. For any short exact sequence of finitely presented O_C-modules 0→A→B→C →0

we havelOC(Btor)≤lOC(Ator)+lOC(Ctor)andlOC(Btor/p^m)≤lOC(Ator/p^m)+lOC(Ctor/p^m) for any positive integerm.

Proof. For the first statement, it is easy to see that M = B_tor/A_tor is a submodule of Ctor, so we have lO_C(M) = lO_C(Btor)−lO_C(Ator) ≤ lO_C(Ctor) by the additivity of the length. For the second one, we have an exact sequence

M[p^m]→Ator/p^m→Btor/p^m →M/p^m→0

So we getlO_C(B_tor/p^m)≤lO_C(A_tor/p^m)+lO_C(M/p^m). Then we need to provelO_C(M/p^m)≤ lOC(Ctor/p^m). More generally, given two finitely presented torsionO_C modulesN1 ⊂N2, there is an exact sequence

N[p^m]→N1/p^m →N2/p^m →N/p^m→0

where N =N2/N1. Note that lOC(N[p^m]) =lOC(N/p^m). In fact, this follows from the exact sequence

0→N[p^m]→N ^p

m

−−→N →N/p^m→0

HencelO_C(N₂/p^m)≥lO_C(N/p^m) +lO_C(N₁/p^m)−lO_C(N[p^m]) =lO_C(N₁/p^m). So finally we getlO_C(Btor/p^m)≤lO_C(Ator/p^m) +lO_C(Ctor/p^m).

Corollary 4.5. For any integerisuch that0≤i≤nand any positive integerm, we have lO_C(F_torⁱ /p^m)≤lO_C(F_torⁱ⁺¹/p^m)+lO_C(E∞^i,n−itor/p^m). In particular,lO_C(H_dRⁿ ( ¯X/O_C)/p^m)≤ Pn

i=0lOC(E∞^i,n−itor/p^m).

Recall that the rational Hodge-to-de Rham spectral sequence degenerates atE1 page:

Theorem 4.6 ([BMS18, Theorem 13.12]). For any proper smooth rigid analytic space X over C, the Hodge-to-de Rham spectral sequence

E₁^i,j =H^j(X,Ωⁱ_X) =⇒H_dR^i+j(X/C) degenerates atE1. Moreover, for all i≥0,

i

X

j=0

dim_CH^i−j(X,Ω^j_X) = dim_CH_dRⁱ (X/C) = dim_QpH_´etⁱ (X,Qp).

As a consequence, we have the following lemma: