2.1 The Robba ring

We quickly remind the reader of the definition of the Robba ring ${\mathcal{R}}$ , setting notation for the most part. We note that $K$ is fixed throughout, but that the definition of ${\mathcal{R}}$ depends on $K$ (we simply suppress it from the notation).

For each pair of rational numbers $0<s\leqslant r\leqslant \infty$ we define a $p$ -adic annulus

over $F^{\prime }$ . When $r=\infty$ this is a $p$ -adic disc. We also let

be the half-open annulus. We denote by ${\mathcal{R}}^{[s,r]}$ the formal substitution of a certain indeterminate $\unicode[STIX]{x1D70B}_{K}$ , arising from the field of norms, for the variable $T$ in the ring of functions on $\mathbf{A}_{/F^{\prime }}^{1}[s,r]$ . This is the ring denoted by ${\mathcal{R}}^{[s,r]}(\unicode[STIX]{x1D70B}_{K})$ in [Reference Kedlaya, Pottharst and XiaoKPX14].

If $A$ is a $\mathbf{Q}_{p}$ -affinoid algebra, then we denote ${\mathcal{R}}_{A}^{[s,r]}={\mathcal{R}}^{[s,r]}\widehat{\otimes }_{\mathbf{Q}_{p}}A$ . Let $X=\operatorname{Sp}(A)$ be the associated affinoid space to $A$ . Then ${\mathcal{R}}_{A}^{[s,r]}$ is abstractly isomorphic to the ring of rigid analytic functions on $X^{[s,r]}:=\mathbf{A}^{1}[s,r]_{/F^{\prime }}\times _{\operatorname{Sp}\mathbf{Q}_{p}}\operatorname{Sp}A$ . Thus, it is a noetherian Banach algebra when equipped with the usual Gauss norm.

If $0<s<s^{\prime }\leqslant r\leqslant \infty$ , then there is an injective restriction morphism ${\mathcal{R}}_{A}^{[s,r]}\rightarrow {\mathcal{R}}_{A}^{[s^{\prime },r]}$ which is flat and has dense image. We then define ${\mathcal{R}}_{A}^{r}:=\bigcap _{0<s\leqslant r}{\mathcal{R}}_{A}^{[s,r]}$ and the relative Robba ring over  $A$  is

The ring ${\mathcal{R}}_{A}^{r}$ is the global sections on the rigid space $X^{r}:=\mathbf{A}_{/F^{\prime }}^{1}(0,r]\times _{\operatorname{Sp}\mathbf{Q}_{p}}X$ , the relative half open annulus. We will also use the notation ${\mathcal{R}}_{X}^{r}$ and ${\mathcal{R}}_{X}$ with the obvious meaning.

Returning to the closed annuli, for any $0<s\leqslant r$ there is a continuous action of the group $\unicode[STIX]{x1D6E4}_{K}{\twoheadrightarrow}\operatorname{Gal}(F^{\prime }/F)$ on the $F^{\prime }$ coefficients in ${\mathcal{R}}^{[s,r]}$ ; we can extend this canonically, up to the choice of $\unicode[STIX]{x1D70B}_{K}$ , to the ring ${\mathcal{R}}^{[s,r]}$ . When $r$ is sufficiently small there is also an operator $\unicode[STIX]{x1D711}:{\mathcal{R}}^{[s,r]}\rightarrow {\mathcal{R}}^{[s/p,r/p]}$ called Frobenius which acts on the coefficients in $F^{\prime }$ via the usual Frobenius action and acts on $\unicode[STIX]{x1D70B}_{K}$ by a choice,Footnote ⁴ again, canonically up to $\unicode[STIX]{x1D70B}_{K}$ . The operator $\unicode[STIX]{x1D711}$ turns ${\mathcal{R}}^{[s/p,r/p]}$ into a finite free ${\mathcal{R}}^{[s,r]}$ -module of rank $p$ . Furthermore, $\unicode[STIX]{x1D711}$ extends to an operator $\unicode[STIX]{x1D711}:{\mathcal{R}}^{r}\rightarrow {\mathcal{R}}^{r/p}$ and thus also extends to an operator on ${\mathcal{R}}$ . When $X$ is an affinoid space we extend the actions of $\unicode[STIX]{x1D6E4}_{K}$ and $\unicode[STIX]{x1D711}$ to the relative Robba rings by acting trivially on the coefficients $A$ .

There are two ways to view ${\mathcal{R}}_{X}^{r/p}$ as a module over ${\mathcal{R}}_{X}^{r}$ , either by the restriction map or the operator $\unicode[STIX]{x1D711}$ . If $Q$ is a module over ${\mathcal{R}}_{X}^{r}$ , then we denote by $\unicode[STIX]{x1D711}^{\ast }Q$ the extension of scalars $\unicode[STIX]{x1D711}^{\ast }Q:=Q\otimes _{{\mathcal{R}}_{X}^{r},\unicode[STIX]{x1D711}}{\mathcal{R}}_{X}^{r/p}$ and $Q|_{(0,r/p]}$ the ${\mathcal{R}}_{X}^{r/p}$ -module obtained by using the restriction map.

We now fix $r_{0}>0$ . Since $X$ is affinoid, so is $X^{[s,r]}$ . Thus, by Kiehl’s theorem [Reference Bosch, Güntzer and RemmertBGR84, Theorem 9.4.3/3], global sections give an equivalence of categories

Since $X^{r_{0}}$ is admissibly covered by affinoid opens $\{X^{[s,r]}\}_{0<s\leqslant r\leqslant r_{0}}$ , a coherent sheaf ${\mathcal{Q}}$ on $X^{r_{0}}$ is the same as a system ${\mathcal{Q}}=(Q^{[s,r]})_{0<s\leqslant r\leqslant r_{0}}$ of finite ${\mathcal{R}}_{X}^{[s,r]}$ -modules satisfying the obvious compatibilities. The global sections of a sheaf ${\mathcal{Q}}$ may be calculated by

If $Q$ is a finitely presented module over $X^{r_{0}}$ , then there is a coherent sheaf defined by the compatible family $Q^{[s,r]}:=Q\otimes _{{\mathcal{R}}_{X}^{r_{0}}}{\mathcal{R}}_{X}^{[s,r]}$ . We pause to include an auxiliary result on finitely presented modules over $X^{r}$ . It applies, in particular, to all generalized $\unicode[STIX]{x1D711}$ -modules.

Returning to $\unicode[STIX]{x1D711}$ -modules, if $Q$ is a generalized $\unicode[STIX]{x1D711}$ -module, then the isomorphism $\unicode[STIX]{x1D711}^{\ast }Q\simeq Q|_{(0,r_{0}/p]}$ translates into the choice of a compatible system of isomorphisms $\unicode[STIX]{x1D711}^{\ast }Q^{[s,r]}\simeq Q^{[s/p,r/p]}$ . In fact, all generalized $\unicode[STIX]{x1D711}$ -modules arise this way.

There is another way one might think about the $\unicode[STIX]{x1D711}$ -pullback condition on generalized $\unicode[STIX]{x1D711}$ -modules. If $Q$ is a generalized $\unicode[STIX]{x1D711}$ -module over $X^{r_{0}}$ then the choice of isomorphism $\unicode[STIX]{x1D711}^{\ast }Q\simeq Q|_{(0,r_{0}/p]}$ defines an operator, by $Q{\hookrightarrow}\unicode[STIX]{x1D711}^{\ast }Q\simeq Q|_{(0,r_{0}/p]}$ which we also denote by $\unicode[STIX]{x1D711}$ . If $f\in {\mathcal{R}}_{X}^{r_{0}}$ and $x\in Q$ , then $\unicode[STIX]{x1D711}(fx)=\unicode[STIX]{x1D711}(f)\unicode[STIX]{x1D711}(x)$ and so this version of $\unicode[STIX]{x1D711}$ is naturally a semilinear operator (albeit with a different source than target).

Note that we insist that $\unicode[STIX]{x1D6E4}_{K}$ preserve the radius $r_{0}$ of the generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module. We are now ready to remove the finite radius assumption.

If $Q$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -module, then there exists an $r_{0}>0$ such that $Q$ arises via base change from a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module $Q^{r_{0}}$ over $X^{r_{0}}$ . Thus, for any $r_{0}>r_{0}^{\prime }$ , $Q$ also arises geometrically from $Q^{r_{0}^{\prime }}:=Q|_{(0,r_{0}^{\prime }]}$ . In particular, if $Q$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module, then we may always take a radius sufficiently small to make sense of the notation $Q^{r_{0}}$ .

By a morphism $f:Q\rightarrow Q^{\prime }$ of generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules we mean a continuous $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -equivariant morphism of ${\mathcal{R}}_{X}$ -modules. By definition, there must exist an $r_{0}>0$ so that $f$ arises from base change of a map $f^{r_{0}}:Q^{r_{0}}\rightarrow (Q^{\prime })^{r_{0}}$ for $r_{0}$ sufficiently small. The space of all morphisms will be denoted by $\operatorname{Hom}(Q,Q^{\prime })$ . This is also a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module in the natural way. Taking $Q^{\prime }={\mathcal{R}}_{X}$ we obtain the dual module $Q^{\vee }$ , which we will only use if $Q$ is a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module.

At various points in §§ 5 and 6 we will need to shrink an affinoid space $X$ to an affinoid subdomain $U=\operatorname{Sp}B\subset X$ . Given such a $U$ and a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module $Q$ over $X$ we denote by $Q|_{U}$ the ${\mathcal{R}}_{U}$ -module defined by

Note since $Q$ is finitely presented over ${\mathcal{R}}_{A}$ , the first part of the definition could equivalently be taken to be $Q|_{U}=Q\otimes _{{\mathcal{R}}_{X}}{\mathcal{R}}_{U}$ . We record the following result for later use.

2.2 Galois representations

It will be useful to remind ourselves of the following connection between Galois representations and $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules. If $A$ is an affinoid $\mathbf{Q}_{p}$ -algebra, then by an $A$ -linear representation of $G_{K}$ we mean a finite projective $A$ -module $V$ together with a continuous $A$ -linear action of the Galois group $G_{K}$ .

The theorem as we have stated it can be read off from [Reference Kedlaya and LiuKL10, Theorems 3.11 and 0.2]. The earliest results were proven when $A$ is a field. For that, Fontaine and separately, Cherbonnier and Colmez, gave proofs with the caveat that the Robba ring is replaced with a different ring of analytic functions on affinoid subdomains of discs (see [Reference FontaineFon90] and [Reference Cherbonnier and ColmezCC98]). The key step in extending the theorem to the Robba ring as we have discussed it was Kedlaya’s theorem on slope filtrations [Reference KedlayaKed04]. The family results are more recent and one does not in general have a description of the essential image.

2.3 Rank-one $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules

Rank-one $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules over $A$ are parametrized, essentially, by continuous characters $\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow A^{\times }$ . Let us recall the construction of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules of character type. Note that by [Reference Kedlaya, Pottharst and XiaoKPX14, Theorem 6.2.14] every rank-one $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ arises, locally on $A$ , from one of character type.

Choose a uniformizer $\unicode[STIX]{x1D71B}_{K}$ of $K$ . We can write $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}^{\operatorname{nr}}\unicode[STIX]{x1D6FF}^{\operatorname{wt}}$ where $\unicode[STIX]{x1D6FF}^{\operatorname{nr}}|_{{\mathcal{O}}_{K}^{\times }}=1$ and $\unicode[STIX]{x1D6FF}^{\operatorname{wt}}(\unicode[STIX]{x1D71B}_{K})=1$ . Then, $\unicode[STIX]{x1D6FF}^{\operatorname{wt}}$ extends in a unique manner to the abelianization of the Galois group $G_{K}^{\operatorname{ab}}$ , using the local Artin map, and we denote $\widehat{\unicode[STIX]{x1D6FF}}^{\operatorname{wt}}$ the corresponding Galois character $\widehat{\unicode[STIX]{x1D6FF}}^{\operatorname{wt}}:G_{K}\rightarrow A^{\times }$ . On the other hand, by [Reference Kedlaya, Pottharst and XiaoKPX14, Lemma 6.2.3] there is a unique rank-one free $(F\otimes _{\mathbf{Q}_{p}}A)$ -module $D_{\unicode[STIX]{x1D6FF}^{\operatorname{nr}}(\unicode[STIX]{x1D71B}_{K})}$ equipped with an operator $\unicode[STIX]{x1D711}$ , semilinear with respect to $\unicode[STIX]{x1D711}\otimes 1$ , such that $\unicode[STIX]{x1D711}^{f_{K}}=1\otimes \unicode[STIX]{x1D6FF}^{\operatorname{nr}}(\unicode[STIX]{x1D71B}_{K})$ . We give it the trivial $\unicode[STIX]{x1D6E4}_{K}$ -action and define the rank-one $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module

This is independent of any choices made and satisfies ${\mathcal{R}}_{A}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FF}^{\prime })\simeq {\mathcal{R}}_{A}(\unicode[STIX]{x1D6FF})\otimes _{{\mathcal{R}}_{A}}{\mathcal{R}}_{A}(\unicode[STIX]{x1D6FF}^{\prime })$ . Thus, it makes sense to define $D(\unicode[STIX]{x1D6FF}):=D\otimes _{{\mathcal{R}}_{A}}{\mathcal{R}}_{A}(\unicode[STIX]{x1D6FF})$ for any generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module over $A$ .

Assume now that $A$ is an $L$ -algebra for $L$ as in § 1.6. If $\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow A^{\times }$ is a character, then we can define its weights as follows. The group $K^{\times }$ , as a group over $\mathbf{Q}_{p}$ , has a Lie algebra of dimension $(K:\mathbf{Q}_{p})=\#\unicode[STIX]{x1D6F4}_{K}$ . The differential action gives rise to a weights $(\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}))_{\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}}\in K\otimes _{\mathbf{Q}_{p}}A\simeq \prod _{\unicode[STIX]{x1D70F}}A_{\unicode[STIX]{x1D70F}}$ such that

It is easy to see that $(\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}))_{\unicode[STIX]{x1D70F}}$ only depends only $\unicode[STIX]{x1D6FF}^{\operatorname{wt}}$ in the decomposition of the previous paragraph (thus the notation). We have normalized the weights so that if $z:K^{\times }\rightarrow K^{\times }$ is the identity character, then $\operatorname{wt}_{\unicode[STIX]{x1D70F}}(z)=-1$ for each $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ .

2.4 $p$ -adic Hodge theory

The definition of weight given above is a special case of extending the usual Fontaine functors $D_{\operatorname{Sen}},D_{\operatorname{dR}},D_{\operatorname{cris}}$ , etc. from $p$ -adic Hodge theory to the category of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ modules. In particular, we have the notions of Hodge–Tate–Sen weights, crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules, etc. We will not recall the definitions and will refer to [Reference BergerBer02, Reference BergerBer08] as needed.

One of Berger’s main results [Reference BergerBer08, Théorème A] is that the functor $D_{\operatorname{pst}}(-)$ induces an equivalence

The subcategory of crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules is equivalent to the full subcategory of filtered $\unicode[STIX]{x1D711}$ -modules over $L$ . The étale $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules (the Galois representations, following Theorem 2.8) correspond to the weakly admissible modules on the right-hand side.

Suppose that $D$ is crystalline. The $L\otimes _{\mathbf{Q}_{p}}K$ -module $D_{\operatorname{cris}}(D)_{K}:=D_{\operatorname{cris}}(D)\otimes _{F}K$ is equipped with an exhaustive and separated decreasing filtration $\operatorname{Fil}^{\bullet }D_{\operatorname{cris}}(D)_{K}$ . We denote by $\operatorname{HT}_{\unicode[STIX]{x1D70F}}(D)$ the multi-set of integers such that the induced filtration on the $L$ -vector space $D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}=e_{\unicode[STIX]{x1D70F}}D_{\operatorname{cris}}(D)_{K}$ has jumps given with multiplicity by $\operatorname{HT}_{\unicode[STIX]{x1D70F}}(D)$ . It is easy to see that if $\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow L^{\times }$ is a character such that ${\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})$ is crystalline then $\operatorname{HT}_{\unicode[STIX]{x1D70F}}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}))=\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF})$ . For example, $D_{\operatorname{cris}}({\mathcal{R}}_{L})=F\otimes _{\mathbf{Q}_{p}}L$ with the trivial $\unicode[STIX]{x1D711}^{f_{K}}$ -action and for all $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ we have $\operatorname{HT}_{\unicode[STIX]{x1D70F}}({\mathcal{R}}_{L})=0$ .

If $\unicode[STIX]{x1D70F}:K{\hookrightarrow}L$ is an embedding then we denote the corresponding character $K^{\times }\rightarrow L^{\times }$ by $z_{\unicode[STIX]{x1D70F}}$ . It happens that $D_{\operatorname{cris}}({\mathcal{R}}_{L}(z_{\unicode[STIX]{x1D70F}}))$ is a filtered $\unicode[STIX]{x1D711}$ -module with trivial $\unicode[STIX]{x1D711}^{f_{K}}$ -action and the Hodge–Tate filtration has weights

In particular, $D_{\operatorname{cris}}({\mathcal{R}}_{L}(z_{\unicode[STIX]{x1D70F}}))\subset D_{\operatorname{cris}}({\mathcal{R}}_{L})$ as filtered $\unicode[STIX]{x1D711}$ -modules. Thus, ${\mathcal{R}}_{L}(z_{\unicode[STIX]{x1D70F}})=t_{\unicode[STIX]{x1D70F}}{\mathcal{R}}_{L}$ for some $t_{\unicode[STIX]{x1D70F}}$ , uniquely determined up to unit in ${\mathcal{R}}_{L}$ .

It is easy to see $t_{\unicode[STIX]{x1D70F}}{\mathcal{R}}_{L}$ and $t_{\unicode[STIX]{x1D70E}}{\mathcal{R}}_{L}$ are maximally coprime if $\unicode[STIX]{x1D70E}\neq \unicode[STIX]{x1D70F}$ . Indeed, $D=t_{\unicode[STIX]{x1D70F}}{\mathcal{R}}_{L}+t_{\unicode[STIX]{x1D70E}}{\mathcal{R}}_{L}$ is $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -submodule of ${\mathcal{R}}_{L}$ whose Hodge–Tate weights, computed by passing to $D_{\operatorname{pst}}(D)$ , are zero (for each $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ ). The element $t=\prod _{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70F}}$ is, up to a unit, the ubiquitous $t$ which plays the role of the $p$ -adic $2\unicode[STIX]{x1D70B}i$ in all of $p$ -adic Hodge theory. The $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -submodule $t{\mathcal{R}}_{L}$ is crystalline, its $\unicode[STIX]{x1D70F}$ -Hodge–Tate weight is $-1$ for each $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D711}(t)=pt$ .

2.5 Galois cohomology

Suppose that $A$ is an affinoid algebra and that $Q$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module over $A$ . Let $\unicode[STIX]{x1D6E5}_{K}\subset \unicode[STIX]{x1D6E4}_{K}$ be the $p$ -torsion subgroup (which only exists if $p=2$ ) and choose a topological generator $\unicode[STIX]{x1D6FE}_{0}\in \unicode[STIX]{x1D6E4}_{K}/\unicode[STIX]{x1D6E5}_{K}$ . One then defines the Herr complex [Reference HerrHer98] as the three-term complex $C_{\unicode[STIX]{x1D6FE}_{0}}^{\bullet }(D)$

The Galois cohomology groups $H^{\bullet }(Q)$ of $Q$ are defined to be the cohomology groups of the complex $C_{\unicode[STIX]{x1D6FE}_{0}}^{\bullet }(Q)$ . The complexes depend on the choice of $\unicode[STIX]{x1D6FE}_{0}$ up to canonical quasi-isomorphism, so the cohomology is well-defined.

When $A$ is finite over $L$ , the Galois cohomology is finite-dimensional and satisfies an Euler–Poincaré formula [Reference LiuLiu07]

where the rank of a torsion module must be suitably interpreted (see § 2.6). The finiteness is true over general affinoid $L$ -algebras $A$ in the case of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules by [Reference Kedlaya, Pottharst and XiaoKPX14]. We will review that result in § 4.

But now let us review the dimensions of the cohomology of rank-one $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules over a field. To shorten notation, if $\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow L^{\times }$ is a continuous character we denote $H^{\bullet }(\unicode[STIX]{x1D6FF}):=H^{\bullet }({\mathcal{R}}_{A}(\unicode[STIX]{x1D6FF}))$ . We let $\widehat{T}(L)$ be the space of continuous characters $\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow L^{\times }$ . Define two special subsets of $\widehat{T}(L)$ by

The elements of $\widehat{T}(L)$ which are not in $\widehat{T}(L)^{+}$ or $\widehat{T}(L)^{-}$ are called generic characters.

Let us finish this subsection with a definition.

To put the previous definition in context, if $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FF}^{\prime }$ are homothetic, then following Proposition 2.10 we can find an integer $r$ such that $t^{r}{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}){\hookrightarrow}{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}^{\prime })$ . Thus, ${\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})$ and ${\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}^{\prime })$ are in a sense commensurable. If $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FF}^{\prime }$ are generic up to homothety, then $H^{2}(\unicode[STIX]{x1D702})=H^{0}(\unicode[STIX]{x1D702})=(0)$ for all characters $\unicode[STIX]{x1D702}$ homothetic to $\unicode[STIX]{x1D6FF}^{\prime }\unicode[STIX]{x1D6FF}^{-1}$ .

2.6 Torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules

Note that

By [Reference BergerBer02, Proposition 4.12], each term in the product is an adequate Bézout domain. In particular, finitely generated ${\mathcal{R}}_{L}$ modules are free if and only if they are torsion free (with respect to the total ring of divisors) and there is a robust theory of elementary divisors over ${\mathcal{R}}_{L}$ . As a consequence, a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module over $L$ is a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module if and only if it is torsion-free as an ${\mathcal{R}}_{L}$ -module.

The element $t\in {\mathcal{R}}_{L}$ is an example of a non-zero divisor. If $S$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module we let $S[t^{\infty }]$ denote the $t$ -power torsion submodule. Since $t\in {\mathcal{R}}$ is an eigenvector for $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D6E4}_{K}$ , $S[t^{\infty }]$ is a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -submodule.

The typical example of a pure torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module is ${\mathcal{R}}_{L}/(\prod _{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70F}}^{r_{\unicode[STIX]{x1D70F}}}){\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})$ for some continuous character $\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow L^{\times }$ and non-negative integers $r_{\unicode[STIX]{x1D70F}}$ .

If $S$ is a torsion module, then the Euler–Poincaré formula [Reference LiuLiu07, Theorem 4.7] says

By Lemma 2.13, the cohomology of torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules reduces to the cohomology of pure torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules and that is explained by the following calculation.

3.1 Parabolizations of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules

Let $A$ be an affinoid $L$ -algebra.

If $P_{\bullet }$ is a triangulation of a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module $D$ of rank $d$ , then each quotient $P_{i}/P_{i-1}$ is of the form ${\mathcal{R}}_{A}(\unicode[STIX]{x1D6FF}_{i})$ for some continuous character $\unicode[STIX]{x1D6FF}_{i}:K^{\times }\rightarrow A^{\times }$ , at least locally on $X=\operatorname{Sp}(A)$ [Reference Kedlaya, Pottharst and XiaoKPX14, Theorem 6.2.14]. We call the $d$ -tuple $(\unicode[STIX]{x1D6FF}_{i})_{i=1}^{d}$ the ordered parameter of the triangulation $P_{\bullet }$ and we say that $D$ is trianguline with ordered parameter $(\unicode[STIX]{x1D6FF}_{i})_{i=1}^{d}$ . If $D$ is trianguline with an ordered parameter $(\unicode[STIX]{x1D6FF}_{i})_{i=1}^{d}$ , then $\operatorname{HT}_{\unicode[STIX]{x1D70F}}(D)=\{\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{i})\}_{i=1}^{d}$ .

For the rest of this section we will take $A$ to be the field $L$ itself. Crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules over ${\mathcal{R}}_{L}$ provide examples of triangulations. For that we have the notion of a refinement, following [Reference LiuLiu15, Definition 5.29].

Suppose that $R_{\bullet }$ is a refinement of a crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module. Then each of the quotients $R_{i}/R_{i-1}$ is a rank-one $L\otimes _{\mathbf{Q}_{p}}F$ -module equipped with a linear operator $\unicode[STIX]{x1D711}^{f_{K}}$ . We denote by $\unicode[STIX]{x1D719}_{i}\in L^{\times }$ the eigenvalue of $\unicode[STIX]{x1D711}^{f_{K}}$ appearing in $R_{i}/R_{i-1}$ . Furthermore, $D_{\operatorname{cris}}(D)_{K}$ is an $L\otimes _{\mathbf{Q}_{p}}K$ -vector space equipped with its Hodge filtration $\operatorname{Fil}^{\bullet }D_{\operatorname{cris}}(D)_{K}$ . Each of the $\unicode[STIX]{x1D711}$ -stable subspaces $(R_{i})_{K}$ has an induced Hodge filtration. We define, for each $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ and $i=1,\ldots ,d$ an integer $s_{i,\unicode[STIX]{x1D70F}}$ so that $\{s_{1,\unicode[STIX]{x1D70F}},\ldots ,s_{i,\unicode[STIX]{x1D70F}}\}$ are the $\unicode[STIX]{x1D70F}$ -Hodge–Tate weights appearing in $(R_{i})_{K,\unicode[STIX]{x1D70F}}$ . In summary, triangulations and refinements have the following invariants:

Note that if $P_{\bullet }$ is a triangulation of a crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module $D$ , then each step $P_{i}$ is a crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module as well.

3.2 Critical and non-critical triangulations

For this subsection we work with a fixed crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module $D$ over the field $L$ .

The definition is only given for crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules as it relies on the correspondence Proposition 3.3. A more general definition will be given later which applies to certain $p$ -adic limits of crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules (see Definition 6.7).

In the case of regular Hodge–Tate weights, we have a convenient way to check whether or not a saturated $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -submodule is critical.

At this point, one could define what it means for a triangulation to be non-critical. More generally, for each parabolization $P_{\bullet }$ of a crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module we define a subparabolization $P_{\bullet }^{\operatorname{nc}}\subset P_{\bullet }$ for which every step is non-critical.

Note that, as suggested by our notation, $I^{\operatorname{nc}}\neq \varnothing$ and $i_{r}=d$ , since $D$ itself is always a non-critical $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -submodule of itself. In the case where $D$ has regular weights, Lemma 3.5 shows that $P_{\bullet }^{\operatorname{nc}}$ is the unique subparabolization of $P_{\bullet }$ consisting of the steps whose Hodge–Tate weights are as low as possible. Furthermore, it is easy to check that $(P_{\bullet }^{\operatorname{nc}})^{\operatorname{nc}}=P_{\bullet }^{\operatorname{nc}}$ , hence the use of the word ‘maximal’. Let us end this subsection with a brief example.

3.3 Generalized triangulations

Recall that a pure torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module over $L$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module that is either zero or free over ${\mathcal{R}}_{L}/\prod _{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70F}}^{r_{\unicode[STIX]{x1D70F}}}$ for some collection $(r_{\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}}$ of non-negative integers, not all zero.

If $Q$ is non-zero and pure torsion of character type, then we refer to the $\unicode[STIX]{x1D6F4}_{K}$ -tuple $(r_{\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}}$ as the torsion exponents of $Q$ and $(\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}))_{\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}}$ as the torsion weights of $Q$ . The zero module $(0)$ has, by definition, torsion exponents $(0,\ldots ,0)$ and torsion weights $(w_{\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}}$ for any collection of integers $w_{\unicode[STIX]{x1D70F}}$ . By Proposition 2.14, these invariants, taken together, completely classify $Q$ among pure torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules of character type.

Note that we allow for consecutive steps $Q_{i}\subset Q_{i+1}$ to be equal, since $(0)$ is a pure torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module of character type under our definition. This has two consequences. First, even if $Q$ is a bona fide $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module, then a generalized triangulation is not a triangulation in the sense of § 3.1. We will deal with this ambiguity in the definition of standard triangulation below. Second, since we can always repeat steps in a generalized triangulation, the length of a generalized triangulation $Q_{\bullet }$ depends on $Q_{\bullet }$ ; it is not intrinsic to $Q$ , unlike lengths of triangulations of bona fide $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules. This remains true even if a generalized triangulation $Q_{\bullet }$ is strictly increasing: the length still cannot be read off from $Q$ since ${\mathcal{R}}_{L}/t_{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70E}}\simeq {\mathcal{R}}_{L}/t_{\unicode[STIX]{x1D70F}}\oplus {\mathcal{R}}_{L}/t_{\unicode[STIX]{x1D70E}}$ if $\unicode[STIX]{x1D70E}\neq \unicode[STIX]{x1D70F}$ .

Note that if $Q$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module, then the torsion submodule $Q_{\operatorname{tor}}=Q[t^{\infty }]\subset Q$ is $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -stable and an ${\mathcal{R}}_{L}$ -module summand. The quotient $Q/Q_{\operatorname{tor}}$ is a bona fide $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module whose rank depends only on $Q$ . We isolate those generalized triangulations which appear in practice.

In the definition, the torsion length $i$ depends on $Q_{\bullet }$ whereas the free length $d-i$ depends only on $Q$ (since it is equal to $\operatorname{rank}_{{\mathcal{R}}_{L}}Q/Q_{\operatorname{tor}}$ ). And now a standard triangulation is closer to a triangulation in the case where $Q$ is a bona fide $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module. Indeed, in that case $Q_{\operatorname{tor}}=(0)$ and so a standard triangulation is of the form

where $P_{j}:=Q_{i+j}$ defines a triangulation of $Q$ as in Definition 3.1.

A standard triangulation of a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module has a number of invariants which we now detail. Suppose that $Q_{\bullet }$ is a standard triangulation.

• ∙ The induced generalized triangulation on $Q/Q_{\operatorname{tor}}$ is an actual triangulation with ordered parameter $(\unicode[STIX]{x1D6FF}_{i+1},\ldots ,\unicode[STIX]{x1D6FF}_{d})$ whose length is the free length of $Q$ .

• ∙ The induced generalized triangulation on $Q_{\operatorname{tor}}$ also has invariants. For one, it has its length $i$ . Second, if $1\leqslant j\leqslant i$ then $Q_{j}/Q_{j-1}$ is pure torsion of character type and thus has exponents $(r_{j,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}}$ and weights $(w_{j,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}}$ .

Note that it may happen that for some $\unicode[STIX]{x1D70F}$ , $r_{j,\unicode[STIX]{x1D70F}}=0$ . For example, if there is an integer $j$ such that $Q_{j}=Q_{j-1}$ , then the corresponding torsion exponents are $r_{j,\unicode[STIX]{x1D70F}}=0$ for all $\unicode[STIX]{x1D70F}$ . However, ranging over $j$ we can a priori predict the frequency at which this happens.

Fix a $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ and a standard triangulation $Q_{\bullet }$ of a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module with torsion length $i$ and exponents $((r_{j,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}})_{1\leqslant j\leqslant i}$ . Since the successive quotients of a generalized triangulation are direct summands as ${\mathcal{R}}_{L}$ -modules, it is easy to see that $\ell _{\unicode[STIX]{x1D70F}}(Q)=\#\{j:r_{j,\unicode[STIX]{x1D70F}}\neq 0\}$ and that $\ell _{\unicode[STIX]{x1D70F}}(Q)\leqslant i$ for all $\unicode[STIX]{x1D70F}$ . To summarize the previous discussion we separate out the following definition.

Finally, we finish this section with a result that explains how standard triangulations of generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules are inherently more flexible than triangulations of bona fide $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules. Recall that we defined the notion of homothety among continuous characters of $K^{\times }$ at the end of § 2.5.

Note that one can always find infinitely many such integers $r_{i+1,\unicode[STIX]{x1D70F}}$ which satisfy the hypotheses of the proposition. Also note that if $r_{i+1,\unicode[STIX]{x1D70F}}=0$ for all $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ , then the standard triangulation we just produced will have two consecutive steps which are equal.

5.1 Triangulated families

We will dwell further beyond these remarks. First, $i(x_{0})$ is the torsion length of the standard triangulation $Q_{x_{0},\bullet }$ but beware that $Q_{x_{0}}$ may actually be torsion free, even if $i(x_{0})>0$ . Indeed, our definitions allow for successive quotients in $Q_{x_{0},\bullet }$ to be zero. In particular, if $i(x_{0})=1$ and $s_{1,\unicode[STIX]{x1D70F}}=\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{1,x_{0}})$ for all $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ , then $Q_{x_{0}}$ is torsion free. This situation does not arise in our applications, but we mention it because the remark applies equally well to $y\in X^{\operatorname{nc}}$ . Indeed, if $y\in X^{\operatorname{nc}}$ , then axiom (TF4) says that the torsion exponents of $Q_{y,1}$ are given by $\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{1,x_{0}})-\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{1,x_{0}})=0$ for all $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ and thus $Q_{1,y}=(0)$ .

More generally, when $i(x_{0})>0$ and $y\in X^{\operatorname{nc}}$ there will be $i(x_{0})-1$ distinct torsion steps in the standard triangulation $Q_{y,\bullet }$ . On the other hand, in practice, $Q_{x_{0},\bullet }$ will have $i(x_{0})$ torsion steps. However, consider the situation where for all $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ there exists a $j$ such that $j\leqslant i(x_{0})$ and $s_{j,\unicode[STIX]{x1D70F}}=\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{1,x_{0}})$ . Then, for each $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ we have an equality of $\unicode[STIX]{x1D70F}$ -torsion lengths $\ell _{\unicode[STIX]{x1D70F}}(Q_{y})=\ell _{\unicode[STIX]{x1D70F}}(Q_{x_{0}})$ at $x_{0}$ versus $y\in X^{\operatorname{nc}}$ . Thus, the discrepancy in the number of torsion steps is really an artifact of how we are doing the bookkeeping. We will see in the course of proving Theorem 6.8 that it can even happen that the $t_{\unicode[STIX]{x1D70F}}$ -torsion submodule of $Q_{x_{0}}$ is isomorphic, as an ${\mathcal{R}}_{L}$ -module, to the $t_{\unicode[STIX]{x1D70F}}$ -torsion submodule of $Q_{y}$ for one (and thus all) $y\in X^{\operatorname{nc}}$ .

Let us continue now with constructing new pointwise triangulated families out of old ones in general.

There were two points unresolved in the previous theorem, both of which we resolve with the following lemma. For the second part, and more in the following results, we will make use of the appendix on Nakayama’s lemma.