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

$$\begin{eqnarray}\displaystyle \mathbf{A}_{/F^{\prime }}^{1}[s,r]=\{T:p^{-r/(p-1)}\leqslant |T|\leqslant p^{-s/(p-1)}\} & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle \mathbf{A}_{/F^{\prime }}^{1}(0,r]=\{T:p^{-r/(p-1)}\leqslant |T|<1\}. & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle {\mathcal{R}}_{A}=\mathop{\bigcup }_{0<r}{\mathcal{R}}_{A}^{r}. & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle \{\text{finite }{\mathcal{R}}_{X}^{[s,r]}\text{-modules}\}\longleftrightarrow \{\text{coherent sheaves on }X^{[s,r]}\}. & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle Q=\unicode[STIX]{x1D6E4}(X^{r_{0}},{\mathcal{Q}})=\mathop{\varprojlim }\nolimits_{0<s}Q^{[s,r_{0}]}. & & \displaystyle \nonumber\end{eqnarray}$$

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.

Proof. Since $Q$ is assumed to be finitely presented over ${\mathcal{R}}_{X}^{r}$ , the same is true for $Q/f$ over ${\mathcal{R}}_{X}^{r}/f$ and thus by [Reference MatsumuraMat89, Corollary 7.12] it suffices to replace ‘projective’ with ‘flat’ in the statement of the lemma.

Even without that, one direction is clear: if $Q/f$ is finite projective over ${\mathcal{R}}_{X}^{r}/f$ , then $Q^{[s,r]}/f=Q/f\otimes _{{\mathcal{R}}_{X}^{r}/f}{\mathcal{R}}^{[s,r]}/f$ is finite projective over ${\mathcal{R}}^{[s,r]}/f$ for each $0<s\leqslant r$ .

We will now prove the reverse direction, so assume that $Q^{[s,r]}/f$ is flat over ${\mathcal{R}}_{X}^{[s,r]}/f$ for each $0<s\leqslant r$ . By [Reference MatsumuraMat89, Theorem 7.7] it suffices to show that

(1) $$\begin{eqnarray}I\otimes _{{\mathcal{R}}_{X}^{r}/f}Q/f\rightarrow Q/f\end{eqnarray}$$

is injective for every finitely generated ideal $I\subset {\mathcal{R}}_{X}^{r}/f$ . Now we use the language of ‘co-admissible’ modules originally due to Schneider and Teitelbaum [Reference Schneider and TeitelbaumST03], and we will reference [Reference Kedlaya, Pottharst and XiaoKPX14, § 2.1]. By [Reference Kedlaya, Pottharst and XiaoKPX14, Lemma 2.1.4(7)] the ${\mathcal{R}}_{X}^{r}$ -module ${\mathcal{R}}_{X}^{r}/f$ is co-admissible, and thus so is $I\subset {\mathcal{R}}_{X}^{r}/f$ by [Reference Kedlaya, Pottharst and XiaoKPX14, Lemma 2.1.4(6)]. Since $Q$ is co-admissible so is $Q/f=\operatorname{coker}(Q\overset{f}{\longrightarrow }Q)$ by [Reference Kedlaya, Pottharst and XiaoKPX14, Lemma 2.1.4(5)]. Thus, (1) is a morphism of co-admissible ${\mathcal{R}}_{X}^{r}$ -modules and hence is injective if and only if

(2) $$\begin{eqnarray}(I\otimes _{{\mathcal{R}}_{X}^{r}/f}Q/f)^{[s,r]}\rightarrow Q^{[s,r]}/f\end{eqnarray}$$

is injective for each $0<s\leqslant r$ . But we see

$$\begin{eqnarray}\displaystyle (I\otimes _{{\mathcal{R}}_{X}^{r}/f}Q/f)^{[s,r]}\simeq (I\otimes _{{\mathcal{R}}_{X}^{r}/f}{\mathcal{R}}_{X}^{[s,r]}/f)\otimes _{{\mathcal{R}}_{X}^{[s,r]}/f}Q^{[s,r]}/f & & \displaystyle \nonumber\end{eqnarray}$$

and then since ${\mathcal{R}}_{X}^{r}/f\rightarrow {\mathcal{R}}_{X}^{[s,r]}/f$ is flat we see that $I\otimes _{{\mathcal{R}}_{X}^{r}/f}{\mathcal{R}}_{X}^{[s,r]}/f=I\cdot {\mathcal{R}}_{X}^{[s,r]}/f$ is an ideal in ${\mathcal{R}}_{X}^{[s,r]}/f$ . In particular, this shows that the map (2) may be identified with the natural map

$$\begin{eqnarray}\displaystyle I\cdot {\mathcal{R}}_{X}^{[s,r]}/f\otimes _{{\mathcal{R}}_{X}^{[s,r]}/f}Q^{[s,r]}/f\rightarrow Q^{[s,r]}/f, & & \displaystyle \nonumber\end{eqnarray}$$

which is injective because $Q^{[s,r]}/f$ is flat over ${\mathcal{R}}_{X}^{[s,r]}/f$ by assumption. This completes the proof.◻

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.

Proposition 2.3. There is an equivalence of categories

$$\begin{eqnarray}\displaystyle \biggl\{\begin{array}{@{}c@{}}\text{generalized }\unicode[STIX]{x1D711}\text{-modules}\\ \text{over }X^{r_{0}}\end{array}\biggr\}\longleftrightarrow \left\{\begin{array}{@{}c@{}}\text{coherent sheaves }{\mathcal{Q}}=(Q^{[s,r]})\text{ on }X^{r_{0}}\text{ equipped}\\ \text{with naturally compatible isomorphisms}\\ \unicode[STIX]{x1D711}^{\ast }Q^{[s,r]}\cong Q^{[s/p,r/p]}\end{array}\right\}. & & \displaystyle \nonumber\end{eqnarray}$$

Moreover, the $\unicode[STIX]{x1D711}$ -modules on the left-hand side correspond to sheaves on the right-hand side for which each $Q^{[s,r]}$ is projective.

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

$$\begin{eqnarray}\displaystyle Q|_{U}:=Q\widehat{\otimes }_{{\mathcal{R}}_{X}}{\mathcal{R}}_{U}=Q\widehat{\otimes }_{A}B. & & \displaystyle \nonumber\end{eqnarray}$$

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 $G_{K}$ .

Theorem 2.8. Let $X=\operatorname{Sp}(A)$ . There is a fully faithful, exact embedding

$$\begin{eqnarray}\displaystyle D_{\operatorname{rig}}:\{A\text{-linear representations }V\text{ of }G_{K}\}{\hookrightarrow}\{(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})\text{-modules over }{\mathcal{R}}_{X}\} & & \displaystyle \nonumber\end{eqnarray}$$

such that:

1. (a) $D_{\operatorname{rig}}$ commutes with base change $A\rightarrow A^{\prime }$ ; and

2. (b) when $A$ is finite over $\mathbf{Q}_{p}$ , $D_{\operatorname{rig}}$ is essentially surjective onto the category of étale $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -modules.

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

$$\begin{eqnarray}\displaystyle {\mathcal{R}}_{A}(\unicode[STIX]{x1D6FF}):=(D_{\unicode[STIX]{x1D6FF}^{\operatorname{nr}}(\unicode[STIX]{x1D71B}_{K})}\otimes _{F\otimes _{\mathbf{Q}_{p}}A}{\mathcal{R}}_{A})\otimes _{{\mathcal{R}}_{A}}D_{\operatorname{rig}}(\widehat{\unicode[STIX]{x1D6FF}}^{\operatorname{wt}}). & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle 0=\lim _{\substack{ a\rightarrow 0 \\ a\in {\mathcal{O}}_{K}}}\frac{|\unicode[STIX]{x1D6FF}(1+a)-1+\mathop{\sum }_{\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}}\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D70F}(a)|}{|a|_{K}}. & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle \{\text{potentially semistable }(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})\text{-modules over }L\}\overset{D_{\operatorname{pst}}}{\longrightarrow }\{\text{filtered }(\unicode[STIX]{x1D711},N,G_{K})\text{-modules over }L\}. & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle \operatorname{HT}_{\unicode[STIX]{x1D70E}}({\mathcal{R}}_{L}(z_{\unicode[STIX]{x1D70F}}))=\left\{\begin{array}{@{}ll@{}}-1\quad & \text{if }\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70F},\\ 0\quad & \text{if }\unicode[STIX]{x1D70E}\neq \unicode[STIX]{x1D70F}.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

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]

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{i=0}^{2}(-1)\dim _{L}H^{i}(Q)=-(K:\mathbf{Q}_{p})\operatorname{rank}_{{\mathcal{R}}_{L}}Q, & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle \widehat{T}(L)^{+} & = & \displaystyle \biggl\{\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow L^{\times }:\unicode[STIX]{x1D6FF}=\mathop{\prod }_{\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}}z_{\unicode[STIX]{x1D70F}}^{r_{\unicode[STIX]{x1D70F}}}\text{ with }r_{\unicode[STIX]{x1D70F}}\leqslant 0\text{ for each }\unicode[STIX]{x1D70F}\biggr\},\nonumber\\ \displaystyle \widehat{T}(L)^{-} & = & \displaystyle \biggl\{\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow L^{\times }:\unicode[STIX]{x1D6FF}=|N_{K/\mathbf{Q}_{p}}|\mathop{\prod }_{\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}}z_{\unicode[STIX]{x1D70F}}^{r_{\unicode[STIX]{x1D70F}}}\text{ with }r_{\unicode[STIX]{x1D70F}}\geqslant 1\text{ for each }\unicode[STIX]{x1D70F}\biggr\}.\nonumber\end{eqnarray}$$

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

Proposition 2.10. Let $\unicode[STIX]{x1D6FF}:K^{\times }\rightarrow L^{\times }$ be a continuous character. Then

$$\begin{eqnarray}\displaystyle \dim _{L}H^{0}(\unicode[STIX]{x1D6FF}) & = & \displaystyle \left\{\begin{array}{@{}ll@{}}1\quad & \text{if }\unicode[STIX]{x1D6FF}\in \widehat{T}(L)^{+},\\ 0\quad & \text{otherwise,}\end{array}\right.\nonumber\\ \displaystyle \dim _{L}H^{1}(\unicode[STIX]{x1D6FF}) & = & \displaystyle \left\{\begin{array}{@{}ll@{}}2\quad & \text{if }\unicode[STIX]{x1D6FF}\in \widehat{T}(L)^{+}\cup \widehat{T}(L)^{-},\\ 1\quad & \text{otherwise,}\end{array}\right.\nonumber\\ \displaystyle \dim _{L}H^{2}(\unicode[STIX]{x1D6FF}) & = & \displaystyle \left\{\begin{array}{@{}ll@{}}1\quad & \text{if }\unicode[STIX]{x1D6FF}\in \widehat{T}(L)^{-},\\ 0\quad & \text{otherwise.}\end{array}\right.\nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle {\mathcal{R}}_{L}={\mathcal{R}}\otimes _{\mathbf{Q}_{p}}L={\mathcal{R}}\otimes _{F}(F\otimes _{\mathbf{Q}_{p}}L)=\mathop{\prod }_{\unicode[STIX]{x1D702}\in \operatorname{Gal}(F/\mathbf{Q}_{p})}{\mathcal{R}}\otimes _{F}L. & & \displaystyle \nonumber\end{eqnarray}$$

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

$$\begin{eqnarray}\displaystyle \dim _{L}H^{0}(S)=\dim _{L}H^{1}(S)\quad \text{and}\quad \dim _{L}H^{2}(S)=0. & & \displaystyle \nonumber\end{eqnarray}$$

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.

Proposition 2.14. Let $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ . Then for each $i=0,1$ we have

$$\begin{eqnarray}\displaystyle \dim _{L}H^{i}(({\mathcal{R}}_{L}/t_{\unicode[STIX]{x1D70F}}^{r_{\unicode[STIX]{x1D70F}}})(\unicode[STIX]{x1D6FF}))=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF})\in \{0,1,\ldots ,r_{\unicode[STIX]{x1D70F}}-1\},\\ 0\quad & \text{otherwise.}\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

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

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

Definition 3.1. If $D$ is a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module over $A$ , then a parabolization $P_{\bullet }$ of $D$ (of length $s$ ) is a filtration

$$\begin{eqnarray}\displaystyle 0=P_{0}\subsetneq P_{1}\subsetneq \cdots \subsetneq P_{s-1}\subsetneq P_{s}=D & & \displaystyle \nonumber\end{eqnarray}$$

such that each:

1. – $P_{i}$ is a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module; and

2. – for each $i=1,\ldots ,s$ we have that $P_{i}/P_{i-1}$ is a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module over $A$ which is a ${\mathcal{R}}_{A}$ -module direct summand of $P_{i}$ .

If $P_{\bullet }$ is a parabolization of the maximal length $s=\operatorname{rank}_{{\mathcal{R}}_{A}}D$ , then we say that $P_{\bullet }$ is a triangulation. We say $D$ is trianguline if, after possibly extending the coefficient field $L$ , there exists a triangulation of $D$ .

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].

Definition 3.2. If $D$ is a crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module of rank $d$ over $L$ , then a partial refinement $R_{\bullet }$ of $D$ is the choice of a $\unicode[STIX]{x1D711}$ -stable $L\otimes _{\mathbf{Q}_{p}}F$ -linear filtration

$$\begin{eqnarray}\displaystyle 0=R_{0}\subsetneq R_{1}\subsetneq \cdots \subsetneq R_{s}=D_{\operatorname{cris}}(D), & & \displaystyle \nonumber\end{eqnarray}$$

whose successive quotients are free $L\otimes _{\mathbf{Q}_{p}}F$ -modules. In the case that $s=d$ we call $R_{\bullet }$ a refinement.

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:

$$\begin{eqnarray}\displaystyle \text{a triangulation }P_{\bullet } & {\rightsquigarrow} & \displaystyle \text{the ordered parameter }(\unicode[STIX]{x1D6FF}_{1},\ldots ,\unicode[STIX]{x1D6FF}_{n});\nonumber\\ \displaystyle \text{a refinement }R_{\bullet } & {\rightsquigarrow} & \displaystyle \text{the ordering of }\unicode[STIX]{x1D711}^{f_{K}}\text{-eigenvalues }(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{n})\nonumber\\ \displaystyle & & \displaystyle \quad \text{and the }\unicode[STIX]{x1D70F}\text{-}\text{Hodge}{-}\text{Tate weights }(s_{1,\unicode[STIX]{x1D70F}},\ldots ,s_{d,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}}.\nonumber\end{eqnarray}$$

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 $L$ .

Definition 3.4. Suppose that $P\subset D$ is a saturated $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -submodule. If $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ , then we say that $P$ is $\unicode[STIX]{x1D70F}$ -non-critical if there exist an integer $k_{\unicode[STIX]{x1D70F}}$ such that

$$\begin{eqnarray}\displaystyle D_{\operatorname{cris}}(P)_{K,\unicode[STIX]{x1D70F}}\oplus \operatorname{Fil}^{k_{\unicode[STIX]{x1D70F}}}D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}=D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}. & & \displaystyle \nonumber\end{eqnarray}$$

We say $P$ is $\unicode[STIX]{x1D70F}$ -critical otherwise. Finally, $P$ is called non-critical if $P$ is $\unicode[STIX]{x1D70F}$ -non-critical for each $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ and $P$ is critical if there exists a $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}_{K}$ such that $P$ is $\unicode[STIX]{x1D70F}$ -critical.

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.

Proof. First, part (b) implies part (a) by definition. Second, parts (b) and (c) are easily equivalent. Now suppose that $P$ is non-critical and choose an integer $k_{\unicode[STIX]{x1D70F}}$ such that $D_{\operatorname{cris}}(P)_{K,\unicode[STIX]{x1D70F}}\oplus \operatorname{Fil}^{k_{\unicode[STIX]{x1D70F}}}D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}=D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}$ . Since $P$ is crystalline, $d-i=\dim _{L_{\unicode[STIX]{x1D70F}}}\operatorname{Fil}^{k_{\unicode[STIX]{x1D70F}}}D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}$ . Since the Hodge–Tate weights are all distinct we conclude $\operatorname{Fil}^{k_{\unicode[STIX]{x1D70F}}}D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}=\operatorname{Fil}^{k_{i+1,\unicode[STIX]{x1D70F}}}D_{\operatorname{cris}}(D)_{K,\unicode[STIX]{x1D70F}}$ . This shows that part (a) implies part (b).

It remains to show that parts (c) and (d) are equivalent. Since $D$ has distinct Hodge–Tate weights, the unique lowest weight of $\wedge ^{i}D$ is $k_{1,\unicode[STIX]{x1D70F}}+\cdots +k_{i,\unicode[STIX]{x1D70F}}$ . The next highest weight is $k_{1,\unicode[STIX]{x1D70F}}+\cdots +k_{i-1,\unicode[STIX]{x1D70F}}+k_{i+1,\unicode[STIX]{x1D70F}}$ . Thus, part (c) is true if and only if

$$\begin{eqnarray}\displaystyle D_{\operatorname{cris}}(\det P)_{K,\unicode[STIX]{x1D70F}}\oplus \operatorname{Fil}^{k_{1,\unicode[STIX]{x1D70F}}+\cdots +k_{i-1,\unicode[STIX]{x1D70F}}+k_{i+1,\unicode[STIX]{x1D70F}}}D_{\operatorname{cris}}(\wedge ^{i}D)_{K,\unicode[STIX]{x1D70F}}=D_{\operatorname{cris}}(\wedge ^{i}D)_{K,\unicode[STIX]{x1D70F}}, & & \displaystyle \nonumber\end{eqnarray}$$

which is part (d). ◻

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.

Definition 3.6. Let $P_{\bullet }$ be a parabolization of a crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module $D$ of rank $d$ . Let

$$\begin{eqnarray}\displaystyle I^{\operatorname{nc}}=\{i:P_{i}\text{ is non-critical}\}=\{0=i_{0}<i_{1}<\cdots <i_{r}=d\}. & & \displaystyle \nonumber\end{eqnarray}$$

The maximal non-critical parabolization $P_{\bullet }^{\operatorname{nc}}$ is the filtration

$$\begin{eqnarray}\displaystyle P_{\bullet }^{\operatorname{nc}}:0=P_{i_{0}}\subsetneq P_{i_{1}}\subsetneq \cdots \subsetneq P_{i_{r}}=D. & & \displaystyle \nonumber\end{eqnarray}$$

We say that $P_{\bullet }$ is non-critical if $P_{\bullet }^{\operatorname{nc}}=P^{\operatorname{nc}}$ , and critical otherwise.

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.

Example 3.7. Suppose now that $K=\mathbf{Q}_{p}$ and that $D$ is a rank-two crystalline $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{\mathbf{Q}_{p}})$ -module over $L$ , with Hodge–Tate weights $k_{1}<k_{2}$ and distinct crystalline eigenvalues $\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}^{\prime }\in L^{\times }$ . Since $\unicode[STIX]{x1D719}\neq \unicode[STIX]{x1D719}^{\prime }$ we assume without loss of generality that $D_{\operatorname{cris}}(D)^{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D719}}\cap \operatorname{Fil}^{k_{2}}D_{\operatorname{cris}}(D)=(0)$ . Thus, there is always a non-critical triangulation ${\mathcal{R}}_{L}(z^{-k_{1}}\operatorname{unr}(\unicode[STIX]{x1D719}))\subset D$ . The ordered parameter is $(z^{-k_{1}}\operatorname{unr}(\unicode[STIX]{x1D719}),z^{-k_{2}}\operatorname{unr}(\unicode[STIX]{x1D719}^{\prime }))$ .

On the other hand, one can use Propositions 3.3 and 2.10 to show that $D$ is split if and only if $D_{\operatorname{cris}}(D)^{\unicode[STIX]{x1D711}=\unicode[STIX]{x1D719}^{\prime }}=\operatorname{Fil}^{k_{2}}D_{\operatorname{cris}}(D)$ (if $D$ is étale, the same statement follows from the weak admissibility of the filtered $\unicode[STIX]{x1D711}$ -module $D_{\operatorname{cris}}(D)$ ). Thus, the triangulation corresponding to the ordering $(\unicode[STIX]{x1D719}^{\prime },\unicode[STIX]{x1D719})$ is given by

$$\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}}{\mathcal{R}}_{L}(z^{-k_{1}}\operatorname{unr}(\unicode[STIX]{x1D719}^{\prime }))\subset D\quad & \text{if }D\text{ is non-split,}\\ {\mathcal{R}}_{L}(z^{-k_{2}}\operatorname{unr}(\unicode[STIX]{x1D719}^{\prime }))\subset D\quad & \text{if }D\text{ is split.}\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

This triangulation is critical if and only if $D$ is split.

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.

Definition 3.9. Let $Q$ be a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module over ${\mathcal{R}}_{L}$ . A generalized triangulation of $Q$ is a filtration $Q_{\bullet }$

$$\begin{eqnarray}\displaystyle Q_{\bullet }:0=Q_{0}\subset Q_{1}\subset Q_{2}\subset \cdots \subset Q_{d-1}\subset Q_{d}=Q & & \displaystyle \nonumber\end{eqnarray}$$

such that for $1\leqslant i\leqslant d$ , $Q_{i}/Q_{i-1}$ is either a rank-one $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module or a pure torsion $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module of character type and in either case $Q_{i}/Q_{i-1}$ is a direct summand of $Q_{i}$ as a ${\mathcal{R}}_{L}$ -module. We say that $Q$ is triangulated if it is equipped with a triangulation and trianguline if it may be triangulated, after possibly extending scalars.

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

$$\begin{eqnarray}\displaystyle 0=0=\cdots =0=Q_{i}\subsetneq Q_{i+1}\subsetneq \cdots \subsetneq Q_{d}=Q & & \displaystyle \nonumber\end{eqnarray}$$

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.

Definition 3.11. If $Q$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module, then its $\unicode[STIX]{x1D70F}$ -torsion length is defined by

$$\begin{eqnarray}\displaystyle \ell _{\unicode[STIX]{x1D70F}}(Q)=\operatorname{rank}_{{\mathcal{R}}_{L}/t_{\unicode[STIX]{x1D70F}}}Q_{\operatorname{tor}}/t_{\unicode[STIX]{x1D70F}}. & & \displaystyle \nonumber\end{eqnarray}$$

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.

Proposition 3.13. Suppose that $Q$ is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module,  $Q_{\bullet }$  is a standard triangulation of torsion length $i$ with torsion exponents $((r_{j,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}})_{1\leqslant j\leqslant i}$ , torsion weights $((w_{j,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}})_{1\leqslant j\leqslant i}$ and free parameter $(\unicode[STIX]{x1D6FF}_{j})_{j>i}$ . Assume furthermore that $\unicode[STIX]{x1D6FF}_{i+1}$ is not homothetic to $\unicode[STIX]{x1D6FF}_{j}$ for $j>i+1$ .

Then, for every $\unicode[STIX]{x1D6F4}_{K}$ -tuple $(r_{i+1,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}}$ of non-negative integers such that

$$\begin{eqnarray}\displaystyle j\leqslant i\;\Longrightarrow \;w_{j,\unicode[STIX]{x1D70F}}-\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{i+1})\notin \{-r_{i+1,\unicode[STIX]{x1D70F}},\ldots ,r_{j,\unicode[STIX]{x1D70F}}-r_{i+1,\unicode[STIX]{x1D70F}}-1\}=\{-r_{i+1,\unicode[STIX]{x1D70F}}+m:0\leqslant m<r_{j,\unicode[STIX]{x1D70F}}\}, & & \displaystyle \nonumber\end{eqnarray}$$

there exists a unique (up to scalar) inclusion $\prod _{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70F}}^{r_{i+1,\unicode[STIX]{x1D70F}}}{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1}){\hookrightarrow}Q$ . Its cokernel is a generalized $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4}_{K})$ -module which is naturally equipped with a standard triangulation having invariants:

1. – torsion length $i+1$ ;

2. – torsion exponents $((r_{j,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}})_{j\leqslant i+1}$ ;

3. – torsion weights $((w_{j,\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}})_{j\leqslant i}\cup (\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{i+1}))_{\unicode[STIX]{x1D70F}}$ ; and

4. – free parameter $(\unicode[STIX]{x1D6FF}_{j})_{j>i+1}$ .

Proof. Let $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{i+1}\prod _{\unicode[STIX]{x1D70F}}z_{\unicode[STIX]{x1D70F}}^{r_{i+1,\unicode[STIX]{x1D70F}}}$ . For each $j\leqslant i$ we choose a character $\unicode[STIX]{x1D6FF}_{j}$ whose $\unicode[STIX]{x1D70F}$ -weight is $w_{j,\unicode[STIX]{x1D70F}}$ and so that $Q_{j}/Q_{j-1}\simeq {\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{j})/\prod _{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70F}}^{r_{j,\unicode[STIX]{x1D70F}}}$ . We quickly calculate

$$\begin{eqnarray}\displaystyle \operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{j}\unicode[STIX]{x1D6FF}^{-1})=w_{j,\unicode[STIX]{x1D70F}}-\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{i+1})+r_{i+1,\unicode[STIX]{x1D70F}}. & & \displaystyle \nonumber\end{eqnarray}$$

By our assumptions, $\operatorname{wt}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FF}_{j}\unicode[STIX]{x1D6FF}^{-1})\notin \{0,1,\ldots ,r_{j,\unicode[STIX]{x1D70F}}-1\}$ for each $\unicode[STIX]{x1D70F}$ . Using Proposition 2.14 we see that

$$\begin{eqnarray}\displaystyle \operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),Q_{j}/Q_{j-1})=H^{0}\biggl(\biggl({\mathcal{R}}_{L}/\mathop{\prod }_{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70F}}^{r_{j,\unicode[STIX]{x1D70F}}}\biggr)(\unicode[STIX]{x1D6FF}_{j}\unicode[STIX]{x1D6FF}^{-1})\biggr)=(0). & & \displaystyle \nonumber\end{eqnarray}$$

By induction on $1\leqslant j\leqslant i$ we see that $\operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),Q_{i})=(0)$ . Since $Q_{i}$ is torsion, the Euler–Poincaré formula for torsion modules implies $H^{1}(Q_{i}(\unicode[STIX]{x1D6FF}^{-1}))=(0)$ as well. We deduce from the long exact sequence in cohomology that the natural map $\operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),Q)\rightarrow \operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),Q/Q_{i})$ is an isomorphism.

On the other hand, $Q/Q_{i}$ is triangulated by a parameter $(\unicode[STIX]{x1D6FF}_{i+1},\ldots )$ whose higher terms are not homothetic to $\unicode[STIX]{x1D6FF}_{i+1}$ . From that we deduce that the inclusion $\operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1})){\hookrightarrow}\operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),Q/Q_{i})$ is an isomorphism also. Putting the two calculations together, we see that

$$\begin{eqnarray}\displaystyle \dim _{L}\operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),Q)=\dim _{L}\operatorname{Hom}({\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}),{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1}))=1. & & \displaystyle \nonumber\end{eqnarray}$$

This shows that the morphism in the lemma exists and is unique up to a scalar.

But, the calculation shows more. We have shown in fact that any non-zero morphism $e:{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})\rightarrow Q$ factors through $Q_{i+1}$ and that $e$ remains non-zero when mapped into the quotient ${\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1})$ of $Q_{i+1}$ . Since both ${\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})$ and ${\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1})$ are rank one, $e$ must be injective and it induces an exact sequence

(3) $$\begin{eqnarray}0\rightarrow Q_{i}\rightarrow Q_{i+1}/{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})\rightarrow {\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1})/{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})\rightarrow 0.\end{eqnarray}$$

Since ${\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1})$ is a direct summand of $Q_{i+1}$ as a ${\mathcal{R}}_{L}$ -module, the sequence (3) is also split as a sequence of ${\mathcal{R}}_{L}$ -modules. This means that the standard triangulation $Q_{\bullet }$ on $Q$ induces a standard triangulation $Q_{\bullet }^{\prime }$ on $Q/{\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF})$ whose successive quotients are given by

$$\begin{eqnarray}\displaystyle Q_{j}^{\prime }/Q_{j-1}^{\prime }=\left\{\begin{array}{@{}ll@{}}Q_{j}/Q_{j-1}\quad & \text{if }j\neq i+1,\\ {\mathcal{R}}_{L}(\unicode[STIX]{x1D6FF}_{i+1})/\mathop{\prod }_{\unicode[STIX]{x1D70F}}t_{\unicode[STIX]{x1D70F}}^{r_{i+1,\unicode[STIX]{x1D70F}}}\quad & \text{if }j=i+1.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

The invariants are of the new standard triangulation are easily calculated from this. ◻

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