Crystalline condition for $\boldsymbol{A_{\mathrm{inf}}}$–cohomology and ramification bounds

Let $k$ be a perfect field of characteristic $p>2$ and $K^{\prime}=W(k)[1/p]$ the associated absolutely unramified field. Let $K/K^{\prime}$ be a totally ramified finite extension with ramification index $e$, and denote by $G_{K}$ its absolute Galois group. The goal of the present paper is to provide new bounds for ramification of the mod $p$ representations of $G_{K}$ that arise as the étale cohomology groups $\mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Z}/p\mathbb{Z})$ in terms of $p,i$ and $e$, where $\mathscr{X}$ is a smooth proper $p$–adic formal scheme over $\mathcal{O}_{K}$ (and $\mathscr{X}_{\overline{\eta}}$ is the geometric adic generic fiber). Concretely, let us denote by $G_{K}^{\mu}$ the $\mu$–th ramification group of $G_{K}$ in the upper numbering (in the standard convention, e.g. [Ser13]) and $G_{K}^{(\mu)}=G_{K}^{\mu-1}$. The main result is as follows.

Let us briefly summarize the history of related results. Questions of this type originate in Fontaine’s paper [Fon85], where he proved that for a finite flat group scheme $\Gamma$ over $\mathcal{O}_{K}$ that is annihilated by $p^{n}$, $G_{K}^{(\mu)}$ acts trivially on $\Gamma(\overline{K})$ when $\mu>e(n+1/(p-1))$; this is a key step in his proof that there are no non–trivial abelian schemes over $\mathbb{Z}$. In the same paper, Fontaine conjectured that general $p^{n}$–torsion cohomology would follow the same pattern: given a proper smooth variety $X$ over $K$ with good reduction, $G_{K}^{(\mu)}$ should act trivially on $\mathrm{H}^{i}_{\text{\'{e}t}}(X_{\overline{K}},\mathbb{Z}/p^{n}\mathbb{Z})$ when $\mu>e(n+i/(p-1))$.

This conjecture has been subsequently proved by Fontaine himself ([Fon93]) in the case when $e=n=1,i<p-1$ and by Abrashkin ([Abr90]; see also [Abr15]) when $e=1,i<p-1$ and $n$ is arbitrary. This is achieved by using Fontaine–Laffaille modules (introduced in [FL82]), which parametrize quotients of pairs of $G_{K}$–stable lattices in crystalline representations with Hodge–Tate weights in $[0,i]$ (such as $\mathrm{H}^{i}_{\text{\'{e}t}}({X}_{\overline{K}},\mathbb{Q}_{p})^{\vee}$). The (duals of the) representations $\mathrm{H}^{i}_{\text{\'{e}t}}({X}_{\overline{K}},\mathbb{Z}/p^{n}\mathbb{Z})$ are included among these thanks to a comparison theorem of Fontaine–Messing ([FM87]). Similarly to the orginal application, these ramification bounds lead to a scarcity result for existence of smooth proper $\mathbb{Z}$–schemes.

Various extensions to the semistable case subsequently followed. Under the asumption $i<p-1$ (and arbitrary $e$), Hattori proved in [Hat09] a ramification bound for $p^{n}$–torsion quotients of lattices in semistable representations with Hodge–Tate weights in the range $[0,i],$ using (a variant of) Breuil’s filtered $(\phi_{r},N)$–modules. Thanks to a comparison result between log–crystalline and étale cohomology by Caruso ([Car08]), this results in a ramification bound for $\mathrm{H}^{i}_{\text{\'{e}t}}({X}_{\overline{K}},\mathbb{Z}/p^{n}\mathbb{Z})$ when ${X}$ is proper with semistable reduction, assuming $ie<p-1$ when $n=1$ and $(i+1)e<p-1$ when $n\geq 2$ ²²2Recently, in [LL20] Li and Liu extended Caruso’s result to the range $ie<p-1$ regardless of $n$, for $\mathscr{X}/\mathcal{O}_{K}$ proper and smooth (formal) scheme. In view of this, results of [Hat09] should apply in these situations as well..

These results were further extended by Caruso and Liu in [CL11] for all $p^{n}$–torsion quotients of pairs of semistable lattices with Hodge–Tate weights in $[0,i]$, without any restriction on $i$ or $e$. The proof uses the theory of $(\varphi,\widehat{G})$–modules, which are objects suitable for description of lattices in semistable representations. Roughly speaking, a $(\varphi,\widehat{G})$–module consists of a Breuil–Kisin module $M$ and the datum of an action of $\widehat{G}=\mathrm{Gal}(K(\mu_{p^{\infty}},\pi^{1/p^{\infty}})/K)$ on $\widehat{M}=M\otimes_{\mathfrak{S},\varphi}\widehat{\mathcal{R}}$ where $\widehat{\mathcal{R}}$ is a suitable subring of Fontaine’s period ring $A_{\mathrm{inf}}=W(\mathcal{O}_{\mathbb{C}_{K}^{\flat}})$ (and $\pi\in K$ is a fixed choice of a uniformizer). An obstacle to applying the results of [CL11] to the torsion étale cohomology groups $\mathrm{H}^{i}_{\text{\'{e}t}}(X_{\overline{K}},\mathbb{Z}/p\mathbb{Z})$ is that it is not quite clear when (duals of) such representations come as a quotient of two semistable lattices with Hodge–Tate weights in $[0,i].$ This is indeed the case in the situation when $e=1$, $i<p-1$ and $X$ has good reduction by the aforementioned Fontaine–Messing theorem, and it was also shown in the case $i=1$ (no restriction on $e,p$) for $X$ with semistable reduction by Emerton and Gee in [EG15], but in general the question seems open.

Nevertheless, the idea of the proof of Theorem 1.1 is to follow the general strategy of Caruso and Liu. While one does not necessarily have semistable lattices and the associated $(\varphi,\widehat{G})$–modules to work with, a suitable replacement comes from the recently developed cohomology theories of Bhatt–Morrow–Scholze and Bhatt–Scholze ([BMS18, BMS19, BS22]). Concretely, to a smooth $p$–adic formal scheme $\mathscr{X}$ one can associate the “$p^{n}$-torsion prismatic cohomologies”

where $\mathsf{R}\Gamma_{{{\mathbbl{\Delta}}}}(\mathscr{X}_{A_{\mathrm{inf}}}/A_{\mathrm{inf}}),\mathsf{R}\Gamma_{{{\mathbbl{\Delta}}}}(\mathscr{X}/\mathfrak{S})$ are the prismatic avatars of the $A_{\mathrm{inf}}$– and Breuil–Kisin cohomologies from [BMS18] and [BMS19], resp. Taking $M_{\mathrm{BK}}=\mathrm{H}^{i}_{{{\mathbbl{\Delta}}},1}(\mathscr{X}/\mathfrak{S})$ and $M_{\inf}=\mathrm{H}^{i}_{{{\mathbbl{\Delta}}},1}(\mathscr{X}/A_{\mathrm{inf}}),$ Li and Liu showed in [LL20] that $M_{\mathrm{BK}}$ is a $p$–torsion Breuil–Kisin module, $M_{\inf}$ is a $p$–torsion Breuil–Kisin–Fargues $G_{K}$–module, and that these modules recover the étale cohomology group $\mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Z}/p\mathbb{Z})$ essentially due to the étale comparison theorem for prismatic cohomology from [BS22]. The pair $(M_{\mathrm{BK}},M_{\inf})$ then serves as a suitable replacement of a $(\varphi,\widehat{G})$–module in our context.

The most significant deviation from the strategy of [CL11] then stems from the fact that the pair $(M_{\mathrm{BK}},M_{\inf})$ obtained this way is “inherently $p$–torsion”, that is, it does not come equipped with any apparent lift to analogous objects in characteristic $0$. This is not the case in [CL11], where all torsion modules ultimately originate from a free $(\varphi,\widehat{G})$–module $(M,\widehat{M})$. A key technical input in loc. cit. is to establish a partial control on the Galois action on $M$ inside $\widehat{M},$ namely, a condition of the form

Here $J_{n,s}\subseteq A_{\mathrm{inf}}$ are certain ideals (that are shrinking with growing $s$). This is a “rational” fact, in the sense that this claim is a consequence of the description of the Galois action in terms of the monodromy operator on the associated Breuil module $\mathcal{D}(\widehat{M})$ (cf. [Bre97], [Liu10, §3.2]), a vector space over the characteristic $0$ field $K^{\prime}$.

As a starting point for replacing (1.1) in our context, we turn to a result by Gee and Liu in [EG23, Appendix F] (see also [Oze18, Theorem 3.8]). Given a finite free Breuil–Kisin module $M_{\mathrm{BK}}$ (of finite height) and a compatible structure of Breuil–Kisin–Fargues $G_{K}$–module on $M_{\inf}=M_{\mathrm{BK}}\otimes_{\mathfrak{S}}A_{\mathrm{inf}},$ such that the image of $M_{\mathrm{BK}}$ under the natural map lands in $(M_{\inf})^{G_{K(\pi^{1/p^{\infty}})}}$, the étale realization of $M_{\inf}$ is crystalline if and only if

Here $[-]$ denotes the Teichmüller lift and $\underline{\varepsilon},\underline{\pi}$ are the elements of $\mathcal{O}_{\mathbb{C}_{K}^{\flat}}$ given by a collection $(\zeta_{p^{n}})_{n}$ of (compatible) $p^{n}$–th roots of unity and a collection $(\pi^{1/p^{n}})_{n}$ of $p^{n}$–th roots of the chosen uniformizer $\pi$, resp. We call condition ($\mathrm{Cr}_{0}$) the crystalline condition. As the considered formal scheme $\mathscr{X}$ is assumed to be smooth over $\mathcal{O}_{K}$, it is reasonable to expect that the same condition applies to the pair $M_{\mathrm{BK}}=\mathrm{H}^{i}_{{{\mathbbl{\Delta}}}}(\mathscr{X}/\mathfrak{S})$ and $M_{\inf}=\mathrm{H}^{i}_{{{\mathbbl{\Delta}}}}(\mathscr{X}_{A_{\mathrm{inf}}}/A_{\mathrm{inf}})$, despite the fact that the Breuil–Kisin and Breuil–Kisin–Fargues modules coming from prismatic cohomology are not necessarily free.

This is indeed the case and, moreover, it can be shown that the crystalline condition even applies to the embedding of the chain complexes $\mathsf{R}\Gamma_{{{\mathbbl{\Delta}}}}(\mathscr{X}/\mathfrak{S})\rightarrow\mathsf{R}\Gamma_{{{\mathbbl{\Delta}}}}(\mathscr{X}_{A_{\mathrm{inf}}}/A_{\mathrm{inf}})$: to make sense of this claim, we model the cohomology theories by their associated Čech–Alexander complexes. These were introduced in [BS22] in the case that $\mathscr{X}$ is affine, but can be extended to (at least) arbitrary separated smooth $p$–adic formal schemes. We are then able to verify the condition termwise for this pair of complexes. More generally, we introduce a decreasing series of ideals $I_{s}$, $s\geq 0$ where $I_{0}=\varphi^{-1}([\underline{\varepsilon}]-1)[\underline{\pi}]A_{\mathrm{inf}},$ and then formulate and prove the analogue of ($\mathrm{Cr}_{0}$) for $I_{s}$ and the action of $G_{K(\pi^{1/p^{s}})}.$ As a consequence, we obtain:

Theorem 1.3 (3) specialized to $n=1$ provides the desired analogue of the property (1.1) of $(\varphi,\widehat{G})$–modules and allows us to carry out the proof of Theorem 1.1.

As a consequence of Theorem 1.3 (2), we obtain a proof of crystallinity of the cohomology groups $\mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Q}_{p})$ in the proper case partially by means of “formal” $p$–adic Hodge theory (Corollary 4.6). This fact is usually established via a a direct comparison between crystalline and étale cohomology, and in this generality is originally due to Bhatt, Morrow and Scholze ([BMS18]). Of course, since our setup relies on the machinery of prismatic cohomology and especially the étale comparison, the proof can be considered independent of the one from [BMS18] only in that it avoids the crystalline comparison theorem for (prismatic) $A_{\mathrm{inf}}$–cohomology.

The bounds of Theorem 1.1 compare to the already known bounds as follows. Whenever the bounds of “semistable type” are known to apply to the situation of $\mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Z}/p\mathbb{Z})$ (e.g. [CL11] when $i=1$, [Hat09] when $ie<p-1$ and $\mathscr{X}$ is a scheme), the bounds from Theorem 1.1 agree with those bounds. The bounds tailored to crystalline representations ([Fon93, Abr90]) are slightly better but their applicability is quite limited ($e=1$ and $i<p-1$).

The fact that the cohomology groups $\mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Z}/p^{n}\mathbb{Z})$ have an associated Breuil–Kisin module yields one more source of ramification estimates: in [Car13], Caruso provides a very general bound for $p^{n}$–torsion $G_{K}$–modules based on their restriction to $G_{K(\pi^{1/p^{\infty}})}$ via Fontaine’s theory of étale $\mathcal{O}_{\mathcal{E}}$–modules. Using the Breuil–Kisin module $\mathrm{H}^{i}_{{{\mathbbl{\Delta}}},n}(\mathscr{X}/\mathfrak{S})$ attached to $\mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Z}/p^{n}\mathbb{Z})$, this bound becomes explicit (as discussed in more detail in Remark 5.6). Comparing this result to Theorem 1.1 is more ambiguous due to somewhat different shapes of the estimates, but roughly speaking, the estimate of Theorem 1.1 is approximately the same for $e\leq p$, becomes worse when $K$ is absolutely tamely ramified with large ramification degree, and is expected to outperform Caruso’s bound in case of large wild absolute ramification (rel. to the tame part of the ramification).

In future work, we intend to extend the result of Theorem 1.1 to the case of arbitrary $n$. This seems plausible thanks to the full statement of Theorem 1.3 (3). In a different direction, we plan to extend the results of the present paper to the case of semistable reduction, using the log–prismatic cohomology developed by Koshikawa in [Kos20]. An important facts in this regard are that the $A_{\mathrm{inf}}$–log–prismatic cohomology groups are still Breuil–Kisin–Fargues $G_{K}$–modules by a result of Česnavičius and Koshikawa ([ČK19]) and that by results of Gao, a variant of the condition ($\mathrm{Cr}_{0}$) might exist in the semistable case ([Gao22]; see Remark 3.12 (3) below for details).

The outline of the paper is as follows. In §2 we establish some necessary technical results. Namely, we discuss non–zero divisors and regular sequences on derived complete and completely flat modules with respect to the weak topology of $A_{\mathrm{inf}}$, and establish Čech–Alexander complexes in the case of a separated and smooth formal scheme. Next, §3 introduces the conditions ($\mathrm{Cr}_{s}$), studies their basic algebraic properties and discusses in particular the crystalline condition ($\mathrm{Cr}_{0}$) in the case of Breuil–Kisin–Fargues $G_{K}$–modules. In §4 we prove the conditions ($\mathrm{Cr}_{s}$) for the Alexander–Čech complexes of a separated smooth $p$–adic scheme $\mathscr{X}$ over $\mathfrak{S}$ and $A_{\mathrm{inf}}$, and draw some consequences for the inidividual cohomology groups (especially when $\mathscr{X}$ is proper), proving Theorem 1.3. Finally, in §5 we establish the ramification bounds for mod $p$ étale cohomology, proving Theorem 1.1. Subsequently, we discuss in detail how the bounds from Theorem 1.1 compare to the various known bounds from the literature.

Let us set up some basic notation used throughout the paper. We fix a perfect field $k$ of characteristic $p>0$ and a finite totally ramified extension $K/K^{\prime}$ of degree $e$ where $K^{\prime}=W(k)[1/p]$. We fix a uniformizer $\pi\in\mathcal{O}_{K}$, and a compatible system $(\pi_{n})_{n}$ of $p^{n}$–th roots of $\pi$ in $\mathbb{C}_{K}$, the completion of algebraic closure of $K$. Setting $\mathfrak{S}=W(k)[[u]]$, the choice of $\pi$ determines a surjective map $\mathfrak{S}\rightarrow\mathcal{O}_{K}$ via $u\mapsto\pi$; the kernel of this map is generated by an Eisenstein polynomial $E(u)$ of degree $e$. $\mathfrak{S}$ is endowed with a Frobenius lift (hence a $\delta$–structure) extending the one on $W(k)$ by $u\mapsto u^{p}$.

Denote $A_{\mathrm{inf}}=\mathbb{A}_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{K}})=W(\mathcal{O}_{\mathbb{C}_{K}^{\flat}})$ where $W(-)$ denotes the Witt vectors construction and $\mathcal{O}_{\mathbb{C}_{K}^{\flat}}=\mathcal{O}_{\mathbb{C}_{K}}^{\flat}$ is the tilt of $\mathcal{O}_{\mathbb{C}_{K}}$, $\mathcal{O}_{\mathbb{C}_{K}}^{\flat}=\varprojlim_{x\mapsto x^{p}}\mathcal{O}_{\mathbb{C}_{K}}/p$. The choice of the system $(\pi_{n})_{n}$ describes an element $\underline{\pi}\in\mathcal{O}_{\mathbb{C}_{K}^{\flat}}\simeq\varprojlim_{x\mapsto x^{p}}\mathcal{O}_{\mathbb{C}_{K}}$, and hence an embedding of $\mathfrak{S}$ into $A_{\mathrm{inf}}$ via $u\mapsto[\underline{\pi}]$ where $[-]$ denotes the Teichmüller lift. Under this embedding, $E(u)$ is sent to a generator $\xi$ of the kernel of the canonical map $\theta:A_{\mathrm{inf}}\rightarrow\mathcal{O}_{\mathbb{C}_{K}}$ that lifts the canonical projection $\mathrm{pr}_{0}:\mathcal{O}_{\mathbb{C}_{K}}^{\flat}=\varprojlim_{\varphi}\mathcal{O}_{\mathbb{C}_{K}}/p\rightarrow\mathcal{O}_{\mathbb{C}_{K}}/p.$ Consequently, $(\mathfrak{S},(E(u)))\rightarrow(A_{\mathrm{inf}},\mathrm{Ker}\,\theta)$ is a map of prisms. It is known that under such embedding, $A_{\mathrm{inf}}$ is faithfully flat over $\mathfrak{S}$ (see e.g. [EG23, Proposition 2.2.13]).

Similarly, we fix a choice of a compatible system of primitive $p^{n}$–th roots of unity $(\zeta_{p^{n}})_{n\geq 0}$. This defines an element $\underline{\varepsilon}$ of $\mathcal{O}_{\mathbb{C}_{K}^{\flat}}$ in an analogous manner, and the embedding $\mathfrak{S}\hookrightarrow A_{\mathrm{inf}}$ extends to a map (actually still an embedding by [Car13, Proposition 1.14]) $W(k)[[u,v]]\rightarrow A_{\mathrm{inf}}$ by additionally setting $v\mapsto[\underline{\varepsilon}]-1$. Additionally, we denote by $\omega$ the element $([\underline{\varepsilon}]-1)/([\underline{\varepsilon}^{1/p}]-1)=[\underline{\varepsilon}^{1/p}]^{p-1}+\dots+[\underline{\varepsilon}^{1/p}]+1$. It is well–known that this is another generator of $\mathrm{Ker}\,\theta$, therefore $\omega/\xi$ is a unit in $A_{\mathrm{inf}}$.

The choices of $\pi,\pi_{n}$ and $\zeta_{p^{n}}$, hence also the maps $\mathfrak{S}\hookrightarrow A_{\mathrm{inf}}$ and $W(k)[[u,v]]\hookrightarrow A_{\mathrm{inf}}$, remain fixed throughout. For this reason, we often refer to $[\underline{\pi}]$ as $u$, $[\underline{\varepsilon}]-1$ as $v$, $\xi$ as $E(u),$ etc.

Throughout the paper, we use freely the language of prisms and $\delta$–rings from [BS22], and we adopt much of the related notation and conventions. In particular, a formal scheme $\mathscr{X}$ over a $p$–adically complete ring $A$ always means a $p$–adic formal scheme, and it is called smooth if it is locally of the form $\mathrm{Spf}\,R$ for a (derived³³3As we will always consider the base $A$ to have bounded $p^{\infty}$–torsion, there is no distinction between derived $p$–completion and $p$–adic completion in this case.) $p$–completely smooth $A$–algebra $R$ – that is, a $p$–complete $A$–algebra such that $R/p$ is a smooth $A/p$–algebra and $\mathrm{Tor}_{i}^{A}(R,A/p)=0$ for all $i>0$. By the results of Elkik [Elk73] and the discussion in [BS22, §1.2], $R$ is equivalently the $p$–adic completion of a smooth $A$-algebra.

Acknowledgements. I would like to express my gratitude to my Ph.D. advisor Tong Liu for suggesting the topic of this paper, his constant encouragement and many comments, suggestions and valuable insights. Many thanks go to Deepam Patel and Shuddhodan Kadattur Vasudevan for organizing the prismatic cohomology learning seminar at Purdue University in Fall 2019, and to Donu Arapura for a useful discussion of Čech theory. I would like to thank Xavier Caruso, Shin Hattori and Shizhang Li for reading an earlier version of this paper, and for providing me with useful comments and questions. The present paper is an adapted version of the author’s Ph.D. thesis at Purdue University. During the preparation of the paper, the author was partially supported by the Ross Fellowship and the Bilsland Fellowship of Purdue University, as well as Graduate School Summer Research Grants of Purdue University during summers 2019–2021.