The A_{\mathrm{inf}}-cohomology in the semistable case

# The $A_{inf}$-cohomology in the semistable case

## Abstract.

For a proper, smooth scheme over a -adic field , we show that any proper, flat, semistable -model of whose logarithmic de Rham cohomology is torsion free determines the same -lattice inside and, moreover, that this lattice is functorial in . For this, we extend the results of Bhatt–Morrow–Scholze on the construction and the analysis of an -valued cohomology theory of -adic formal, proper, smooth -schemes to the semistable case. The relation of the -cohomology to the -adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.

###### Key words and phrases:
Comparison isomorphism, integral cohomology, -adic Hodge theory
###### 2010 Mathematics Subject Classification:
Primary 14G20; Secondary 14G22, 14F20, 14F30, 14F40.

## 1. Introduction

###### 1.1. Integral relations between p-adic cohomology theories.

For a proper, smooth scheme over a complete, discretely valued extension of with a perfect residue field , comparison isomorphisms of -adic Hodge theory relate the -adic étale, de Rham, and, in the case of semistable reduction, also crystalline cohomologies of . For instance, they show that for , the -representation functorially determines the filtered -vector space . Even though the integral analogues of these isomorphisms are known to fail in general, one may still consider their hypothetical consequences, for instance, one may ask the following.

• For proper, flat, semistable -models and of endowed with their standard log structures, do the images of and in agree?

One of the goals of the present paper is to show that the answer is positive if the logarithmic de Rham cohomology of the models and is torsion free (see (8.6.2) and Theorem 8.7): in this case, both and agree with the -lattice in that is functorially determined by . The good reduction case of this result may be derived from the work of Bhatt–Morrow–Scholze [BMS16] on integral -adic Hodge theory, and our approach, as well as the bulk of this paper, is concerned with extending the framework of op. cit. to the semistable case.

###### 1.2. The Ainf-cohomology in the semistable case.

To approach the question above, we set , let be the basic period ring of Fontaine, and, for a semistable -model of , similarly to the smooth case treated in [BMS16], construct the -cohomology object

 RΓAinf(X)∈D[0,2dim(X)](Ainf)

that is quasi-isomorphic to a bounded complex of finite free -modules and has finitely presented cohomology . We show that base changes of recover other cohomology theories:

 RΓAinf(X)⊗LAinfW(C♭) ≅RΓ´et(X¯¯¯¯K,Zp)⊗LZpW(C♭), (1.2.1) RΓAinf(X)⊗LAinf,θOC ≅RΓlogdR(X/OK)⊗LOKOC, RΓAinf(X)⊗LAinfW(¯¯¯k) ≅RΓlogcris(Xk/W(k))⊗LW(k)W(¯¯¯k),

see §7.2; here denotes the logarithmic crystalline (that is, Hyodo–Kato) cohomology, (resp., ) is endowed with the log structure associated to (resp., ), and is endowed with the base change of the standard log structure of .

If the cohomology of is torsion free, then each is -free and the base changes (1.2.1) hold in each individual cohomological degree (see §7.6 and Proposition 7.7). In this case, the Fargues equivalence for Breuil–Kisin–Fargues -modules allows us to prove that

 the Gal(¯¯¯¯¯K/K)-representationHi´et(X¯¯¯¯K,Zp)determinesHiAinf(X)

(see Theorem 8.7). Then also determines1 (and ) and, since the same reasoning applies to another model , the result claimed in §1.1 follows.

The base changes (1.2.1) also allow us to extend the cohomology specialization results obtained in the good reduction case in [BMS16]. Qualitatively, in Proposition 7.7 we show that is torsion free if and only if so is , in which case is torsion free. Quantitatively, in Theorems 7.12 and 7.9 we show that for every ,

 lengthZp((Hi´et(X¯¯¯¯K,Zp)tors)/pn) ≤lengthW(k)((Hilogcris(Xk/W(k))tors)/pn), lengthZp((Hi´et(X¯¯¯¯K,Zp)tors)/pn) ≤1lengthOK(OK/p)⋅lengthOK((HilogdR(X/OK)tors)/pn).
###### 1.3. The semistable comparison isomorphism.

The analysis of , specifically, its relation to the -adic étale and the logarithmic crystalline cohomologies, permits us to reprove in Theorem 9.5 the semistable conjecture of Fontaine–Jansen [Kat94a]*Conj. 1.1:

 (1.3.1)

Other proofs of this conjecture have been given in [Tsu99], [Fal02], [Niz08], [Bha12], [Bei13], and [CN17], whereas [BMS16] used to reprove the crystalline conjecture. Similarly to [CN17], we establish (1.3.1) for -adic formal -schemes that are proper, flat, and “semistable.”

A key result that leads to (1.3.1) is the absolute crystalline comparison isomorphism

 RΓAinf(X)⊗LAinfAcris≅RΓlogcris(XO¯¯¯¯K/p/Acris) (1.3.2)

of Corollary 5.43, whose construction in §5 forms the technical core of this paper. This construction is based on an “all possible coordinates” technique that is a variant of its analogue used to establish (1.3.2) in the smooth case in [BMS16]*§12. The presence of singularities and log structures creates additional complications that do not appear in the smooth case and are overviewed in §5.

Using the absolute crystalline comparison isomorphism, in Theorem 6.6 we compare the -cohomology of with the -cohomology of defined by Bhatt–Morrow–Scholze in [BMS16]*§13:

The identification (1.3.3) is important for ensuring that the semistable comparison (1.3.1) is compatible with the de Rham comparison proved in [Sch13], and hence that it respects filtrations.

As for the question posed in §1.1, even though it only involves the étale and the de Rham cohomologies, the resolution of its torsion free case outlined in §1.2 uses both (1.3.2) and (1.3.3) (so also the bulk of the material of this paper). This is because we need to ensure that the determination of by via the de Rham comparison of -adic Hodge theory is compatible with the determination of and by via -cohomology and Breuil–Kisin–Fargues modules. In fact, even for showing that the cohomology modules of are Breuil–Kisin–Fargues, we already use the absolute crystalline comparison (1.3.2).

###### 1.4. The object AΩX and its base changes.

Even though above we have focused on schemes, the construction and the analysis of works for any -adic formal -scheme that is semistable in the sense described in §1.5 (see (1.5.1)) and that, whenever needed, is assumed to be proper. Specifically, for such an , in §2.2 we use the (variant for the étale topology of the) definition of Bhatt–Morrow–Scholze from [BMS16] to build an object

 AΩX∈D≥0(X´et,Ainf)and to setRΓAinf(X)\colonequalsRΓ(X´et,AΩX).

As in the smooth case of [BMS16], the relation of to the -adic étale cohomology of the adic generic fiber of follows from the results of [Sch13] (see §2). In turn, the relations to the logarithmic de Rham and crystalline cohomologies are the subjects of §4 and §5, respectively, and rest on the following identifications established in Theorems 5.4 and 4.17:

 (1.4.1)

where is the forgetful map of topoi. The arguments for (1.4.1) build on the same general skeleton as in [BMS16] but differ, among other aspects, in how they handle the interaction of the Deligne–Berthelot–Ogus décalage functor used in the definition of with the intervening base changes and with the almost isomorphisms supplied by the almost purity theorem. Namely, for this, the nonflatness over the singular points of of the explicit perfectoid proétale covers that we construct makes it difficult to directly adapt the arguments from op. cit. Instead, we take advantage of several general results about from [Bha16]. Verifying their assumptions in our case amounts to the analysis in §3 of continuous group cohomology modules built using the aforementioned perfectoid cover. The typical conclusion of this analysis is that these modules have no nonzero “almost torsion” and that the element kills their “nonintegral parts.”

Further and more specific overviews of our arguments are given in the beginning parts of the sections that follow. In the rest of this introduction, we fix the precise notational setup for the rest of the paper (see §1.5), discuss the logarithmic structure on that we later use without notational explication (see §1.6), and review the relevant general notational conventions (see §1.7).

###### 1.5. The setup.

In what follows, we fix the following notational setup.

• We fix an algebraically closed field of characteristic , let be the completed algebraic closure of , and let be the maximal ideal in the valuation ring of .

• For convenience, we fix an embedding , that is, for every prime , we fix a system of compatible -power roots of in .

• We fix a -adic formal scheme over that in the étale topology may be covered by open affines which admit an étale -morphism

 U=Spf(R)→Spf(R□)withR□\colonequalsOC{t0,…,tr,t±1r+1,…,t±1d}/(t0⋯tr−pq) (1.5.1)

for some , some , and some (where , , and may depend on ).

For example, could be the completed algebraic closure of any discretely valued field of mixed characteristic with a perfect residue field. In addition, no generality is gained by replacing in (1.5.1) by any nonunit . The role of the embedding is to simplify arguments with explicit charts for the log structure on (defined in §1.6); this is particularly useful in §5, especially in §§5.255.26. Our is less general than in [BMS16], where any complete algebraically closed nonarchimedean extension of is typically allowed. One of the main reasons for this is that we want to be able to apply, especially in §5, certain auxiliary results from [Bei13a] (besides, relations in which has a nonrational valuation go beyond “semistable reduction”).

The existence of étale local semistable coordinates (1.5.1) implies that each is flat and locally of finite presentation over and is dense in . By [SP]*04D1 and limit arguments, (1.5.1) is the formal -adic completion of the -base change of an étale -morphism

 U→Spec(O[t0,…,tr,t±1r+1,…,t±1d]/(t0⋯tr−pq)) (1.5.2)

for some discrete valuation subring that contains . Loc. cit. and [GR03]*7.1.6 (i) also imply that is -flat. In addition, if is not -smooth, then determines .2

Any smooth -adic formal -scheme meets the requirements above: indeed, then the cover exists already for the Zariski topology with and for all , see [FK14]*I.5.3.18. Another key example is

 X=ˆXOC (1.5.3)

for some discrete valuation subring with a perfect residue field and a uniformizer and a locally of finite type, flat -scheme that is semistable in the sense that is a normal crossings divisor in (as defined in [SP]*0BSF), so that, in particular, is regular at every point of .3 Moreover, if is even strictly semistable in the sense that is even a strict normal crossings divisor in (as defined in [SP]*0BI9), then the étale maps (1.5.4) exist even Zariski locally on , and so also the cover exists already for the Zariski topology of .

• We let denote the adic generic fiber of . By (1.5.1) and [Hub96]*3.5.1, the adic space is smooth over ; by [Hub96]*1.3.18 ii), if is -proper, then is -proper.

• We let denote the proétale site of (reviewed in [BMS16]*§5.1 and defined in [Sch13]*3.9 and [Sch13e]*(1)) and let

be the morphism to the étale site of that sends any étale to the constant pro-system associated to its adic generic fiber. By [SP]*00X6, this functor indeed defines a morphism of sites: by [Hub96]*3.5.1, it preserves coverings, commutes with fiber products, and respects final objects. Thus, induces a morphism of topoi (see [SP]*00XC).

###### 1.6. The logarithmic structure on X.

Unless noted otherwise, we always equip

1. the ring (resp.,  or ) with the log structure (resp., its pullback);

2. the formal scheme (resp.,  or ) with the log structure given by the subsheaf associated to the subpresheaf4 (resp., its pullback log structure).

Both 1 and 2 determine the same log structure on , so the map is that of log formal schemes. Moreover, étale locally on , the log structure may be made explicit: in the presence of a coordinate morphism (1.5.1), Claims 1.6.3 and 1.6.1 below give an explicit chart for the log structure of , namely, the chart (1.6.2) in which we replace by , replace by , and set . This chart shows, in particular, that and may be endowed with fine log structures whose base changes along a “change of log structure” self-map of recover the log structures described in 12. In practice this means that we may deal with the log structures in 12 as if they were fine and, in particular, we may cite [Kat89] for certain purposes.

By the preceding discussion, all the log structures above are quasi-coherent and integral. Moreover, by [Kat89]*3.7 (2), each is log smooth over , so that, by [Kat89]*3.10, the -module of logarithmic differentials is finite locally free. We set

 ΩiX/OC,log\colonequals⋀iΩ1X/OC,log,

let denote the logarithmic de Rham complex, and set

 RΓlogdR(X/OC)\colonequalsRΓ(X´et,Ω∙X/OC,log).
###### Claim 1.6.1.

For a valuation subring and an -scheme that has an étale morphism

 U→Spec(O[t0,…,tr,t±1r+1,…,t±1d]/(t0⋯tr−π))for% some nonunitπ∈O∖{0},

the log structure on associated to has the chart

 Nr+1≥0⊔N≥0(O∖{0})→Γ(U,OU) (1.6.2)

given by on , the diagonal and on , and the structure map on .

###### Proof.

Without loss of generality, is affine, so, by a limit argument, we may assume that is discretely valued. Then , endowed with the log structure associated to (1.6.2), is logarithmically regular in the sense of [Kat94b]*2.1 (compare with [Bei12]*§4.1, proof of Lemma). Therefore, since the locus of triviality of this log structure is , the claim follows from [Kat94b]*11.6. ∎

###### Claim 1.6.3.

For as in Claim 1.6.1, a flat -scheme (resp., and its formal -adic completion ) endowed with the log structure associated to (resp., ),

 the formal p-adic completion morphism j:U→Uof log ringed \'{e}tale sites is strict. (1.6.4)
###### Proof.

For a geometric point of , due to [SP]*04D1, the stalk map induces an isomorphism for every . We consider the stalk map

 (1.6.5)

Every element of the target of (1.6.5) satisfies the equation for some . We choose an congruent to modulo , so that for some . Since , we adjust to get , which shows that and . Thus, the image of in and generate the same ideal, and hence are unit multiples of each other. Conversely, if are unit multiples of each other in , then, by reducing modulo for a large enough , we see that they generate the same ideal in , so are unit multiples of each other already in . In conclusion, the map (1.6.5) induces an isomorphism

 (OU,¯¯¯u∩(OU,¯¯¯u[1p])×)/O×U,¯¯¯u∼⟶(OU,¯¯¯u∩(OU,¯¯¯u[1p])×)/O×U,¯¯¯u,

to the effect that the map (1.6.4) is indeed strict, as claimed. ∎

###### 1.7. Conventions and additional notation.

For a field , we let be its algebraic closure (taken inside if is given as a subfield of ). If has a valuation, we let be its valuation subring and write for the integral closure of in . In mixed characteristic, we normalize the valuations by requiring that . We let denote the smooth locus of a (formal) scheme over an implicitly understood base. For power series rings, we use to indicate decaying coefficients. For a topological ring , we let denote the subset of powerbounded elements.

We let (resp. ) denote -typical Witt vectors (resp., their length truncation), and let denote Teichmüller representatives. We let be the localization of at , let be the group scheme of -th roots of unity, and let denote a primitive -th root of unity. For brevity, we set . We let denote the (by default, -adic) completion of a module and, similarly, let denote the completion of a direct sum. Unless specified otherwise, we endow a -adically complete module with the inverse limit of the discrete topologies.

We use the definition of a perfectoid ring given in [BMS16]*3.5 (the compatibility with prior definitions is discussed in [BMS16]*3.20). Explicitly, by [BMS16]*3.9 and 3.10, a -torsion free ring is perfectoid if and only if is -adically complete and the divisor has a -power root in the sense that there is a with and In particular, for such an , any -adically formally étale -algebra that is -adically complete is also perfectoid.

For a ring object of a topos , we write , or simply , for the derived category of -modules. For an object of a derived category, we denote its derived -adic completion by

 ˆM\colonequalsRlimn(M⊗LZZ/pnZ),and also set∗ˆ⊗L⋅−\colonequalsRlimn((∗⊗L⋅−)⊗LZZ/pnZ) (1.7.1)

(see [SP]*0940 for the definition of ). For a morphism of ringed topoi, we use the commutation of the functor with derived limits and derived completions, see [SP]*0A07 and 0944.

For a profinite group and a continuous -module , we write for the continuous cochain complex. Whenever convenient, we also view as the derived global sections functor of the site of profinite -sets (see [Sch13]*3.7 (iii) and [Sch13e]*(1)).

For commuting endomorphisms of an abelian group , we recall the Koszul complex:

 KA(f1,…,fn)\colonequalsA⊗Z[x1,…,xn]⨂ni=1(Z[x1,…,xn]xi−→Z[x1,…,xd]), (1.7.2)

where is regarded as a -module by letting act as , the tensor products are over , and the factor complexes are concentrated in degrees and .

For an ideal of a ring and an -module complex with for , the subcomplex

 ηI(M∙)⊂M∙is defined by(ηI(M∙))j\colonequals{m∈IjMj|dj(m)∈Ij+1Mj+1}. (1.7.3)

We will mostly (but a priori not always, see Proposition 5.34) use as in [BMS16]*6.2, namely, when is generated by a nonzerodivisor and the have no nonzero -torsion.

A logarithmic divided power thickening (or, for brevity, a log PD thickening) is an exact closed immersion of logarithmic (often abbreviated to log) schemes equipped with a divided power structure on the quasi-coherent sheaf of ideals that defines the underlying closed immersion of schemes.

### Acknowledgements

We thank Bhargav Bhatt and Matthew Morrow for writing the surveys [Bha16] and [Mor16], which have been useful for preparing this paper. We thank Bhargav Bhatt, Pierre Colmez, Ravi Fernando, Luc Illusie, Arthur-César Le Bras, Matthew Morrow, Wiesława Nizioł, Arthur Ogus, Peter Scholze, Joseph Stahl, Jakob Stix, Jan Vonk, and Olivier Wittenberg for helpful conversations or correspondence. We thank the Kyoto Top Global University program for providing the framework in which this collaboration started. We thank the Miller Institute at the University of California Berkeley, the Research Institute for Mathematical Sciences at Kyoto University, and the University of Bonn for their support during the preparation of this article.

## 2. The object AΩX and the p-adic étale cohomology of X

As in the case when is smooth treated in [BMS16], the eventual construction of the -cohomology modules of rests on the object that lives in a derived category of -module sheaves on . In this short section, we review the definition of in §2.2 and then, in the case when is proper, review the connection between and the integral -adic étale cohomology of in Theorem 2.3. We begin by fixing the basic notation that concerns the ring of integral -adic Hodge theory. The setup of §§2.12.2 will be used freely in the rest of the paper.

###### 2.1. The ring Ainf.

We denote the tilt of by

as multiplicative monoids (see [Sch12]*3.4 (i)). We regard fixed in §1.5 as an element of . Due to the fixed embedding , this element comes equipped with well-defined powers for . For each , we let denote its preimage in . The map makes a complete valuation ring of height whose fraction field is algebraically closed (see [Sch12]*3.4 (iii), 3.7 (ii)). We let denote the maximal ideal of .

The basic period ring of Fontaine is defined by

 Ainf\colonequalsW(O♭C)and comes equipped with the Witt vector Frobeniusφ:Ainf∼⟶Ainf.

We equip the local domain with the product of the valuation topologies via the Witt coordinate bijection . Then is complete and its topology agrees with the -adic topology for any nonzero nonunit . We fix (once and for all) a compatible system of -power roots of unity in , so that , and set

 μ\colonequals[ϵ]−1∈Ainf. (2.1.1)

Since , the topology of is -adic. By forming the limit of the sequences

 0→Wn(O♭C)μ→Wn(O♭C)→Wn(O♭C)/μ→0, (2.1.2)

we see that is -adically complete and that the ideal does not depend on the choice of (use the fact that the valuation of does not depend on ).

The assignment extends uniquely to a ring homomorphism

 θ:Ainf↠OC,%the\emphdeRhamspecializationmapof$Ainf$, (2.1.3)

which is surjective, as indicated, and intertwines the Frobenius of with the absolute Frobenius of . Its kernel is principal and generated by the element

 ξ\colonequals∑p−1i=0[ϵi/p] (2.1.4)

(see [BMS16]*3.16). Analogues of the sequences (2.1.2) show that each is -adically complete. In fact, the map identifies with the initial -adically complete infinitesimal thickening of of order , see [SZ17]*3.13. The composition

 θ∘φ−1:Ainf↠OCis the \emph{Hodge--Tate specialization} map of Ainf,

and its kernel is generated by the element .

Due to the nature of our (see §1.5), the ring is a -algebra, so is a -algebra.

###### 2.2. The object AΩX.

The operations that define and make sense on the proétale site : namely, as in [Sch13]*4.1, 5.10, and 6.1, we have the integral completed structure sheaf

and the basic period sheaf

For brevity, we often denote these sheaves simply by , , and . Affinoid perfectoids form a basis for (see [Sch13]*4.7) and the construction of the map of (2.1.3) makes sense for any perfectoid -algebra (see [BMS16]*§3). In particular, comes equipped with the map

 (2.2.2)

which, by construction, is compatible with the map , intertwines the Witt vector Frobenius of with the absolute Frobenius of , and, by [Sch13]*6.3 and 6.5, is surjective with (in addition, is not a zero divisor in ).

The key object that we are going to study in this paper is

where the décalage functor of [BMS16]*§6 is formed with respect to the ideal of the constant sheaf of (the definition of builds on the formula (1.7.3) for ). The formula (2.2.3) may also be executed with the Zariski site as the target of , and it then defines the object

 AΩXZar∈D≥0(XZar,Ainf), (2.2.4)

which is the that was used in [BMS16]. We will only use in Corollary 4.21 (and in some results that lead to it) for comparison to .

Since , the Frobenius automorphism of gives the Frobenius morphism

 AΩX⊗LAinf,φAinf≅Lη(φ(ξ))(AΩX) \@@cite[cite]{[\@@bibref{}{BMS16}{}{}]}*{6.11, 6.10, and 3% .17~{}(ii)} −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→AΩXinD≥0(X´et,Ainf), (2.2.5)

which, by [BMS16]*6.14, induces an isomorphism

 (AΩX⊗LAinf,φAinf)[1φ(ξ)]∼⟶(AΩX)[1φ(ξ)]. (2.2.6)

In addition, by loc. cit., we also have

so a result of Scholze [BMS16]*5.6 supplies the following relation to integral -adic étale cohomology:

###### Theorem 2.3.

If is proper over , then there is an identification

In broad strokes, the proof of Theorem 2.3 given in loc. cit. goes as follows: one considers the map

 (2.3.2)

induced by the inclusion and deduces from the almost purity theorem with, for instance, Lemma 3.17 below that the ideal

 W(m♭)\colonequalsKer(W(O♭C)↠W(k))ofAinf (2.3.3)

kills the cohomology of its cone. Since lies in and we have the identification (2.2.7), it follows that the map (2.3.2) induces the identification (2.3.1).

###### Remark 2.4.

In practice, often arises as the formal -adic completion of a proper, finitely presented -scheme . In this situation, agrees with the adic space associated to (see [Con99]*5.3.1 4., [Hub94]*4.6 (i), and [Hub96]*1.9.2 ii)) and, by [Hub96]*3.7.2, we have