# Components of moduli stacks of two-dimensional Galois representations

Ana Caraiani<EMAIL_ADDRESS>Department of Mathematics, Imperial
College London, London SW7 2AZ, UK  ,  Matthew Emerton
<EMAIL_ADDRESS>Department of Mathematics, University of Chicago,
5734 S. University Ave., Chicago, IL 60637, USA  ,  Toby Gee
<EMAIL_ADDRESS>Department of Mathematics, Imperial College London,
London SW7 2AZ, UK  and  David Savitt<EMAIL_ADDRESS>Department of
Mathematics, Johns Hopkins University

###### Abstract.

In the article [CEGS20b] we introduced various moduli stacks of two-
dimensional tamely potentially Barsotti–Tate representations of the absolute
Galois group of a $p$-adic local field, as well as related moduli stacks of
Breuil–Kisin modules with descent data. We study the irreducible components of
these stacks, establishing in particular that the components of the former are
naturally indexed by certain Serre weights.

The first author was supported in part by NSF grant DMS-1501064, by a Royal
Society University Research Fellowship, and by ERC Starting Grant 804176. The
second author was supported in part by the NSF grants DMS-1303450,
DMS-1601871, DMS-1902307, DMS-1952705, and DMS-2201242. The third author was
supported in part by a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie
Career Integration Grant 303605, ERC Starting Grant 306326, and a Royal
Society Wolfson Research Merit Award. The fourth author was supported in part
by NSF CAREER grant DMS-1564367 and NSF grant DMS-1702161.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Families of extensions of Breuil–Kisin modules
  4. 4 Extensions of rank one Breuil–Kisin modules
  5. 5 Components of Breuil–Kisin and Galois moduli stacks
  6. A Serre weights and tame types

##  1\. Introduction

Fix a prime number $p$, and let $K/{\mathbf{Q}}_{p}$ be a finite extension
with residue field $k$ and absolute Galois group
$G_{K}:=\operatorname{Gal}(\overline{K}/K)$. In the paper [CEGS20b], inspired
by a construction of Kisin [Kis09] in the setting of formal deformations, we
constructed and began to study the geometry of certain moduli stacks
${\mathcal{Z}}^{\mathrm{dd}}$. The stacks ${\mathcal{Z}}^{\mathrm{dd}}$ can be
thought of as moduli of two-dimensional tamely potentially Barsotti–Tate
representations of $G_{K}$; they are in fact moduli stacks of étale
$\varphi$-modules with descent data, and by construction are equipped with a
partial resolution

${\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}\to{\mathcal{Z}}^{\mathrm{dd}}$

where ${\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}$ is a moduli stack of
rank two Breuil–Kisin modules with tame descent data and height one.

The purpose of this paper is to make an explicit study of the morphism
${\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}\to{\mathcal{Z}}^{\mathrm{dd}}$
at the level of irreducible components. To be precise, for each two-
dimensional tame inertial type $\tau$ there are closed substacks
${\mathcal{C}}^{\tau,\operatorname{BT}}\subset{\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}$
and ${\mathcal{Z}}^{\tau}\subset{\mathcal{Z}}^{\mathrm{dd}}$ corresponding to
representations having inertial type $\tau$, and a morphism
${\mathcal{C}}^{\tau,\operatorname{BT}}\to{\mathcal{Z}}^{\tau}$. These are
$p$-adic formal algebraic stacks; let
${\mathcal{C}}^{\tau,\operatorname{BT},1}$ be the special fibre of
${\mathcal{C}}^{\tau,\operatorname{BT}}$, and ${\mathcal{Z}}^{\tau,1}$ its
scheme-theoretic image in ${\mathcal{Z}}^{\tau}$ (in the sense of [EG21]).
These were proved in [CEGS20b] to be equidimensional of dimension
$[K:{\mathbf{Q}}_{p}]$. Moreover the finite type points
$\operatorname{Spec}({\mathbf{F}})\to{\mathcal{Z}}^{\tau,1}$ are in bijection
with Galois representations $G_{K}\to\operatorname{GL}_{2}({\mathbf{F}})$
admitting a potentially Barsotti–Tate lift of type $\tau$.

(In fact ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ is shown in [CEGS20b] to
be reduced, from which it follows that ${\mathcal{Z}}^{\tau,1}$ is also
reduced. The special fibre of ${\mathcal{Z}}^{\tau}$ need not be reduced, so
it need not equal ${\mathcal{Z}}^{\tau,1}$, but it will be proved in the
sequel [CEGS20a] that it is _generically reduced_ , using the results of this
paper as input.)

Much of the work in our study of the irreducible components of
${\mathcal{Z}}^{\tau,1}$ involves an explicit construction of families of
extensions of characters. Intuitively, a natural source of “families” of
representations
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$ is
given by the extensions of two fixed characters. Indeed, given two characters
$\chi_{1},\chi_{2}:G_{K}\to\overline{{\mathbf{F}}}^{\times}_{p}$, the
$\overline{{\mathbf{F}}}_{p}$-vector space
$\operatorname{Ext}^{1}_{G_{K}}(\chi_{2},\chi_{1})$ is usually
$[K:{\mathbf{Q}}_{p}]$-dimensional, and a back of the envelope calculation
suggests that as a stack the collection of these representations should have
dimension $[K:{\mathbf{Q}}_{p}]-2$: the difference between an extension and a
representation counts for a $-1$, as does the ${\mathbf{G}}_{m}$ of
endomorphisms. Twisting $\chi_{1},\chi_{2}$ independently by unramified
characters gives a candidate for a $[K:{\mathbf{Q}}_{p}]$-dimensional family;
if contained in ${\mathcal{Z}}^{\tau}$, then since ${\mathcal{Z}}^{\tau}$ is
equidimensional of dimension $[K:{\mathbf{Q}}_{p}]$, the closure of such a
family should be an irreducible component of ${\mathcal{Z}}^{\tau}$.

Since there are only finitely many possibilities for the restrictions of the
$\chi_{i}$ to the inertia subgroup $I_{K}$, this gives a finite list of
maximal-dimensional families. On the other hand, there are up to unramified
twist only finitely many irreducible two-dimensional representations of
$G_{K}$, which suggests that the irreducible representations should correspond
to $0$-dimensional substacks. Together these considerations suggest that the
irreducible components of our moduli stack should be given by the closures of
the families of extensions considered in the previous paragraph, and in
particular that the irreducible representations should arise as limits of
reducible representations. This could not literally be the case for families
of Galois representations, rather than families of étale $\varphi$-modules,
and may seem surprising at first glance, but it is indeed what happens.

In the body of the paper we make this analysis rigorous, and we show that the
different families that we have constructed exhaust the irreducible
components. We can therefore label the irreducible components of
${\mathcal{Z}}^{\tau,1}$ as follows. A component is specified by an ordered
pair of characters $I_{K}\to\overline{{\mathbf{F}}}_{p}^{\times}$, which via
local class field theory corresponds to a pair of characters
$k^{\times}\to\overline{{\mathbf{F}}}^{\times}_{p}$. Such a pair can be
thought of as the highest weight of a _Serre weight_ : an irreducible
$\overline{{\mathbf{F}}}_{p}$-representation of $\operatorname{GL}_{2}(k)$. To
each irreducible component we have thus associated a Serre weight. (In fact,
we need to make a shift in this dictionary, corresponding to half the sum of
the positive roots of $\operatorname{GL}_{2}(k)$, but we ignore this for the
purposes of this introduction.)

This might seem artificial, but in fact it is completely natural, for the
following reason. Following the pioneering work of Serre [Ser87] and
Buzzard–Diamond–Jarvis [BDJ10] (as extended in [Sch08] and [Gee11]), we now
know how to associate a set $W(\overline{r})$ of Serre weights to each
continuous representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$,
with the property that if $F$ is a totally real field and
$\overline{\rho}:G_{F}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$
is an irreducible representation coming from a Hilbert modular form, then the
possible weights of Hilbert modular forms giving rise to $\overline{\rho}$ are
precisely determined by the sets $W(\overline{\rho}|_{G_{F_{v}}})$ for places
$v|p$ of $F$ (see for example [BLGG13, GK14, GLS15]).

Going back to our labelling of irreducible components above, we have
associated a Serre weight $\overline{\sigma}$ to each irreducible component of
${\mathcal{Z}}^{\tau,1}$. The inertial local Langlands correspondence assigns
a finite set of Serre weights $\operatorname{JH}(\overline{\sigma}(\tau))$ to
$\tau$, the Jordan–Hölder factors of the reduction mod $p$ of the
representation $\sigma(\tau)$ of $\operatorname{GL}_{2}({\mathcal{O}}_{K})$
corresponding to $\tau$. One of our main theorems is that the components of
${\mathcal{Z}}^{\tau,1}$ are labeled precisely by the Serre weights
$\overline{\sigma}\in\operatorname{JH}(\overline{\sigma}(\tau))$. Furthermore
the component labeled by $\overline{\sigma}$ has a dense set of finite type
points $\overline{r}$ with $\overline{\sigma}\in W(\overline{r})$. In the
sequel [CEGS20a] this will be strengthened to the statement that the
representations $\overline{r}$ on the irreducible component labelled by
$\overline{\sigma}$ are precisely the representations with
$\overline{\sigma}\in W(\overline{r})$,

We also study the irreducible components of the stack
${\mathcal{C}}^{\tau,\operatorname{BT},1}$. If $\tau$ is a non-scalar
principal series type then the set
$\operatorname{JH}(\overline{\sigma}(\tau))$ can be identified with a subset
of the power set ${\mathcal{S}}$ of the set of embeddings
$k\hookrightarrow\overline{{\mathbf{F}}}_{p}$ (hence, after fixing one such
embedding, with a subset ${\mathcal{P}}_{\tau}$ of
${\mathbf{Z}}/f{\mathbf{Z}}$). For generic choices of $\tau$, this subset is
the whole of ${\mathcal{S}}$. We are able to show, using the theory of
Dieudonné modules, that for any non-scalar principal series type $\tau$ the
irreducible components of ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ can be
identified with ${\mathcal{S}}$, and those irreducible components not
corresponding to elements of $\operatorname{JH}(\overline{\sigma}(\tau))$ have
image in ${\mathcal{Z}}^{\tau}$ of positive codimension. There is an analogous
statement for cuspidal types, while for scalar types, both
${\mathcal{C}}^{\tau,\operatorname{BT},1}$ and ${\mathcal{Z}}^{\tau,1}$ are
irreducible.

To state our main results precisely we must first introduce a bit more
notation. Fix a tame inertial type $\tau$ and a uniformiser $\pi$ of $K$. Let
$L$ be the unramified quadratic extension of $K$, and write $f$ for the
inertial degree of $K/{\mathbf{Q}}_{p}$. We set
$K^{\prime}=K(\pi^{1/p^{f}-1})$ if $\tau$ is principal series, and set
$K^{\prime}=L(\pi^{1/(p^{2f}-1)})$ if $\tau$ is cuspidal. Our moduli stacks of
$p$-adic Hodge theoretic objects with descent data will have descent data from
$K^{\prime}$ to $K$. Let $f^{\prime}$ be the inertial degree of
$K^{\prime}/{\mathbf{Q}}_{p}$, so that $f^{\prime}=f$ if the type $\tau$ is
principal series, while $f^{\prime}=2f$ if the type $\tau$ is cuspidal.

We say that a subset $J\subset{\mathbf{Z}}/f^{\prime}{\mathbf{Z}}$ is a
_shape_ if:

  * •

$\tau$ is scalar and $J=\varnothing$,

  * •

$\tau$ is a non-scalar principal series type and $J$ is arbitrary, or

  * •

$\tau$ is cuspidal and $J$ has the property that $i\in J$ if and only if
$i+f\not\in J$.

If $\tau$ is non-scalar then there are exactly $2^{f}$ shapes.

As above, write $\sigma(\tau)$ for the representation of
$\operatorname{GL}_{2}({\mathcal{O}}_{K})$ corresponding to $\tau$ under the
inertial local Langlands correspondence of Henniart. The Jordan–Hölder factors
of the reduction mod $p$ of $\sigma(\tau)$ are parameterized by an explicit
set of shapes ${\mathcal{P}}_{\tau}$, and we write
$\overline{\sigma}(\tau)_{J}$ for the factor corresponding to $J$.

To each shape $J$, we will associate a closed substack
$\overline{\mathcal{C}}(J)$ of ${\mathcal{C}}^{\tau,\operatorname{BT},1}$. The
stack $\overline{{\mathcal{Z}}}(J)$ is then defined to be the scheme-theoretic
image of $\overline{{\mathcal{C}}}(J)$ under the map
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{Z}}^{\tau,1}$, in the
sense of [EG21]. Then the following is our main result.

######  Theorem 1.1.

The irreducible components of ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ and
${\mathcal{Z}}^{\tau,1}$ are as follows.

  1. (1)

The irreducible components of ${\mathcal{C}}^{\tau,1}$ are precisely the
$\overline{{\mathcal{C}}}(J)$ for shapes $J$, and if $J\neq J^{\prime}$ then
$\overline{{\mathcal{C}}}(J)\neq\overline{{\mathcal{C}}}(J^{\prime})$.

  2. (2)

The irreducible components of ${\mathcal{Z}}^{\tau,1}$ are precisely the
$\overline{{\mathcal{Z}}}(J)$ for shapes $J\in{\mathcal{P}}_{\tau}$, and if
$J\neq J^{\prime}$ then
$\overline{{\mathcal{Z}}}(J)\neq\overline{{\mathcal{Z}}}(J^{\prime})$.

  3. (3)

For each $J\in{\mathcal{P}}_{\tau}$, there is a dense open substack
${\mathcal{U}}$ of $\overline{{\mathcal{C}}}(J)$ such that the map
$\overline{{\mathcal{C}}}(J)\to\overline{{\mathcal{Z}}}(J)$ restricts to an
open immersion on ${\mathcal{U}}$.

  4. (4)

For each $J\in{\mathcal{P}}_{\tau}$, there is a dense set of finite type
points of $\overline{{\mathcal{Z}}}(J)$ with the property that the
corresponding Galois representations have $\overline{\sigma}(\tau)_{J}$ as a
Serre weight, and which furthermore admit a unique Breuil–Kisin model of type
$\tau$.

######  Remark 1.2.

We emphasize in Theorem 1.1 that the components of ${\mathcal{Z}}^{\tau,1}$
are indexed by shapes $J\in{\mathcal{P}}_{\tau}$, _not_ by all shapes. If
$J\not\in{\mathcal{P}}_{\tau}$, then the stack $\overline{{\mathcal{Z}}}(J)$
has dimension strictly smaller than $[K:{\mathbf{Q}}_{p}]$, and so is properly
contained in some component of ${\mathcal{Z}}^{\tau,1}$. We anticipate that
the loci $\overline{{\mathcal{Z}}}(J)$ will nevertheless be of interest when
$J\not\in{\mathcal{P}}_{\tau}$: we expect that they will correspond to
“phantom” (partial weight one) Serre weights of relevance to the geometric
variant of the weight part of Serre’s conjecture proposed by Diamond–Sasaki
[DS17]. This will be the subject of future work.

We assume that $p>2$ in much of the paper; while we expect that our results
should also hold if $p=2$, there are several reasons to exclude this case. We
are frequently able to considerably simplify our arguments by assuming that
the extension $K^{\prime}/K$ is not just tamely ramified, but in fact of
degree prime to $p$; this is problematic when $p=2$, as the consideration of
cuspidal types involves a quadratic unramified extension. Furthermore, in the
sequel [CEGS20a] we will use results on the Breuil–Mézard conjecture which
ultimately depend on automorphy lifting theorems that are not available in the
case $p=2$ at present (although it is plausible that the methods of [Tho17]
could be used to prove them).

We conclude this introduction by discussing the relationship between our
results and those of [EG22]. Two of us (M.E. and T.G.) have constructed moduli
stacks ${\mathcal{X}}_{d}$ of rank $d$ étale $(\varphi,\Gamma)$-modules for
$K$, as well as substacks ${\mathcal{X}}_{d}^{\lambda,\tau}$ which may be
regarded as stacks of potentially crystalline representations of $G_{K}$ with
inertial type $\tau$ and Hodge type $\lambda$. When $d=2$ and $\lambda$ is the
trivial Hodge type, these are stacks
${\mathcal{X}}_{2}^{\tau,\operatorname{BT}}$ of potentially Barsotti–Tate
representations of $G_{K}$ of inertial type $\tau$, and we anticipate that
${\mathcal{X}}_{2}^{\tau,\operatorname{BT}}$ is isomorphic to
${\mathcal{Z}}^{\tau,\operatorname{BT}}$ (but since we do not need this fact,
we have not proved it).

One of the main results of the book [EG22] is that the irreducible components
of the underlying reduced stacks ${\mathcal{X}}_{d,\operatorname{red}}$ are in
bijection with the irreducible representations of $\operatorname{GL}_{d}(k)$.
This bijection is characterised in essentially exactly the same way as our
description of the components of ${\mathcal{Z}}^{\tau,1}$ in this paper: a
Serre weight has a highest weight, which corresponds to a tuple of inertial
characters, which gives rise to a family of successive extensions of
$1$-dimensionals representations. Then the closure of this family is a
component of ${\mathcal{X}}_{d,\operatorname{red}}$.

The crucial difference between our setting and that of [EG22] is that we could
prove in [CEGS20b] that the stacks ${\mathcal{Z}}^{\tau,1}$ are reduced. The
proof makes use of the resolution
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{Z}}^{\tau,1}$ and the
fact that we are able to relate the stack
${\mathcal{C}}^{\tau,\operatorname{BT}}$ to a local model at Iwahori level,
whose special fibre is known to be reduced. In the sequel [CEGS20a] we combine
the characterisation of the components of ${\mathcal{Z}}^{\tau,1}$ from this
paper with the reducedness of ${\mathcal{Z}}^{\tau,1}$ from [CEGS20b] to prove
that the special fibre of ${\mathcal{Z}}^{\tau}$ is _generically_ reduced.
This will then allow us to completely characterise _all_ of the finite type
points on each component of ${\mathcal{Z}}^{\tau,1}$ (not just a dense set of
points), and to prove geometrisations of the Breuil–Mézard conjecture and of
the weight part of Serre’s conjecture for the stacks
${\mathcal{Z}}^{\mathrm{dd},1}$. Furthermore, by means of a comparison of
versal rings, these results can be transported to the stacks
${\mathcal{X}}^{\tau,\operatorname{BT}}_{2}$ of [EG22] as well.

###  1.3. Acknowledgements

We would like to thank Kalyani Kansal for helpful comments.

###  1.4. Notation and conventions

#### Topological groups

If $M$ is an abelian topological group with a linear topology, then as in
[Sta13, Tag 07E7] we say that $M$ is complete if the natural morphism
$M\to\varinjlim_{i}M/U_{i}$ is an isomorphism, where $\\{U_{i}\\}_{i\in I}$ is
some (equivalently any) fundamental system of neighbourhoods of $0$ consisting
of subgroups. Note that in some other references this would be referred to as
being complete and separated. In particular, any $p$-adically complete ring
$A$ is by definition $p$-adically separated.

#### Galois theory and local class field theory

If $M$ is a field, we let $G_{M}$ denote its absolute Galois group. If $M$ is
a global field and $v$ is a place of $M$, let $M_{v}$ denote the completion of
$M$ at $v$. If $M$ is a local field, we write $I_{M}$ for the inertia subgroup
of $G_{M}$.

Let $p$ be a prime number. Fix a finite extension $K/{\mathbf{Q}}_{p}$, with
ring of integers ${\mathcal{O}}_{K}$ and residue field $k$. Let $e$ and $f$ be
the ramification and inertial degrees of $K$, respectively, and write
$\\#k=p^{f}$ for the cardinality of $k$. Let $K^{\prime}/K$ be a finite tamely
ramified Galois extension. Let $k^{\prime}$ be the residue field of
$K^{\prime}$, and let $e^{\prime},f^{\prime}$ be the ramification and inertial
degrees of $K^{\prime}$ respectively.

Our representations of $G_{K}$ will have coefficients in
$\overline{{\mathbf{Q}}}_{p}$, a fixed algebraic closure of ${\mathbf{Q}}_{p}$
whose residue field we denote by $\overline{{\mathbf{F}}}_{p}$. Let $E$ be a
finite extension of ${\mathbf{Q}}_{p}$ contained in
$\overline{{\mathbf{Q}}}_{p}$ and containing the image of every embedding of
$K^{\prime}$ into $\overline{{\mathbf{Q}}}_{p}$. Let ${\mathcal{O}}$ be the
ring of integers in $E$, with uniformiser $\varpi$ and residue field
${\mathbf{F}}\subset\overline{{\mathbf{F}}}_{p}$.

Fix an embedding $\sigma_{0}:k^{\prime}\hookrightarrow{\mathbf{F}}$, and
recursively define $\sigma_{i}:k^{\prime}\hookrightarrow{\mathbf{F}}$ for all
$i\in{\mathbf{Z}}$ so that $\sigma_{i+1}^{p}=\sigma_{i}$; of course, we have
$\sigma_{i+f^{\prime}}=\sigma_{i}$ for all $i$. We let $e_{i}\in
k^{\prime}\otimes_{{\mathbf{F}}_{p}}{\mathbf{F}}$ denote the idempotent
satisfying $(x\otimes 1)e_{i}=(1\otimes\sigma_{i}(x))e_{i}$ for all $x\in
k^{\prime}$; note that $\varphi(e_{i})=e_{i+1}$. We also denote by $e_{i}$ the
natural lift of $e_{i}$ to an idempotent in
$W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}{\mathcal{O}}$. If $M$ is an
$W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}{\mathcal{O}}$-module, then we write
$M_{i}$ for $e_{i}M$.

We write ${\operatorname{Art}}_{K}\colon K^{\times}\to W_{K}^{\mathrm{ab}}$
for the isomorphism of local class field theory, normalised so that
uniformisers correspond to geometric Frobenius elements.

######  Lemma 1.4.1.

Let $\pi$ be any uniformiser of ${\mathcal{O}}_{K}$. The composite
$I_{K}\to{\mathcal{O}}_{K}^{\times}\to k^{\times}$, where the map
$I_{K}\to{\mathcal{O}}_{K}^{\times}$ is induced by the restriction of
${\operatorname{Art}}_{K}^{-1}$, sends an element $g\in I_{K}$ to the image in
$k^{\times}$ of $g(\pi^{1/(p^{f}-1)})/\pi^{1/(p^{f}-1)}$.

###### Proof.

This follows (for example) from the construction in [Yos08, Prop. 4.4(iii),
Prop. 4.7(ii), Cor. 4.9, Def. 4.10]. ∎

For each $\sigma\in\operatorname{Hom}(k,\overline{{\mathbf{F}}}_{p})$ we
define the fundamental character $\omega_{\sigma}$ to $\sigma$ to be the
composite

$\textstyle{I_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{K}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\overline{{\mathbf{F}}}_{p}^{\times},}$

where the map $I_{K}\to{\mathcal{O}}_{K}^{\times}$ is induced by the
restriction of ${\operatorname{Art}}_{K}^{-1}$. Let $\varepsilon$ denote the
$p$-adic cyclotomic character and $\overline{\varepsilon}$ the mod $p$
cyclotomic character, so that
$\prod_{\sigma\in\operatorname{Hom}(k,\overline{{\mathbf{F}}}_{p})}\omega_{\sigma}^{e}=\overline{\varepsilon}$.
We will often identify characters of $I_{K}$ with characters of $k^{\times}$
via the Artin map.

#### Inertial local Langlands

A two-dimensional _tame inertial type_ is (the isomorphism class of) a tamely
ramified representation
$\tau:I_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Z}}}_{p})$ that extends
to a representation of $G_{K}$ and whose kernel is open. Such a representation
is of the form $\tau\simeq\eta\oplus\eta^{\prime}$, and we say that $\tau$ is
a _tame principal series type_ if $\eta,\eta^{\prime}$ both extend to
characters of $G_{K}$. Otherwise, $\eta^{\prime}=\eta^{q}$, and $\eta$ extends
to a character of $G_{L}$, where $L/K$ is a quadratic unramified extension. In
this case we say that $\tau$ is a _tame cuspidal type_.

Henniart’s appendix to [BM02] associates a finite dimensional irreducible
$E$-representation $\sigma(\tau)$ of
$\operatorname{GL}_{2}({\mathcal{O}}_{K})$ to each inertial type $\tau$; we
refer to this association as the inertial local Langlands correspondence.
Since we are only working with tame inertial types, this correspondence can be
made very explicit as follows.

If $\tau\simeq\eta\oplus\eta^{\prime}$ is a tame principal series type, then
we also write $\eta,\eta^{\prime}:k^{\times}\to{\mathcal{O}}^{\times}$ for the
multiplicative characters determined by
$\eta\circ{\operatorname{Art}}_{K}|_{{\mathcal{O}}_{K}^{\times}},\eta^{\prime}\circ{\operatorname{Art}}_{K}|_{{\mathcal{O}}_{K}^{\times}}$
respectively. If $\eta=\eta^{\prime}$, then we set
$\sigma(\tau)=\eta\circ\det$. Otherwise, we write $I$ for the Iwahori subgroup
of $\operatorname{GL}_{2}({\mathcal{O}}_{K})$ consisting of matrices which are
upper triangular modulo a uniformiser $\varpi_{K}$ of $K$, and write
$\chi=\eta^{\prime}\otimes\eta:I\to{\mathcal{O}}^{\times}$ for the character

$\begin{pmatrix}a&b\\\
\varpi_{K}c&d\end{pmatrix}\mapsto\eta^{\prime}(\overline{a})\eta(\overline{d}).$

Then
$\sigma(\tau):=\operatorname{Ind}_{I}^{\operatorname{GL}_{2}({\mathcal{O}}_{K})}\chi$.

If $\tau=\eta\oplus\eta^{q}$ is a tame cuspidal type, then as above we write
$L/K$ for a quadratic unramified extension, and $l$ for the residue field of
${\mathcal{O}}_{L}$. We write $\eta:l^{\times}\to{\mathcal{O}}^{\times}$ for
the multiplicative character determined by
$\eta\circ{\operatorname{Art}}_{L}|_{{\mathcal{O}}_{L}^{\times}}$; then
$\sigma(\tau)$ is the inflation to $\operatorname{GL}_{2}({\mathcal{O}}_{K})$
of the cuspidal representation of $\operatorname{GL}_{2}(k)$ denoted by
$\Theta(\eta)$ in [Dia07].

####  $p$-adic Hodge theory

We normalise Hodge–Tate weights so that all Hodge–Tate weights of the
cyclotomic character are equal to $-1$. We say that a potentially crystalline
representation
$\rho:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Q}}}_{p})$ has _Hodge
type_ $0$, or is _potentially Barsotti–Tate_ , if for each
$\varsigma:K\hookrightarrow\overline{{\mathbf{Q}}}_{p}$, the Hodge–Tate
weights of $\rho$ with respect to $\varsigma$ are $0$ and $1$. (Note that this
is a more restrictive definition of potentially Barsotti–Tate than is
sometimes used; however, we will have no reason to deal with representations
with non-regular Hodge-Tate weights, and so we exclude them from
consideration. Note also that it is more usual in the literature to say that
$\rho$ is potentially Barsotti–Tate if it is potentially crystalline, and
$\rho^{\vee}$ has Hodge type $0$.)

We say that a potentially crystalline representation
$\rho:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Q}}}_{p})$ has
_inertial type_ $\tau$ if the traces of elements of $I_{K}$ acting on $\tau$
and on

$\operatorname{D_{pcris}}(\rho)=\varinjlim_{K^{\prime}/K}(\mathrm{B}_{\mathrm{cris}}\otimes_{{\mathbf{Q}}_{p}}V_{\rho})^{G_{K^{\prime}}}$

are equal (here $V_{\rho}$ is the underlying vector space of $V_{\rho}$). A
representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$ _has
a potentially Barsotti–Tate lift of type $\tau$_ if and only if $\overline{r}$
admits a lift to a representation
$r:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Z}}}_{p})$ of Hodge type
$0$ and inertial type $\tau$.

#### Serre weights

By definition, a _Serre weight_ is an irreducible
${\mathbf{F}}$-representation of $\operatorname{GL}_{2}(k)$. Concretely, such
a representation is of the form

$\overline{\sigma}_{\vec{t},\vec{s}}:=\otimes^{f-1}_{j=0}(\det{\\!}^{t_{j}}\operatorname{Sym}^{s_{j}}k^{2})\otimes_{k,\sigma_{j}}{\mathbf{F}},$

where $0\leq s_{j},t_{j}\leq p-1$ and not all $t_{j}$ are equal to $p-1$. We
say that a Serre weight is _Steinberg_ if $s_{j}=p-1$ for all $j$, and _non-
Steinberg_ otherwise.

#### A remark on normalisations

Given a continuous representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$,
there is an associated (nonempty) set of Serre weights $W(\overline{r})$ whose
precise definition we will recall in Appendix A. There are in fact several
different definitions of $W(\overline{r})$ in the literature; as a result of
the papers [BLGG13, GK14, GLS15], these definitions are known to be equivalent
up to normalisation.

However, the normalisations of Hodge–Tate weights and of inertial local
Langlands used in [GK14, GLS15, EGS15] are not all the same, and so for
clarity we lay out how they differ, and how they compare to the normalisations
of this paper.

Our conventions for Hodge–Tate weights and inertial types agree with those of
[GK14, EGS15], but our representation $\sigma(\tau)$ is the representation
$\sigma(\tau^{\vee})$ of [GK14, EGS15] (where
$\tau^{\vee}=\eta^{-1}\oplus(\eta^{\prime})^{-1}$); to see this, note the dual
in the definition of $\sigma(\tau)$ in [GK14, Thm. 2.1.3] and the discussion
in §1.9 of [EGS15].111However, this dual is erroneously omitted when the
inertial local Langlands correspondence is made explicit at the end of [EGS15,
§3.1]. See Remark A.1.

In all cases one chooses to normalise the set of Serre weights so that the
condition of Lemma A.5(1) holds. Consequently, our set of weights
$W(\overline{r})$ is the set of duals of the weights $W(\overline{r})$
considered in [GK14]. In turn, the paper [GLS15] has the opposite convention
for the signs of Hodge–Tate weights to our convention (and to the convention
of [GK14]), so we find that our set of weights $W(\overline{r})$ is the set of
duals of the weights $W(\overline{r}^{\vee})$ considered in [GLS15].

#### Stacks

We follow the terminology of [Sta13]; in particular, we write “algebraic
stack” rather than “Artin stack”. More precisely, an algebraic stack is a
stack in groupoids in the _fppf_ topology, whose diagonal is representable by
algebraic spaces, which admits a smooth surjection from a scheme. See [Sta13,
Tag 026N] for a discussion of how this definition relates to others in the
literature, and [Sta13, Tag 04XB] for key properties of morphisms
representable by algebraic spaces.

For a commutative ring $A$, an _fppf stack over $A$_ (or _fppf_ $A$-stack) is
a stack fibred in groupoids over the big _fppf_ site of
$\operatorname{Spec}A$.

##  2\. Preliminaries

We begin by reviewing the various constructions and results that we will need
from [CEGS20b]. Section 2.1 recalls the definition and a few basic algebraic
properties of Breuil–Kisin modules with coefficients and descent data, while
Section 2.2 does the same for étale $\varphi$-modules. In Section 2.3 we
define the stacks ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ and
${\mathcal{Z}}^{\tau,1}$ (as well as various other related stacks) and state
the main results of [CEGS20b]. Finally, in Section 2.4 we introduce and study
stacks of Dieudonné modules that will be used at the end of the paper to
determine the irreducible components of
${\mathcal{C}}^{\tau,\operatorname{BT},1}$.

###  2.1. Breuil–Kisin modules with descent data

Recall that we have a finite tamely ramified Galois extension $K^{\prime}/K$.
Suppose further that there exists a uniformiser $\pi^{\prime}$ of
${\mathcal{O}}_{K^{\prime}}$ such that $\pi:=(\pi^{\prime})^{e(K^{\prime}/K)}$
is an element of $K$, where $e(K^{\prime}/K)$ is the ramification index of
$K^{\prime}/K$. Recall that $k^{\prime}$ is the residue field of $K^{\prime}$,
while $e^{\prime},f^{\prime}$ are the ramification and inertial degrees of
$K^{\prime}$ respectively. Let $E(u)$ be the minimal polynomial of
$\pi^{\prime}$ over $W(k^{\prime})[1/p]$.

Let $\varphi$ denote the arithmetic Frobenius automorphism of $k^{\prime}$,
which lifts uniquely to an automorphism of $W(k^{\prime})$ that we also denote
by $\varphi$. Define $\mathfrak{S}:=W(k^{\prime})[[u]]$, and extend $\varphi$
to $\mathfrak{S}$ by

$\varphi\left(\sum a_{i}u^{i}\right)=\sum\varphi(a_{i})u^{pi}.$

By our assumptions that $(\pi^{\prime})^{e(K^{\prime}/K)}\in K$ and that
$K^{\prime}/K$ is Galois, for each $g\in\operatorname{Gal}(K^{\prime}/K)$ we
can write $g(\pi^{\prime})/\pi^{\prime}=h(g)$ with
$h(g)\in\mu_{e(K^{\prime}/K)}(K^{\prime})\subset W(k^{\prime})$, and we let
$\operatorname{Gal}(K^{\prime}/K)$ act on $\mathfrak{S}$ via

$g\left(\sum a_{i}u^{i}\right)=\sum g(a_{i})h(g)^{i}u^{i}.$

Let $A$ be a $p$-adically complete ${\mathbf{Z}}_{p}$-algebra, set
$\mathfrak{S}_{A}:=(W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}A)[[u]]$, and
extend the actions of $\varphi$ and $\operatorname{Gal}(K^{\prime}/K)$ on
$\mathfrak{S}$ to actions on $\mathfrak{S}_{A}$ in the obvious ($A$-linear)
fashion.

######  Lemma 2.1.1.

An $\mathfrak{S}_{A}$-module is projective if and only if it is projective as
an $A[[u]]$-module.

###### Proof.

Suppose that $\mathfrak{M}$ is an $\mathfrak{S}_{A}$-module that is projective
as an $A[[u]]$-module. Certainly
$W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}\mathfrak{M}$ is projective over
$\mathfrak{S}_{A}$, and we claim that it has $\mathfrak{M}$ as an
$\mathfrak{S}_{A}$-module direct summand. Indeed, this follows by rewriting
$\mathfrak{M}$ as $W(k^{\prime})\otimes_{W(k^{\prime})}\mathfrak{M}$ and
noting that $W(k^{\prime})$ is a $W(k^{\prime})$-module direct summand of
$W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}W(k^{\prime})$. ∎

The actions of $\varphi$ and $\operatorname{Gal}(K^{\prime}/K)$ on
$\mathfrak{S}_{A}$ extend to actions on
$\mathfrak{S}_{A}[1/u]=(W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}A)((u))$ in the
obvious way. It will sometimes be necessary to consider the subring
$\mathfrak{S}_{A}^{0}:=(W(k)\otimes_{{\mathbf{Z}}_{p}}A)[[v]]$ of
$\mathfrak{S}_{A}$ consisting of power series in $v:=u^{e(K^{\prime}/K)}$, on
which $\operatorname{Gal}(K^{\prime}/K)$ acts trivially.

######  Definition 2.1.2.

Fix a $p$-adically complete ${\mathbf{Z}}_{p}$-algebra $A$. A _Breuil–Kisin
module with $A$-coefficients and descent data from $K^{\prime}$ to $K$_ (or
often simply a _Breuil–Kisin module_) is a triple
$(\mathfrak{M},\varphi_{\mathfrak{M}},\\{\hat{g}\\}_{g\in\operatorname{Gal}(K^{\prime}/K)})$
consisting of a $\mathfrak{S}_{A}$-module $\mathfrak{M}$ and a
$\varphi$-semilinear map $\varphi_{\mathfrak{M}}:\mathfrak{M}\to\mathfrak{M}$
such that:

  * •

the $\mathfrak{S}_{A}$-module $\mathfrak{M}$ is finitely generated and
projective, and

  * •

the induced map
$\Phi_{\mathfrak{M}}=1\otimes\varphi_{\mathfrak{M}}:\varphi^{*}\mathfrak{M}\to\mathfrak{M}$
is an isomorphism after inverting $E(u)$ (here as usual we write
$\varphi^{*}\mathfrak{M}:=\mathfrak{S}_{A}\otimes_{\varphi,\mathfrak{S}_{A}}\mathfrak{M}$),

together with additive bijections $\hat{g}:\mathfrak{M}\to\mathfrak{M}$,
satisfying the further properties that the maps $\hat{g}$ commute with
$\varphi_{\mathfrak{M}}$, satisfy
$\hat{g_{1}}\circ\hat{g_{2}}=\widehat{g_{1}\circ g_{2}}$, and have
$\hat{g}(sm)=g(s)\hat{g}(m)$ for all $s\in\mathfrak{S}_{A}$,
$m\in\mathfrak{M}$. We say that $\mathfrak{M}$ is has _height at most $h$_ if
the cokernel of $\Phi_{\mathfrak{M}}$ is killed by $E(u)^{h}$.

The Breuil–Kisin module $\mathfrak{M}$ is said to be of rank $d$ if the
underlying finitely generated projective $\mathfrak{S}_{A}$-module has
constant rank $d$. It is said to be free if the underlying
$\mathfrak{S}_{A}$-module is free.

A morphism of Breuil–Kisin modules with descent data is a morphism of
$\mathfrak{S}_{A}$-modules that commutes with $\varphi$ and with the
$\hat{g}$. In the case that $K^{\prime}=K$ the data of the $\hat{g}$ is
trivial, so it can be forgotten, giving the category of _Breuil–Kisin modules
with $A$-coefficients._ In this case it will sometimes be convenient to elide
the difference between a Breuil–Kisin module with trivial descent data, and a
Breuil–Kisin module without descent data, in order to avoid making separate
definitions in the case of Breuil–Kisin modules without descent data.

######  Remark 2.1.3.

We refer the reader to [EG21, §5.1] for a discussion of foundational results
concerning finitely generated modules over the power series ring $A[[u]]$. In
particular (using Lemma 2.1.1) we note the following.

  1. (1)

An $\mathfrak{S}_{A}$-module $\mathfrak{M}$ is finitely generated and
projective if and only if it is $u$-torsion free and $u$-adically complete,
and $\mathfrak{M}/u\mathfrak{M}$ is a finitely generated projective $A$-module
([EG21, Prop. 5.1.8]).

  2. (2)

If the $\mathfrak{S}_{A}$-module $\mathfrak{M}$ is projective of rank $d$,
then it is Zariski locally free of rank $d$ in the sense that there is a cover
of $\operatorname{Spec}A$ by affine opens $\operatorname{Spec}B_{i}$ such that
each of the base-changed modules
$\mathfrak{M}\otimes_{\mathfrak{S}_{A}}\mathfrak{S}_{B_{i}}$ is free of rank
$d$ ([EG21, Prop. 5.1.9]).

  3. (3)

If $A$ is coherent (so in particular, if $A$ is Noetherian), then $A[[u]]$ is
faithfully flat over $A$, and so $\mathfrak{S}_{A}$ is faithfully flat over
$A$, but this need not hold if $A$ is not coherent.

######  Definition 2.1.4.

If $Q$ is any (not necessarily finitely generated) $A$-module, and
$\mathfrak{M}$ is an $A[[u]]$-module, then we let
$\mathfrak{M}\,\widehat{\otimes}_{A}Q$ denote the $u$-adic completion of
$\mathfrak{M}\otimes_{A}Q$.

######  Lemma 2.1.5.

If $\mathfrak{M}$ is a Breuil–Kisin module and $B$ is an $A$-algebra, then the
base change $\mathfrak{M}\,\widehat{\otimes}_{A}B$ is a Breuil–Kisin module.

###### Proof.

This is [CEGS20b, Lem. 2.1.4]. ∎

We make the following two further remarks concerning base change.

######  Remark 2.1.6.

(1) If $A$ is Noetherian, if $Q$ is finitely generated over $A$, and if
$\mathfrak{N}$ is finitely generated over $A[[u]]$, then
$\mathfrak{N}\otimes_{A}Q$ is finitely generated over $A[[u]]$, and hence (by
the Artin–Rees lemma) is automatically $u$-adically complete. Thus in this
case the natural morphism
$\mathfrak{N}\otimes_{A}Q\to\mathfrak{N}\,\widehat{\otimes}_{A}Q$ is an
isomorphism.

(2) Note that $A[[u]]\,\widehat{\otimes}_{A}Q=Q[[u]]$ (the $A[[u]]$-module
consisting of power series with coefficients in the $A$-module $Q$), and so if
$\mathfrak{N}$ is Zariski locally free on $\operatorname{Spec}A$, then
$\mathfrak{N}\,\widehat{\otimes}_{A}Q$ is Zariski locally isomorphic to a
direct sum of copies of $Q[[u]]$, and hence is $u$-torsion free (as well as
being $u$-adically complete). In particular, by Remark 2.1.3(2), this holds if
$\mathfrak{N}$ is projective.

Let $A$ be a ${\mathbf{Z}}_{p}$-algebra. We define a _Dieudonné module of rank
$d$ with $A$-coefficients and descent data from $K^{\prime}$ to $K$_ to be a
finitely generated projective
$W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}A$-module $D$ of constant rank $d$ on
$\operatorname{Spec}A$, together with:

  * •

$A$-linear endomorphisms $F,V$ satisfying $FV=VF=p$ such that $F$ is
$\varphi$-semilinear and $V$ is $\varphi^{-1}$-semilinear for the action of
$W(k^{\prime})$, and

  * •

a $W(k^{\prime})\otimes_{{\mathbf{Z}}_{p}}A$-semilinear action of
$\operatorname{Gal}(K^{\prime}/K)$ which commutes with $F$ and $V$.

######  Definition 2.1.7.

If $\mathfrak{M}$ is a Breuil–Kisin module of height at most $1$ and rank $d$
with descent data, then there is a corresponding Dieudonné module
$D=D(\mathfrak{M})$ of rank $d$ defined as follows. We set
$D:=\mathfrak{M}/u\mathfrak{M}$ with the induced action of
$\operatorname{Gal}(K^{\prime}/K)$, and $F$ given by the induced action of
$\varphi$. The endomorphism $V$ is determined as follows. Write $E(0)=cp$, so
that we have $p\equiv c^{-1}E(u)\pmod{u}$. The condition that the cokernel of
$\varphi^{*}\mathfrak{M}\to\mathfrak{M}$ is killed by $E(u)$ allows us to
factor the multiplication-by-$E(u)$ map on $\mathfrak{M}$ uniquely as
$\mathfrak{V}\circ\varphi$, and $V$ is defined to be $c^{-1}\mathfrak{V}$
modulo $u$.

###  2.2. Étale $\varphi$-modules and Galois representations

######  Definition 2.2.1.

Let $A$ be a ${\mathbf{Z}}/p^{a}{\mathbf{Z}}$-algebra for some $a\geq 1$. A
_weak étale $\varphi$-module_ with $A$-coefficients and descent data from
$K^{\prime}$ to $K$ is a triple $(M,\varphi_{M},\\{\hat{g}\\})$ consisting of:

  * •

a finitely generated $\mathfrak{S}_{A}[1/u]$-module $M$;

  * •

a $\varphi$-semilinear map $\varphi_{M}:M\to M$ with the property that the
induced map

$\Phi_{M}=1\otimes\varphi_{M}:\varphi^{*}M:=\mathfrak{S}_{A}[1/u]\otimes_{\varphi,\mathfrak{S}_{A}[1/u]}M\to
M$

is an isomorphism,

together with additive bijections $\hat{g}:M\to M$ for
$g\in\operatorname{Gal}(K^{\prime}/K)$, satisfying the further properties that
the maps $\hat{g}$ commute with $\varphi_{M}$, satisfy
$\hat{g_{1}}\circ\hat{g_{2}}=\widehat{g_{1}\circ g_{2}}$, and have
$\hat{g}(sm)=g(s)\hat{g}(m)$ for all $s\in\mathfrak{S}_{A}[1/u]$, $m\in M$.

If $M$ as above is projective as an $\mathfrak{S}_{A}[1/u]$-module then we say
simply that $M$ is an étale $\varphi$-module. The étale $\varphi$-module $M$
is said to be of rank $d$ if the underlying finitely generated projective
$\mathfrak{S}_{A}[1/u]$-module has constant rank $d$.

######  Remark 2.2.2.

We could also consider étale $\varphi$-modules for general $p$-adically
complete ${\mathbf{Z}}_{p}$-algebras $A$, but we would need to replace
$\mathfrak{S}_{A}[1/u]$ by its $p$-adic completion. As we will not need to
consider these modules in this paper, we do not do so here, but we refer the
interested reader to [EG22].

A morphism of weak étale $\varphi$-modules with $A$-coefficients and descent
data from $K^{\prime}$ to $K$ is a morphism of $\mathfrak{S}_{A}[1/u]$-modules
that commutes with $\varphi$ and with the $\hat{g}$. Again, in the case
$K^{\prime}=K$ the descent data is trivial, and we obtain the usual category
of étale $\varphi$-modules with $A$-coefficients.

Note that if $A$ is a ${\mathbf{Z}}/p^{a}{\mathbf{Z}}$-algebra, and
$\mathfrak{M}$ is a Breuil–Kisin module with descent data, then
$\mathfrak{M}[1/u]$ naturally has the structure of an étale $\varphi$-module
with descent data.

Suppose that $A$ is an ${\mathcal{O}}$-algebra (where ${\mathcal{O}}$ is as in
Section 1.4). In making calculations, it is often convenient to use the
idempotents $e_{i}$ (again as in Section 1.4). In particular if $\mathfrak{M}$
is a Breuil–Kisin module, then writing as usual
$\mathfrak{M}_{i}:=e_{i}\mathfrak{M}$, we write
$\Phi_{\mathfrak{M},i}:\varphi^{*}(\mathfrak{M}_{i-1})\to\mathfrak{M}_{i}$ for
the morphism induced by $\Phi_{\mathfrak{M}}$. Similarly if $M$ is an étale
$\varphi$-module then we write $M_{i}:=e_{i}M$, and we write
$\Phi_{M,i}:\varphi^{*}(M_{i-1})\to M_{i}$ for the morphism induced by
$\Phi_{M}$.

To connect étale $\varphi$-modules to $G_{K_{\infty}}$-representations we
begin by recalling from [Kis09] some constructions arising in $p$-adic Hodge
theory and the theory of fields of norms, which go back to [Fon90]. Following
Fontaine, we write $R:=\varprojlim_{x\mapsto x^{p}}{\mathcal{O}}_{\bar{K}}/p$.
Fix a compatible system $(\\!\sqrt[p^{n}]{\pi}\,)_{n\geq 0}$ of $p^{n}$th
roots of $\pi$ in $\bar{K}$ (compatible in the obvious sense that
$\bigl{(}\\!\sqrt[p^{n+1}]{\pi}\,\bigr{)}^{p}=\sqrt[p^{n}]{\pi}\,$), and let
$K_{\infty}:=\cup_{n}K(\sqrt[p^{n}]{\pi})$, and also
$K^{\prime}_{\infty}:=\cup_{n}K^{\prime}(\sqrt[p^{n}]{\pi})$. Since
$(e(K^{\prime}/K),p)=1$, the compatible system
$(\\!\sqrt[p^{n}]{\pi}\,)_{n\geq 0}$ determines a unique compatible system
$(\\!\sqrt[p^{n}]{\pi^{\prime}}\,)_{n\geq 0}$ of $p^{n}$th roots of
$\pi^{\prime}$ such that
$(\\!\sqrt[p^{n}]{\pi^{\prime}}\,)^{e(K^{\prime}/K)}=\sqrt[p^{n}]{\pi}$. Write
$\underline{\pi}^{\prime}=(\sqrt[p^{n}]{\pi^{\prime}})_{n\geq 0}\in R$, and
$[\underline{\pi}^{\prime}]\in W(R)$ for its image under the natural
multiplicative map $R\to W(R)$. We have a Frobenius-equivariant inclusion
$\mathfrak{S}\hookrightarrow W(R)$ by sending
$u\mapsto[\underline{\pi}^{\prime}]$. We can naturally identify
$\operatorname{Gal}(K^{\prime}_{\infty}/K_{\infty})$ with
$\operatorname{Gal}(K^{\prime}/K)$, and doing this we see that the action of
$g\in G_{K_{\infty}}$ on $u$ is via $g(u)=h(g)u$.

We let ${\mathcal{O}}_{{\mathcal{E}}}$ denote the $p$-adic completion of
$\mathfrak{S}[1/u]$, and let ${\mathcal{E}}$ be the field of fractions of
${\mathcal{O}}_{\mathcal{E}}$. The inclusion $\mathfrak{S}\hookrightarrow
W(R)$ extends to an inclusion ${\mathcal{E}}\hookrightarrow
W(\operatorname{Frac}(R))[1/p]$. Let ${\mathcal{E}}^{\text{nr}}$ be the
maximal unramified extension of ${\mathcal{E}}$ in
$W(\operatorname{Frac}(R))[1/p]$, and let
${\mathcal{O}}_{{\mathcal{E}}^{\text{nr}}}\subset W(\operatorname{Frac}(R))$
denote its ring of integers. Let
${\mathcal{O}}_{\widehat{{\mathcal{E}}^{\text{nr}}}}$ be the $p$-adic
completion of ${\mathcal{O}}_{{\mathcal{E}}^{\text{nr}}}$. Note that
${\mathcal{O}}_{\widehat{{\mathcal{E}}^{\text{nr}}}}$ is stable under the
action of $G_{K_{\infty}}$.

######  Definition 2.2.3.

Suppose that $A$ is a ${\mathbf{Z}}/p^{a}{\mathbf{Z}}$-algebra for some $a\geq
1$. If $M$ is a weak étale $\varphi$-module with $A$-coefficients and descent
data, set
$T_{A}(M):=\left({\mathcal{O}}_{\widehat{{\mathcal{E}}^{\text{nr}}}}\otimes_{\mathfrak{S}[1/u]}M\right)^{\varphi=1}$,
an $A$-module with a $G_{K_{\infty}}$-action (via the diagonal action on
${\mathcal{O}}_{\widehat{{\mathcal{E}}^{\text{nr}}}}$ and $M$, the latter
given by the $\hat{g}$). If $\mathfrak{M}$ is a Breuil–Kisin module with
$A$-coefficients and descent data, set
$T_{A}(\mathfrak{M}):=T_{A}(\mathfrak{M}[1/u])$.

######  Lemma 2.2.4.

Suppose that $A$ is a local ${\mathbf{Z}}_{p}$-algebra and that $|A|<\infty$.
Then $T_{A}$ induces an equivalence of categories from the category of weak
étale $\varphi$-modules with $A$-coefficients and descent data to the category
of continuous representations of $G_{K_{\infty}}$ on finite $A$-modules. If
$A\to A^{\prime}$ is finite, then there is a natural isomorphism
$T_{A}(M)\otimes_{A}A^{\prime}\buildrel\sim\over{\longrightarrow}T_{A^{\prime}}(M\otimes_{A}A^{\prime})$.
A weak étale $\varphi$-module with $A$-coefficients and descent data $M$ is
free of rank $d$ if and only if $T_{A}(M)$ is a free $A$-module of rank $d$.

###### Proof.

This is due to Fontaine [Fon90], and can be proved in exactly the same way as
[Kis09, Lem. 1.2.7]. ∎

We will frequently simply write $T$ for $T_{A}$. Note that if we let
$M^{\prime}$ be the étale $\varphi$-module obtained from $M$ by forgetting the
descent data, then by definition we have
$T(M^{\prime})=T(M)|_{G_{K^{\prime}_{\infty}}}$.

######  Remark 2.2.5.

Although étale $\varphi$-modules naturally give rise to representations of
$G_{K_{\infty}}$, those coming from Breuil–Kisin modules of height at most $1$
admit canonical extensions to $G_{K}$ by [Kis09, Prop. 1.1.13].

######  Lemma 2.2.6.

If
$\overline{r},\overline{r}^{\prime}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$
are continuous representations, both of which arise as the reduction mod $p$
of potentially Barsotti–Tate representations of tame inertial type, and there
is an isomorphism
$\overline{r}|_{G_{K_{\infty}}}\cong\overline{r}^{\prime}|_{G_{K_{\infty}}}$,
then $\overline{r}\cong\overline{r}^{\prime}$.

###### Proof.

The extension $K_{\infty}/K$ is totally wildly ramified. Since the irreducible
$\overline{{\mathbf{F}}}_{p}$-representations of $G_{K}$ are induced from
tamely ramified characters, we see that $\overline{r}|_{G_{K_{\infty}}}$ is
irreducible if and only if $\overline{r}$ is irreducible, and if
$\overline{r}$ or $\overline{r}^{\prime}$ is irreducible then we are done. In
the reducible case, we see that $\overline{r}$ and $\overline{r}^{\prime}$ are
extensions of the same characters, and the result then follows from [GLS15,
Lem. 5.4.2] and Lemma A.5 (2). ∎

###  2.3. Recollections from [CEGS20b]

The main objects of study in this paper are certain algebraic stacks
${\mathcal{C}}^{\tau,\operatorname{BT}}$ and ${\mathcal{Z}}^{\tau,1}$, of rank
two Breuil–Kisin modules and étale $\varphi$-modules respectively, that were
introduced and studied in [CEGS20b]. We review their definitions now, and
recall the main properties of these stacks that were established in [CEGS20b].

To define ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ we first introduce stacks
of Breuil–Kisin modules with descent data; then we impose two conditions on
them, corresponding (in the analogy with Galois representations) to fixing an
inertial type $\tau$ and requiring all pairs Hodge–Tate weights to be
$\\{0,1\\}$.

Take $K^{\prime}/K$ to be any Galois extension such that $[K^{\prime}:K]$ is
prime to $p$, and write $N=K\cdot W(k^{\prime})[1/p]$.

######  Definition 2.3.1.

For each integer $a\geq 1$, we let ${\mathcal{C}}_{d,h}^{\mathrm{dd},a}$ be
the fppf stack over ${\mathcal{O}}/\varpi^{a}$ which associates to any
${\mathcal{O}}/\varpi^{a}$-algebra $A$ the groupoid
${\mathcal{C}}_{d,h}^{\mathrm{dd},a}(A)$ of rank $d$ Breuil–Kisin modules of
height at most $h$ with $A$-coefficients and descent data from $K^{\prime}$ to
$K$.

By [Sta13, Tag 04WV], we may also regard each of the stacks
${\mathcal{C}}_{d,h}^{\mathrm{dd},a}$ as an fppf stack over ${\mathcal{O}}$,
and we then write
${\mathcal{C}}_{d,h}^{\mathrm{dd}}:=\varinjlim_{a}{\mathcal{C}}_{d,h}^{\mathrm{dd},a}$;
this is again an fppf stack over ${\mathcal{O}}$. We will omit the subscripts
$d,h$ from this notation when doing so will not cause confusion.

######  Definition 2.3.2.

Let $\tau$ be a $d$-dimensional $E$-representation of $I(K^{\prime}/K)$. We
say that an object $\mathfrak{M}$ of ${\mathcal{C}}^{\mathrm{dd},a}$ is has
_type_ $\tau$ if Zariski locally on $\operatorname{Spec}A$ there is an
$I(K^{\prime}/K)$-equivariant isomorphism
$\mathfrak{M}_{i}/u\mathfrak{M}_{i}\cong A\otimes_{{\mathcal{O}}}\tau^{\circ}$
for each $i$. (Here we recall that $\mathfrak{M}_{i}:=e_{i}\mathfrak{M}$, and
$\tau^{\circ}$ denotes an ${\mathcal{O}}$-lattice in $\tau$.)

######  Definition 2.3.3.

Let ${\mathcal{C}}^{\tau}$ be the étale substack of
${\mathcal{C}}^{\mathrm{dd}}$ consisting of the objects of type $\tau$. This
is an open and closed substack of ${\mathcal{C}}^{\mathrm{dd}}$ (see [CEGS20b,
Prop. 3.3.5]).

For the remainder of this section we fix $d=2$ and $h=1$. Suppose that $A$ is
an ${\mathcal{O}}/\varpi^{a}$-algebra and consider a pair
$(\mathfrak{L},\mathfrak{L}^{+})$, where:

  * •

$\mathfrak{L}$ is a rank $2$ projective
${\mathcal{O}}_{K^{\prime}}\otimes_{{\mathbf{Z}}_{p}}A$-module, with a
$\operatorname{Gal}(K^{\prime}/K)$-semilinear, $A$-linear action of
$\operatorname{Gal}(K^{\prime}/K)$;

  * •

$\mathfrak{L}^{+}$ is an
${\mathcal{O}}_{K^{\prime}}\otimes_{{\mathbf{Z}}_{p}}A$-submodule of
$\mathfrak{L}$, which is locally on $\operatorname{Spec}A$ a direct summand of
$\mathfrak{L}$ as an $A$-module (or equivalently, for which
$\mathfrak{L}/\mathfrak{L}^{+}$ is projective as an $A$-module), and is
preserved by $\operatorname{Gal}(K^{\prime}/K)$.

For each character $\xi:I(K^{\prime}/K)\to{\mathcal{O}}^{\times}$, let
$\mathfrak{L}_{\xi}$ (resp. $\mathfrak{L}^{+}_{\xi}$) be the
${\mathcal{O}}_{N}\otimes_{{\mathbf{Z}}_{p}}A$-submodule of $\mathfrak{L}$
(resp. $\mathfrak{L}^{+}$) on which $I(K^{\prime}/K)$ acts through $\xi$. We
say that the pair $(\mathfrak{L},\mathfrak{L}^{+})$ _satisfies the strong
determinant condition_ if Zariski locally on $\operatorname{Spec}A$ the
following condition holds: for all $\alpha\in{\mathcal{O}}_{N}$ and all $\xi$,
we have

(2.3.4)
$\det{\\!}_{A}(\alpha{}|\mathfrak{L}^{+}_{\xi})=\prod_{\psi:N\hookrightarrow
E}\psi(\alpha{})$

as polynomial functions on ${\mathcal{O}}_{N}$ in the sense of [Kot92, §5].

######  Definition 2.3.5.

An object $\mathfrak{M}$ of ${\mathcal{C}}^{\mathrm{dd},a}$ _satisfies the
strong determinant condition_ if the pair
$(\mathfrak{M}/E(u)\mathfrak{M},\operatorname{im}\Phi_{\mathfrak{M}}/E(u)\mathfrak{M})$
satisfies the strong determinant condition as in the previous paragraph.

We define ${\mathcal{C}}^{\tau,\operatorname{BT}}$ to be the substack of
${\mathcal{C}}^{\tau}$ of objects satisfying the strong determinant condition.
This is a $\varpi$-adic formal algebraic stack of finite presentation over
${\mathcal{O}}$ by [CEGS20b, Prop. 4.2.7], and so its special fibre
${\mathcal{C}}^{\tau,\operatorname{BT},1}$ is an algebraic stack, locally of
finite type over ${\mathbf{F}}$. The
$\operatorname{Spf}({\mathcal{O}}_{E^{\prime}})$-points of
${\mathcal{C}}^{\tau,\operatorname{BT}}$, for any finite extension
$E^{\prime}/E$, correspond to potentially Barsotti–Tate Galois representations
$G_{K}\to\operatorname{GL}_{2}({\mathcal{O}}_{E^{\prime}})$ of inertial type
$\tau$ ([CEGS20b, Lem. 4.2.16]).

The following result combines [CEGS20b, Cor. 4.5.3, Prop. 5.2.21].

######  Theorem 2.3.6.

We have:

  1. (1)

${\mathcal{C}}^{\tau,\operatorname{BT}}$ is analytically normal, and
Cohen–Macaulay.

  2. (2)

The special fibre ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ is reduced and
equidimensional of dimension equal to $[K:{\mathbf{Q}}_{p}]$.

  3. (3)

${\mathcal{C}}^{\tau,\operatorname{BT}}$ is flat over ${\mathcal{O}}$.

We now introduce our stacks of étale $\varphi$-modules.

######  Definition 2.3.7.

Let ${\mathcal{R}}^{\mathrm{dd},1}$ be the _fppf_ ${\mathbf{F}}$-stack which
associates to any ${\mathbf{F}}$-algebra $A$ the groupoid
${\mathcal{R}}^{\mathrm{dd},1}(A)$ of rank $2$ étale $\varphi$-modules with
$A$-coefficients and descent data from $K^{\prime}$ to $K$.

Inverting $u$ gives a proper morphism
${\mathcal{C}}^{\mathrm{dd},1}\to{\mathcal{R}}^{\mathrm{dd},1}$, which then
restricts to a proper morphism
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{R}}^{\mathrm{dd},1}$ for
each $\tau$.

We now briefly remind the reader of some definitions from [EG21, §3.2]. Let
${\mathcal{X}}\to{\mathcal{F}}$ be a proper morphism of stacks over a locally
Noetherian base-scheme $S$, where ${\mathcal{X}}$ is an algebraic stack which
is locally of finite presentation over $S$, and the diagonal of
${\mathcal{F}}$ is representable by algebraic spaces and locally of finite
presentation.

We refer to [EG21, Defn. 3.2.8] for the definition of the _scheme-theoretic
image_ ${\mathcal{Z}}$ of the proper morphism ${\mathcal{X}}\to{\mathcal{F}}$.
By definition, it is a full subcategory in groupoids of ${\mathcal{F}}$, and
in fact by [EG21, Lem. 3.2.9] it is a Zariski substack of ${\mathcal{F}}$. By
[EG21, Lem. 3.2.14], the finite type points of ${\mathcal{Z}}$ are precisely
the finite type points of ${\mathcal{F}}$ for which the corresponding fibre of
${\mathcal{X}}$ is nonzero.

The results of [EG21, §3.2] give criteria for ${\mathcal{Z}}$ to be an
algebraic stack, and prove a number of associated results (such as universal
properties of the morphism ${\mathcal{Z}}\to{\mathcal{F}}$, and a description
of versal deformation rings for ${\mathcal{Z}}$). This formalism applies in
particular to the proper morphism
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{R}}^{\mathrm{dd},1}$,
and so we make the following definition.

######  Definition 2.3.8.

We define ${\mathcal{Z}}^{\tau,1}$ to be the scheme-theoretic image (in the
sense of [EG21, Defn. 3.2.8]) of the morphism
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{R}}^{\mathrm{dd},1}$.

In [CEGS20b, Thm. 5.1.2, Prop. 5.2.20] we established the following properties
of this construction.

######  Proposition 2.3.9.

  1. (1)

${\mathcal{Z}}^{\tau,1}$ is an algebraic stack of finite presentation over
${\mathbf{F}}$, and is a closed substack of ${\mathcal{R}}^{\mathrm{dd},1}$.

  2. (2)

The morphism
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{R}}^{\mathrm{dd},1}$
factors through a morphism
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{Z}}^{\mathrm{dd},1}$
which is representable by algebraic spaces, scheme-theoretically dominant, and
proper.

  3. (3)

The $\overline{{\mathbf{F}}}_{p}$-points of ${\mathcal{Z}}^{\tau,1}$ are
naturally in bijection with the continuous representations
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$
which have a potentially Barsotti–Tate lift of type $\tau$.

######  Theorem 2.3.10.

The algebraic stacks ${\mathcal{Z}}^{\tau,1}$ are equidimensional of dimension
equal to $[K:{\mathbf{Q}}_{p}]$.

###  2.4. Dieudonné and gauge stacks

We now specialise the choice of $K^{\prime}$ in the following way. Choose a
tame inertial type $\tau=\eta\oplus\eta^{\prime}$. Fix a uniformiser $\pi$ of
$K$. If $\tau$ is a tame principal series type, we take
$K^{\prime}=K(\pi^{1/(p^{f}-1)})$, while if $\tau$ is a tame cuspidal type, we
let $L$ be an unramified quadratic extension of $K$, and set
$K^{\prime}=L(\pi^{1/(p^{2f}-1)})$. Let $N$ be the maximal unramified
extension of $K$ in $K^{\prime}$. In either case $K^{\prime}/K$ is a Galois
extension; in the principal series case, we have $e^{\prime}=(p^{f}-1)e$,
$f^{\prime}=f$, and in the cuspidal case we have $e^{\prime}=(p^{2f}-1)e$,
$f^{\prime}=2f$. We refer to this choice of extension as the _standard choice_
(for the fixed type $\tau$ and uniformiser $\pi$).

For the rest of this section we assume that $\eta\neq\eta^{\prime}$ (we will
not need to consider Dieudonné modules for scalar types). Let $\mathfrak{M}$
be an object of ${\mathcal{C}}^{\tau,\operatorname{BT}}(A)$, and let
$D:=\mathfrak{M}/u\mathfrak{M}$ be its corresponding Dieudonné module as in
Definition 2.1.7. The group $I(K^{\prime}/K)$ is abelian of order prime to
$p$, and so we can write $D=D_{\eta}\oplus D_{\eta^{\prime}}$, where
$D_{\eta}$ is the submodule on which $I(K^{\prime}/K)$ acts via $\eta$.
Setting $D_{\eta,j}:=e_{j}D_{\eta}$, it follows from the projectivity of
$\mathfrak{M}$ that each $D_{\eta,j}$ is an invertible $A$-module. The maps
$F,V$ induce linear maps $F:D_{\eta,j}\to D_{\eta,j+1}$ and $V:D_{\eta,j+1}\to
D_{\eta,j}$ such that $FV=VF=p$.

######  Definition 2.4.1.

If $\tau$ is a principal series type we define a stack

${\mathcal{D}}_{\eta}:=\Big{[}\bigl{(}\operatorname{Spec}W(k)[X_{0},Y_{0},\ldots,X_{f-1},Y_{f-1}]/(X_{j}Y_{j}-p)_{j=0,\ldots,f-1})\bigr{)}/\mathbf{G}_{m}^{f}\big{]},$

where the $f$ copies of $\mathbf{G}_{m}$ act as
$(u_{0},\ldots,u_{f-1})\cdot(X_{j},Y_{j})\mapsto(u_{j}u_{j+1}^{-1}X_{j},u_{j+1}u_{j}^{-1}Y_{j}).$

If instead $\tau$ is a cuspidal type we define

${\mathcal{D}}_{\eta}:=\Big{[}\bigl{(}\operatorname{Spec}W(k)[X_{0},Y_{0},\ldots,X_{f-1},Y_{f-1}]/(X_{j}Y_{j}-p)_{j=0,\ldots,f-1})\times{\mathbf{G}}_{m}\bigr{)}/\mathbf{G}_{m}^{f+1}\big{]},$

where the $f+1$ copies of $\mathbf{G}_{m}$ act as

$(u_{0},\ldots,u_{f-1},u_{f})\cdot((X_{j},Y_{j}),\alpha)\mapsto((u_{j}u_{j+1}^{-1}X_{j},u_{j+1}u_{j}^{-1}Y_{j}),\alpha).$

In [CEGS20b, Sec. 4.6] we explained how the stack ${\mathcal{D}}_{\eta}$
classifies the line bundles $D_{\eta,j}$ together with the maps $F,V$, so that
in either case (principal series or cuspidal) there is a natural map
${\mathcal{C}}^{\tau,\operatorname{BT}}\to{\mathcal{D}}_{\eta}$.

It will be helpful to introduce another stack, the stack
${\mathcal{G}}_{\eta}$ of $\eta$-gauges. This classifies $f$-tuples of line
bundles ${\mathcal{D}}_{j}$ ($j=0,\ldots,f-1$) equipped with sections
$X_{j}\in{\mathcal{D}}_{j}$ and $Y_{j}\in{\mathcal{D}}_{j}^{-1}$. Explicitly,
it can be written as the quotient stack

${\mathcal{G}}_{\eta}:=\Big{[}\bigl{(}\operatorname{Spec}W(k)[X_{0},Y_{0},\ldots,X_{f-1},Y_{f-1}]/(X_{j}Y_{j}-p)_{j=0,\ldots,f-1})\bigr{)}/\mathbf{G}_{m}^{f}\big{]},$

where the $f$ copies of $\mathbf{G}_{m}$ act as follows:

$(v_{0},\ldots,v_{f-1})\cdot(X_{j},Y_{j})\mapsto(v_{j}X_{j},v_{j}^{-1}Y_{j}).$

There is a morphism of stacks ${\mathcal{D}}_{\eta}\to{\mathcal{G}}_{\eta}$
which we can define explicitly using their descriptions as quotient stacks.
Indeed, in the principal series case we have a morphism
${\mathbf{G}}_{m}^{f}\to{\mathbf{G}}_{m}^{f}$ given by
$(u_{j})_{j=0,\ldots,f-1}\mapsto(u_{j}u_{j+1}^{-1})_{j=0,\ldots,f-1}$, which
is compatible with the actions of these two groups on
$\operatorname{Spec}W(k)[(X_{j},Y_{j})_{j=0,\ldots,f-1}]/(X_{j}Y_{j}-p)_{j=0,\ldots,f-1},$
and we are just considering the map from the quotient by the first
${\mathbf{G}}_{m}^{f}$ to the quotient by the second ${\mathbf{G}}_{m}^{f}$.
In the cuspidal case we have a morphism
${\mathbf{G}}_{m}^{f+1}\to{\mathbf{G}}_{m}^{f}$ given by
$(u_{j})_{j=0,\ldots,f}\mapsto(u_{j}u_{j+1}^{-1})_{j=0,\ldots,f-1}$, and the
morphism ${\mathcal{D}}_{\eta}\to{\mathcal{G}}_{\eta}$ is the obvious one
which forgets the factor of ${\mathbf{G}}_{m}$ coming from $\alpha$.

Composing our morphism
${\mathcal{C}}^{\tau,\operatorname{BT}}\to{\mathcal{D}}_{\eta}$ with the
forgetful morphism ${\mathcal{D}}_{\eta}\to{\mathcal{G}}_{\eta}$, we obtain a
morphism ${\mathcal{C}}^{\tau,\operatorname{BT}}\to{\mathcal{G}}_{\eta}$.

For our analysis of the irreducible components of the stacks
${\mathcal{C}}^{\tau,\operatorname{BT},1}$ at the end of Section 3, it will be
useful to have a more directly geometric interpretation of a morphism
$S\to{\mathcal{G}}_{\eta}$, in the case that the source is a flat
$W(k)$-scheme, or, more generally, a flat $p$-adic formal algebraic stack over
$\operatorname{Spf}W(k)$. In order to do this we will need some basic material
on effective Cartier divisors for (formal) algebraic stacks; while it is
presumably possible to develop this theory in considerable generality, we only
need a very special case, and we limit ourselves to this setting.

The property of a closed subscheme being an effective Cartier divisor is not
preserved under arbitrary pull-back, but it is preserved under flat pull-back.
More precisely, we have the following result.

######  Lemma 2.4.2.

If $X$ is a scheme, and $Z$ is a closed subscheme of $X$, then the following
are equivalent:

  1. (1)

$Z$ is an effective Cartier divisor on $X$.

  2. (2)

For any flat morphism of schemes $U\to X$, the pull-back $Z\times_{X}U$ is an
effective Cartier divisor on $U$.

  3. (3)

For some fpqc covering $\\{X_{i}\to X\\}$ of $X$, each of the pull-backs
$Z\times_{X}X_{i}$ is an effective Cartier divisor on $X_{i}$.

###### Proof.

Since $Z$ is an effective Cartier divisor if and only if its ideal sheaf
${\mathcal{I}}_{Z}$ is an invertible sheaf on $X$, this follows from the fact
that the invertibility of a quasi-coherent sheaf is a local property in the
fpqc topology. ∎

######  Lemma 2.4.3.

If $A$ is a Noetherian adic topological ring, then pull-back under the natural
morphism $\operatorname{Spf}A\to\operatorname{Spec}A$ induces a bijection
between the closed subschemes of $\operatorname{Spec}A$ and the closed
subspaces of $\operatorname{Spf}A$.

###### Proof.

It follows from [Sta13, Tag 0ANQ] that closed immersions
$Z\to\operatorname{Spf}A$ are necessarily of the form
$\operatorname{Spf}B\to\operatorname{Spf}A$, and correspond to continuous
morphisms $A\to B$, for some complete linearly topologized ring $B$, which are
taut (in the sense of [Sta13, Tag 0AMX]), have closed kernel, and dense image.
Since $A$ is adic, it admits a countable basis of neighbourhoods of the
origin, and so it follows from [Sta13, Tag 0APT] (recalling also [Sta13, Tag
0AMV]) that $A\to B$ is surjective. Because any ideal of definition $I$ of $A$
is finitely generated, it follows from [Sta13, Tag 0APU] that $B$ is endowed
with the $I$-adic topology. Finally, since $A$ is Noetherian, any ideal in $A$
is $I$-adically closed. Thus closed immersions
$\operatorname{Spf}B\to\operatorname{Spf}A$ are determined by giving the
kernel of the corresponding morphism $A\to B$, which can be arbitrary. The
same is true of closed immersions
$\operatorname{Spec}B\to\operatorname{Spec}A$, and so the lemma follows. ∎

######  Definition 2.4.4.

If $A$ is a Noetherian adic topological ring, then we say that a closed
subspace of $\operatorname{Spf}A$ is an effective Cartier divisor on
$\operatorname{Spf}A$ if the corresponding closed subscheme of
$\operatorname{Spec}A$ is an effective Cartier divisor on
$\operatorname{Spec}A$.

######  Lemma 2.4.5.

Let $\operatorname{Spf}B\to\operatorname{Spf}A$ be a flat adic morphism of
Noetherian affine formal algebraic spaces. If
$Z\hookrightarrow\operatorname{Spf}A$ is a Cartier divisor, then
$Z\times_{\operatorname{Spf}A}\operatorname{Spf}B\hookrightarrow\operatorname{Spf}B$
is a Cartier divisor. Conversely, if
$\operatorname{Spf}B\to\operatorname{Spf}A$ is furthermore surjective, and if
$Z\hookrightarrow\operatorname{Spf}A$ is a closed subspace for which the base-
change
$Z\times_{\operatorname{Spf}A}\operatorname{Spf}B\hookrightarrow\operatorname{Spf}B$
is a Cartier divisor, then $Z$ is a Cartier divisor on $\operatorname{Spf}A$.

###### Proof.

The morphism $\operatorname{Spf}B\to\operatorname{Spf}A$ corresponds to an
adic flat morphism $A\to B$ ([Sta13, Tag 0AN0] and [Eme, Lem. 8.18]) and hence
is induced by a flat morphism $\operatorname{Spec}B\to\operatorname{Spec}A$,
which is furthermore faithfully flat if and only if
$\operatorname{Spf}B\to\operatorname{Spf}A$ is surjective (again by [Eme, Lem.
8.18]). The present lemma thus follows from Lemma 2.4.2. ∎

The preceding lemma justifies the following definition.

######  Definition 2.4.6.

We say that a closed substack ${\mathcal{Z}}$ of a locally Noetherian formal
algebraic stack ${\mathcal{X}}$ is an effective Cartier divisor on
${\mathcal{X}}$ if for any morphism $U\to{\mathcal{X}}$ whose source is a
Noetherian affine formal algebraic space, and which is representable by
algebraic spaces and flat, the pull-back
${\mathcal{Z}}\times_{{\mathcal{X}}}U$ is an effective Cartier divisor on $U$.

We consider the $W(k)$-scheme $\operatorname{Spec}W(k)[X,Y]/(XY-p)$, which we
endow with a ${\mathbf{G}}_{m}$-action via $u\cdot(X,Y):=(uX,u^{-1}Y).$ There
is an obvious morphism

$\operatorname{Spec}W(k)[X,Y]/(XY-p)\to\operatorname{Spec}W(k)[X]=\mathbf{A}^{1}$

given by $(X,Y)\to X$, which is ${\mathbf{G}}_{m}$-equivariant (for the action
of ${\mathbf{G}}_{m}$ on $\mathbf{A}^{1}$ given by $u\cdot X:=uX$), and so
induces a morphism

(2.4.7)
$[\bigl{(}\operatorname{Spec}W(k)[X,Y]/(XY-p)\bigr{)}/{\mathbf{G}}_{m}]\to[\mathbf{A}^{1}/{\mathbf{G}}_{m}].$

######  Lemma 2.4.8.

If ${\mathcal{X}}$ is a locally Noetherian $p$-adic formal algebraic stack
which is furthermore flat over $\operatorname{Spf}W(k)$, then the groupoid of
morphisms

${\mathcal{X}}\to[\operatorname{Spec}W(k)[X,Y]/(XY-p)/{\mathbf{G}}_{m}]$

is in fact a setoid, and is equivalent to the set of effective Cartier
divisors on ${\mathcal{X}}$ that are contained in the effective Cartier
divisor $(\operatorname{Spec}k)\times_{\operatorname{Spf}W(k)}{\mathcal{X}}$
on ${\mathcal{X}}$.

###### Proof.

Essentially by definition (and taking into account [Eme, Lem. 8.18]), it
suffices to prove this in the case when ${\mathcal{X}}=\operatorname{Spf}B$,
where $B$ is a flat Noetherian adic $W(k)$-algebra admitting $(p)$ as an ideal
of definition. In this case, the restriction map

$[\operatorname{Spec}W(k)[X,Y]/(XY-p)/{\mathbf{G}}_{m}](\operatorname{Spec}B)\to[\operatorname{Spec}W(k)[X,Y]/(XY-p)/{\mathbf{G}}_{m}](\operatorname{Spf}B)$

is an equivalence of groupoids. Indeed, the essential surjectivity follows
from the (standard and easily verified) fact that if $\\{M_{i}\\}$ is a
compatible family of locally free $B/p^{i}B$-modules of rank one, then
$M:=\varprojlim M_{i}$ is a locally free $B$-module of rank one, for which
each of the natural morphisms $M/p^{i}M\to M_{i}$ is an isomorphism. The full
faithfulness follows from the fact that a locally free $B$-module of rank one
is $p$-adically complete, and so is recovered as the inverse limit of its
compatible family of quotients $\\{M/p^{i}M\\}.$

We are therefore reduced to the same statement with
${\mathcal{X}}=\operatorname{Spec}B$. The composite morphism
$\operatorname{Spec}B\to[\mathbf{A}^{1}/{\mathbf{G}}_{m}]$ induced by (2.4.7)
corresponds to giving a pair $({\mathcal{D}},X)$ where ${\mathcal{D}}$ is a
line bundle on $\operatorname{Spec}B$, and $X$ is a global section of
${\mathcal{D}}^{-1}$. Indeed, giving a morphism
$\operatorname{Spec}B\to[\mathbf{A}^{1}/{\mathbf{G}}_{m}]$ is equivalent to
giving a ${\mathbf{G}}_{m}$-torsor $P\to\operatorname{Spec}B$, together with a
${\mathbf{G}}_{m}$-equivariant morphism $P\to\mathbf{A}^{1}$. Giving a
${\mathbf{G}}_{m}$-torsor $P$ over $\operatorname{Spec}B$ is equivalent to
giving an invertible sheaf ${\mathcal{D}}$ on $\operatorname{Spec}B$ (the
associated ${\mathbf{G}}_{m}$-torsor is then obtained by deleting the zero
section from the line bundle $D\to X$ corresponding to ${\mathcal{D}}$), and
giving a ${\mathbf{G}}_{m}$-equivariant morphism $P\to\mathbf{A}^{1}$ is
equivalent to giving a global section of ${\mathcal{D}}^{-1}$.

It follows that giving a morphism
$\operatorname{Spec}B\to[\operatorname{Spec}W(k)[X,Y]/(XY-p)/{\mathbf{G}}_{m}]$
corresponds to giving a line bundle ${\mathcal{D}}$ and sections
$X\in{\mathcal{D}}^{-1}$, $Y\in{\mathcal{D}}$ satisfying $XY=p$. To say that
$B$ is flat over $W(k)$ is just to say that $p$ is a regular element on $B$,
and so we see that $X$ (resp. $Y$) is a regular section of
${\mathcal{D}}^{-1}$ (resp. ${\mathcal{D}}$). Again, since $p$ is a regular
element on $B$, we see that $Y$ is uniquely determined by $X$ and the equation
$XY=p$, and so giving a morphism
$\operatorname{Spec}B\to[\operatorname{Spec}W(k)[X,Y]/(XY-p)/{\mathbf{G}}_{m}]$
is equivalent to giving a line bundle ${\mathcal{D}}$ and a regular section
$X$ of ${\mathcal{D}}^{-1}$, such that $pB\subset
X\otimes_{B}{\mathcal{D}}\subset{\mathcal{D}}^{-1}\otimes_{B}{\mathcal{D}}\buildrel\sim\over{\longrightarrow}B$;
this last condition guarantees the existence of the (then uniquely determined)
$Y$.

Now giving a line bundle ${\mathcal{D}}$ on $\operatorname{Spec}B$ and a
regular section $X\in{\mathcal{D}}^{-1}$ is the same as giving the zero locus
$D$ of $X$, which is a Cartier divisor on $\operatorname{Spec}B$. (There is a
canonical isomorphism
$({\mathcal{D}},X)\cong\bigl{(}{\mathcal{I}}_{D},1\bigr{)}$, where
${\mathcal{I}}_{D}$ denotes the ideal sheaf of $D$.) The condition that
$pB\subset X\otimes_{B}{\mathcal{D}}$ is equivalent to the condition that
$p\in{\mathcal{I}}_{D}$, i.e. that $D$ be contained in
$\operatorname{Spec}B/pB$, and we are done. ∎

######  Lemma 2.4.9.

If ${\mathcal{S}}$ is a locally Noetherian $p$-adic formal algebraic stack
which is flat over $W(k)$, then giving a morphism
${\mathcal{S}}\to{\mathcal{G}}_{\eta}$ over $W(k)$ is equivalent to giving a
collection of effective Cartier divisors ${\mathcal{D}}_{j}$ on
${\mathcal{S}}$ ($j=0,\ldots,f-1$), with each ${\mathcal{D}}_{j}$ contained in
the Cartier divisor $\overline{{\mathcal{S}}}$ cut out by the equation $p=0$
on ${\mathcal{S}}$ (i.e. the special fibre of ${\mathcal{S}}$).

###### Proof.

This follows immediately from Lemma 2.4.8, by the definition of
${\mathcal{G}}_{\eta}$. ∎

##  3\. Families of extensions of Breuil–Kisin modules

The goal of the next two sections is to construct certain universal families
of extensions of rank one Breuil–Kisin modules over ${\mathbf{F}}$ with
descent data; these families will be used in Section 5 to describe the generic
behaviour of the various irreducible components of the special fibres of
${\mathcal{C}}^{\tau,\operatorname{BT}}$ and ${\mathcal{Z}}^{\tau}$.

In Subsections 3.1 and 3.2 we present some generalities on extensions of
Breuil–Kisin modules and on families of these extensions, respectively. In
Subsection 3.3 we specialize the discussion of Subsection 3.2 to the case of
extensions of two rank one Breuil–Kisin modules, and thus explain how to
construct our desired families of extensions. In Section 4 we recall the
fundamental computations related to extensions of rank one Breuil–Kisin
modules from [DS15], to which the results of Subsection 3.3 will be applied at
the end of Subsection 4.2 to construct the components
$\overline{{\mathcal{C}}}(J)$ and $\overline{{\mathcal{Z}}}(J)$ of Theorem
1.1.

We assume throughout this section that $[K^{\prime}:K]$ is not divisible by
$p$; since we are assuming throughout the paper that $K^{\prime}/K$ is tamely
ramified, this is equivalent to assuming that $K^{\prime}$ does not contain an
unramified extension of $K$ of degree $p$. In our final applications
$K^{\prime}/K$ will contain unramified extensions of degree at most $2$, and
$p$ will be odd, so this assumption will be satisfied. (In fact, we specialize
to such a context begining in Subsection 5.2.)

###  3.1. Extensions of Breuil–Kisin modules with descent data

When discussing the general theory of extensions of Breuil–Kisin modules, it
is convenient to embed the category of Breuil–Kisin modules in a larger
category which is abelian, contains enough injectives and projectives, and is
closed under passing to arbitrary limits and colimits. The simplest way to
obtain such a category is as the category of modules over some ring, and so we
briefly recall how a Breuil–Kisin module with $A$-coefficients and descent
data can be interpreted as a module over a certain $A$-algebra.

Let $\mathfrak{S}_{A}[F]$ denote the twisted polynomial ring over
$\mathfrak{S}_{A}$, in which the variable $F$ obeys the following commutation
relation with respect to elements $s\in\mathfrak{S}_{A}$:

$F\cdot s=\varphi(s)\cdot F.$

Let $\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]$ denote the twisted
group ring over $\mathfrak{S}_{A}[F]$, in which the elements
$g\in\operatorname{Gal}(K^{\prime}/K)$ commute with $F$, and obey the
following commutation relations with elements $s\in\mathfrak{S}_{A}$:

$g\cdot s=g(s)\cdot g.$

One immediately confirms that giving a left
$\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]$-module $\mathfrak{M}$
is equivalent to equipping the underlying $\mathfrak{S}_{A}$-module
$\mathfrak{M}$ with a $\varphi$-linear morphism
$\varphi:\mathfrak{M}\to\mathfrak{M}$ and a semi-linear action of
$\operatorname{Gal}(K^{\prime}/K)$ which commutes with $\varphi$.

In particular, if we let $\mathcal{K}(A)$ denote the category of left
$\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]$-modules, then a
Breuil–Kisin module with descent data from $K^{\prime}$ to $K$ may naturally
be regarded as an object of $\mathcal{K}(A)$. In the following lemma, we
record the fact that extensions of Breuil–Kisin modules with descent data may
be computed as extensions in the category $\mathcal{K}(A)$.

######  Lemma 3.1.1.

If $0\to\mathfrak{M}^{\prime}\to\mathfrak{M}\to\mathfrak{M}^{\prime\prime}\to
0$ is a short exact sequence in $\mathcal{K}(A)$, such that
$\mathfrak{M}^{\prime}$ (resp. $\mathfrak{M}^{\prime\prime}$) is a
Breuil–Kisin module with descent data of rank $d^{\prime}$ and height at most
$h^{\prime}$ (resp. of rank $d^{\prime\prime}$ and height at most
$h^{\prime\prime}$_)_ , then $\mathfrak{M}$ is a Breuil–Kisin module with
descent data of rank $d^{\prime}+d^{\prime\prime}$ and height at most
$h^{\prime}+h^{\prime\prime}$.

More generally, if
$E(u)^{h}\in\operatorname{Ann}_{\mathfrak{S}_{A}}(\operatorname{coker}\Phi_{\mathfrak{M}^{\prime}})\operatorname{Ann}_{\mathfrak{S}_{A}}(\operatorname{coker}\Phi_{\mathfrak{M}^{\prime\prime}})$,
then $\mathfrak{M}$ is a Breuil–Kisin module with descent data of height at
most $h$.

###### Proof.

Note that since $\Phi_{\mathfrak{M}^{\prime}}[1/E(u)]$ and
$\Phi_{\mathfrak{M}^{\prime\prime}}[1/E(u)]$ are both isomorphisms by
assumption, it follows from the snake lemma that $\Phi_{\mathfrak{M}}[1/E(u)]$
is isomorphism. Similarly we have a short exact sequence of
$\mathfrak{S}_{A}$-modules

$0\to\operatorname{coker}\Phi_{\mathfrak{M}^{\prime}}\to\operatorname{coker}\Phi_{\mathfrak{M}}\to\operatorname{coker}\Phi_{\mathfrak{M}^{\prime\prime}}\to
0.$

The claims about the height and rank of $\mathfrak{M}$ follow immediately. ∎

We now turn to giving an explicit description of the functors
$\operatorname{Ext}^{i}(\mathfrak{M},\text{--}\,)$ for a Breuil–Kisin module
with descent data $\mathfrak{M}$.

######  Definition 3.1.2.

Let $\mathfrak{M}$ be a Breuil–Kisin module with $A$-coefficients and descent
data (of some height). If $\mathfrak{N}$ is any object of $\mathcal{K}(A)$,
then we let $C^{\bullet}_{\mathfrak{M}}(\mathfrak{N})$ denote the complex

$\operatorname{Hom}_{\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]}(\mathfrak{M},\mathfrak{N})\to\operatorname{Hom}_{\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]}(\varphi^{*}\mathfrak{M},\mathfrak{N}),$

with differential being given by

$\alpha\mapsto\Phi_{\mathfrak{N}}\circ\varphi^{*}\alpha-\alpha\circ\Phi_{\mathfrak{M}}.$

Also let $\Phi_{\mathfrak{M}}^{*}$ denote the map
$C^{0}_{\mathfrak{M}}(\mathfrak{N})\to C^{1}_{\mathfrak{M}}(\mathfrak{N})$
given by $\alpha\mapsto\alpha\circ\Phi_{\mathfrak{M}}$. When $\mathfrak{M}$ is
clear from the context we will usually suppress it from the notation and write
simply $C^{\bullet}(\mathfrak{N})$.

Each $C^{i}(\mathfrak{N})$ is naturally an $\mathfrak{S}_{A}^{0}$-module. The
formation of $C^{\bullet}(\mathfrak{N})$ is evidently functorial in
$\mathfrak{N}$, and is also exact in $\mathfrak{N}$, since $\mathfrak{M}$, and
hence also $\varphi^{*}\mathfrak{M}$, is projective over $\mathfrak{S}_{A}$,
and since $\operatorname{Gal}(K^{\prime}/K)$ has prime-to-$p$ order. Thus the
cohomology functors $H^{0}\bigl{(}C^{\bullet}(\text{--})\bigr{)}$ and
$H^{1}\bigl{(}C^{\bullet}(\text{--})\bigr{)}$ form a $\delta$-functor on
$\mathcal{K}(A)$.

######  Lemma 3.1.3.

There is a natural isomorphism

$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\text{--}\,)\cong
H^{0}\bigl{(}C^{\bullet}(\text{--}\,)\bigr{)}.$

###### Proof.

This is immediate. ∎

It follows from this lemma and a standard dimension shifting argument (or,
equivalently, the theory of $\delta$-functors) that there is an embedding of
functors

(3.1.4)
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\text{--}\,)\hookrightarrow
H^{1}\bigl{(}C^{\bullet}(\text{--})\bigr{)}.$

######  Lemma 3.1.5.

The embedding of functors (3.1.4) is an isomorphism.

###### Proof.

We first describe the embedding (3.1.4) explicitly. Suppose that

$0\to\mathfrak{N}\to\mathfrak{E}\to\mathfrak{M}\to 0$

is an extension in $\mathcal{K}(A)$. Since $\mathfrak{M}$ is projective over
$\mathfrak{S}_{A}$, and since $\operatorname{Gal}(K^{\prime}/K)$ is of prime-
to-$p$ order, we split this short exact sequence over the twisted group ring
$\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)],$ say via some element
$\sigma\in\operatorname{Hom}_{\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]}(\mathfrak{M},\mathfrak{E}).$
This splitting is well-defined up to the addition of an element
$\alpha\in\operatorname{Hom}_{\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]}(\mathfrak{M},\mathfrak{N}).$

This splitting is a homomorphism in $\mathcal{K}(A)$ if and only if the
element

$\Phi_{\mathfrak{E}}\circ\varphi^{*}\sigma-\sigma\circ\Phi_{\mathfrak{M}}\in\operatorname{Hom}_{\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]}(\varphi^{*}\mathfrak{M},\mathfrak{N})$

vanishes. If we replace $\sigma$ by $\sigma+\alpha,$ then this element is
replaced by

$(\Phi_{\mathfrak{E}}\circ\varphi^{*}\sigma-\sigma\circ\Phi_{\mathfrak{M}})+(\Phi_{\mathfrak{N}}\circ\varphi^{*}\alpha-\alpha\circ\Phi_{\mathfrak{M}}).$

Thus the coset of
$\Phi_{\mathfrak{E}}\circ\varphi^{*}\sigma-\sigma\circ\Phi_{\mathfrak{M}}$ in
$H^{1}\bigl{(}C^{\bullet}(\mathfrak{N})\bigr{)}$ is well-defined, independent
of the choice of $\sigma,$ and this coset is the image of the class of the
extension $\mathfrak{E}$ under the embedding

(3.1.6)
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\hookrightarrow
H^{1}\bigl{(}C^{\bullet}(\mathfrak{N})\bigr{)}$

(up to a possible overall sign, which we ignore, since it doesn’t affect the
claim of the lemma).

Now, given any element
$\nu\in\operatorname{Hom}_{\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]}(\varphi^{*}\mathfrak{M},\mathfrak{N})$,
we may give the $\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]$-module
$\mathfrak{E}:=\mathfrak{N}\oplus\mathfrak{M}$ the structure of a
$\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]$-module as follows: we
need to define a $\varphi$-linear morphism $\mathfrak{E}\to\mathfrak{E}$, or
equivalently a linear morphism
$\Phi_{\mathfrak{E}}:\varphi^{*}\mathfrak{E}\to\mathfrak{E}$. We do this by
setting

$\Phi_{\mathfrak{E}}:=\begin{pmatrix}\Phi_{\mathfrak{N}}&\nu\\\
0&\Phi_{\mathfrak{M}}\end{pmatrix}.$

Then $\mathfrak{E}$ is an extension of $\mathfrak{M}$ by $\mathfrak{N}$, and
if we let $\sigma$ denote the obvious embedding of $\mathfrak{M}$ into
$\mathfrak{E}$, then one computes that

$\nu=\Phi_{\mathfrak{E}}\circ\varphi^{*}\sigma-\sigma\circ\Phi_{\mathfrak{M}}.$

This shows that (3.1.6) is an isomorphism, as claimed. ∎

Another dimension shifting argument, taking into account the preceding lemma,
shows that $\operatorname{Ext}^{2}_{\mathcal{K}(A)}(\mathfrak{M},\text{--}\,)$
embeds into $H^{2}\bigl{(}C^{\bullet}(\text{--})\bigr{)}.$ Since the target of
this embedding vanishes, we find that the same is true of the source. This
yields the following corollary.

######  Corollary 3.1.7.

If $\mathfrak{M}$ is a Breuil–Kisin module with $A$-coefficients and descent
data, then
$\operatorname{Ext}^{2}_{\mathcal{K}(A)}(\mathfrak{M},\text{--}\,)=0.$

We summarise the above discussion in the following corollary.

######  Corollary 3.1.8.

If $\mathfrak{M}$ is a Breuil–Kisin module with $A$-coefficients and descent
data, and $\mathfrak{N}$ is an object of $\mathcal{K}(A)$, then we have a
natural short exact sequence

$0\to\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\to
C^{0}(\mathfrak{N})\to
C^{1}(\mathfrak{N})\to\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\to
0.$

The following lemma records the behaviour of these complexes with respect to
base change.

######  Lemma 3.1.9.

Suppose that $\mathfrak{M}$, $\mathfrak{N}$ are Breuil–Kisin modules with
descent data and $A$-coefficients, that $B$ is an $A$-algebra, and that $Q$ is
a $B$-module. Then the complexes
$C^{\bullet}_{\mathfrak{M}}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)$ and
$C^{\bullet}_{\mathfrak{M}\,\widehat{\otimes}_{A}B}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)$
coincide, the former complex formed with respect to $\mathcal{K}(A)$ and the
latter with respect to $\mathcal{K}(B)$.

###### Proof.

Indeed, there is a natural isomorphism

$\operatorname{Hom}_{\mathfrak{S}_{A}[\operatorname{Gal}(K^{\prime}/K)]}(\mathfrak{M},\mathfrak{N}\,\widehat{\otimes}_{A}Q)\cong\operatorname{Hom}_{\mathfrak{S}_{B}[\operatorname{Gal}(K^{\prime}/K)]}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}Q),$

and similarly with $\varphi^{*}\mathfrak{M}$ in place of $\mathfrak{M}$. ∎

The following slightly technical lemma is crucial for establishing finiteness
properties, and also base-change properties, of Exts of Breuil–Kisin modules.

######  Lemma 3.1.10.

Let $A$ be a ${\mathcal{O}}/\varpi^{a}$-algebra for some $a\geq 1$, suppose
that $\mathfrak{M}$ is a Breuil–Kisin module with descent data and
$A$-coefficients, of height at most $h$, and suppose that $\mathfrak{N}$ is a
$u$-adically complete, $u$-torsion free object of $\mathcal{K}(A)$.

Let $C^{\bullet}$ be the complex defined in _Definition 3.1.2_, and write
$\delta$ for its differential. Suppose that $Q$ is an $A$-module with the
property that $C^{i}\otimes_{A}Q$ is $v$-torsion free for $i=0,1$ and
$v$-adically separated for $i=0$.

Then:

  1. (1)

For any integer $M\geq(eah+1)/(p-1)$,
$\ker(\delta\otimes\operatorname{id}_{Q})\cap v^{M}C^{0}\otimes_{A}Q=0$.

  2. (2)

For any integer $N\geq(peah+1)/(p-1)$, $\delta\otimes\operatorname{id}_{Q}$
induces an isomorphism

$(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}\otimes_{A}Q)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}v^{N}(C^{1}\otimes_{A}Q).$

Consequently, for $N$ as in _(2)_ the natural morphism of complexes of
$A$-modules

$[C^{0}\otimes_{A}Q\buildrel\delta\otimes\operatorname{id}_{Q}\over{\longrightarrow}C^{1}\otimes_{A}Q]\to[C^{0}\otimes_{A}Q/\bigl{(}(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}\otimes_{A}Q)\bigr{)}\buildrel\delta\otimes\operatorname{id}_{Q}\over{\longrightarrow}C^{1}\otimes_{A}Q/v^{N}C^{1}\otimes_{A}Q]$

is a quasi-isomorphism.

Since we are assuming that the $C^{i}\otimes_{A}Q$ are $v$-torsion free, the
expression $v^{r}C^{i}(\mathfrak{N})\otimes_{A}Q$ may be interpreted as
denoting either $v^{r}\bigl{(}C^{i}(\mathfrak{N})\otimes_{A}Q\bigr{)}$ or
$\bigl{(}v^{r}C^{i}(\mathfrak{N})\bigr{)}\otimes_{A}Q$, the two being
naturally isomorphic.

######  Remark 3.1.11.

Before giving the proof of Lemma 3.1.10, we observe that the hypotheses on the
$C^{i}\otimes_{A}Q$ are satisfied if either $Q=A$, or else $\mathfrak{N}$ is a
projective $\mathfrak{S}_{A}$-module and $Q$ is a finitely generated
$B$-module for some finitely generated $A$-algebra $B$. (Indeed
$C^{1}\otimes_{A}Q$ is $v$-adically separated as well in these cases.)

(1) Since $\mathfrak{M}$ is projective of finite rank over $A[[u]]$, and since
$\mathfrak{N}$ is $u$-adically complete and $u$-torsion free, each $C^{i}$ is
$v$-adically separated and $v$-torsion free. In particular the hypothesis on
$Q$ is always satisfied by $Q=A$. (In fact since $\mathfrak{N}$ is
$u$-adically complete it also follows that the $C^{i}$ are $v$-adically
complete. Here we use that $\operatorname{Gal}(K^{\prime}/K)$ has order prime
to $p$ to see that $C^{0}$ is an $\mathfrak{S}_{A}^{0}$-module direct summand
of $\operatorname{Hom}_{\mathfrak{S}_{A}}(\mathfrak{M},\mathfrak{N})$, and
similarly for $C^{1}$.)

(2) Suppose $\mathfrak{N}$ is a projective $\mathfrak{S}_{A}$-module. Then the
$C^{i}$ are projective $\mathfrak{S}_{A}^{0}$-modules, again using that
$\operatorname{Gal}(K^{\prime}/K)$ has order prime to $p$. Since each
$C^{i}(\mathfrak{N})/vC^{i}(\mathfrak{N})$ is $A$-flat, it follows that
$C^{i}(\mathfrak{N})\otimes_{A}Q$ is $v$-torsion free. If furthermore $B$ is a
finitely generated $A$-algebra, and $Q$ is a finitely generated $B$-module,
then the $C^{i}(\mathfrak{N})\otimes_{A}Q$ are $v$-adically separated (being
finitely generated modules over the ring $A[[v]]\otimes_{A}B$, which is a
finitely generated algebra over the Noetherian ring $A[[v]]$, and hence is
itself Noetherian).

###### Proof of Lemma 3.1.10.

Since $p^{a}=0$ in $A$, there exists $H(u)\in\mathfrak{S}_{A}$ with
$u^{e^{\prime}ah}=E(u)^{h}H(u)$ in $\mathfrak{S}_{A}$. Thus the image of
$\Phi_{\mathfrak{M}}$ contains
$u^{e^{\prime}ah}\mathfrak{M}=v^{eah}\mathfrak{M}$, and there exists a map
$\Upsilon:\mathfrak{M}\to\varphi^{*}\mathfrak{M}$ such that
$\Phi_{\mathfrak{M}}\circ\Upsilon$ is multiplication by $v^{eah}$.

We begin with (1). Suppose that
$f\in\ker(\delta\otimes\operatorname{id}_{Q})\cap v^{M}C^{0}\otimes_{A}Q$.
Since $C^{0}\otimes_{A}Q$ is $v$-adically separated, it is enough, applying
induction on $M$, to show that $f\in v^{M+1}C^{0}\otimes_{A}Q$. Since
$f\in\ker(\delta\otimes\operatorname{id}_{Q})$, we have
$f\circ\Phi_{\mathfrak{M}}=\Phi_{\mathfrak{N}}\circ\varphi^{*}f$. Since $f\in
v^{M}C^{0}\otimes_{A}Q$, we have
$f\circ\Phi_{\mathfrak{M}}=\Phi_{\mathfrak{N}}\circ\varphi^{*}f\in
v^{pM}C^{1}\otimes_{A}Q$. Precomposing with $\Upsilon$ gives $v^{eah}f\in
v^{pM}C^{0}\otimes_{A}Q$. Since $C^{0}\otimes_{A}Q$ is $v$-torsion free, it
follows that $f\in v^{pM-eah}C^{0}\otimes_{A}Q\subseteq
u^{M+1}C^{0}\otimes_{A}Q$, as required.

We now move on to (2). Set $M=N-eah$. By precomposing with $\Upsilon$ we see
that $\alpha\circ\Phi_{\mathfrak{M}}\in v^{N}C^{1}\otimes_{A}Q$ implies
$\alpha\in v^{M}C^{0}\otimes_{A}Q$; from this, together with the inequality
$pM\geq N$, it is straightforward to check that

$(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}\otimes_{A}Q)=(\delta\otimes\operatorname{id}_{Q})^{-1}(v^{N}C^{1}\otimes_{A}Q)\cap
v^{M}C^{0}\otimes_{A}Q.$

Note that $M$ satisfies the condition in (1). To complete the proof we will
show that for any $M$ as in (1) and any $N\geq M+eah$ the map $\delta$ induces
an isomorphism

$(\delta\otimes\operatorname{id}_{Q})^{-1}(v^{N}C^{1}\otimes_{A}Q)\cap
v^{M}C^{0}\otimes_{A}Q\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}v^{N}C^{1}\otimes_{A}Q.$

By (1), $\delta\otimes\operatorname{id}_{Q}$ induces an injection
$(\delta\otimes\operatorname{id}_{Q})^{-1}(v^{N}C^{1}\otimes_{A}Q)\cap
v^{M}C^{0}\otimes_{A}Q\hookrightarrow v^{N}C^{1}\otimes_{A}Q$, so it is enough
to show that
$(\delta\otimes\operatorname{id}_{Q})(v^{M}C^{0}\otimes_{A}Q)\supseteq
v^{N}C^{1}\otimes_{A}Q$. Equivalently, we need to show that

$v^{N}C^{1}\otimes_{A}Q\to(C^{1}\otimes_{A}Q)/(\delta\otimes\operatorname{id}_{Q})\bigl{(}v^{M}C^{0}\otimes_{A}Q)$

is identically zero. Since the formation of cokernels is compatible with
tensor products, we see that this morphism is obtained by tensoring the
corresponding morphism

$v^{N}C^{1}\to C^{1}/\delta\bigl{(}v^{M}C^{0}\bigr{)}$

with $Q$ over $A$, so we are reduced to the case $Q=A$. (Recall from Remark
3.1.11(1) that the hypotheses of the Lemma are satisfied in this case, and
that $C^{1}$ is $v$-adically separated.)

We claim that for any $g\in v^{N}C^{1}$, we can find an $f\in v^{N-eah}C^{0}$
such that $\delta(f)-g\in v^{p(N-eah)}C^{1}$. Admitting the claim, given any
$g\in v^{N}C^{1}$, we may find $h\in v^{M}C^{0}$ with $\delta(h)=g$ by
successive approximation in the following way: Set $h_{0}=f$ for $f$ as in the
claim; then $h_{0}\in v^{N-eah}C^{0}\subseteq v^{M}C^{0}$, and
$\delta(h_{0})-g\in v^{p(N-eah)}C^{1}\subseteq v^{N+1}C^{1}$. Applying the
claim again with $N$ replaced by $N+1$, and $g$ replaced by $g-\delta(h_{0})$,
we find $f\in v^{N+1-eah}C^{0}\subseteq v^{M+1}C^{0}$ with
$\delta(f)-g+\delta(h_{0})\in v^{p(N+1-eah)}C^{1}\subseteq v^{N+1}C^{1}$.
Setting $h_{1}=h_{0}+f$, and proceeding inductively, we obtain a Cauchy
sequence converging (in the $v$-adically complete $A[[v]]$-module $C^{0}$) to
the required element $h$.

It remains to prove the claim. Since
$\delta(f)=\Phi_{\mathfrak{N}}\circ\varphi^{*}f-f\circ\Phi_{\mathfrak{M}}$,
and since if $f\in v^{N-eah}C^{0}$ then
$\Phi_{\mathfrak{N}}\circ\varphi^{*}f\in v^{p(N-eah)}C^{1}$, it is enough to
show that we can find an $f\in v^{N-eah}C^{0}$ with
$f\circ\Phi_{\mathfrak{M}}=-g$. Since $\Phi_{\mathfrak{M}}$ is injective, the
map $\Upsilon\circ\Phi_{\mathfrak{M}}$ is also multiplication by $v^{eah}$,
and so it suffices to take $f$ with $v^{eah}f=-g\circ\Upsilon\in v^{N}C^{0}$.
∎

######  Corollary 3.1.12.

Let $A$ be a Noetherian ${\mathcal{O}}/\varpi^{a}$-algebra, and let
$\mathfrak{M}$, $\mathfrak{N}$ be Breuil–Kisin modules with descent data and
$A$-coefficients. If $B$ is a finitely generated $A$-algebra, and $Q$ is a
finitely generated $B$-module, then the natural morphism of complexes of
$B$-modules

$[C^{0}(\mathfrak{N})\otimes_{A}Q\buildrel{\delta\otimes\operatorname{id}_{Q}}\over{\longrightarrow}C^{1}(\mathfrak{N})\otimes_{A}Q]\to[C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)\buildrel\delta\over{\longrightarrow}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)]$

is a quasi-isomorphism.

###### Proof.

By Remarks 3.1.11 and 2.1.6(2) we can apply Lemma 3.1.10 to both
$C^{i}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)$ and
$C^{i}(\mathfrak{N})\otimes_{A}Q$, and we see that it is enough to show that
the natural morphism of complexes

${{[\bigl{(}C^{0}(\mathfrak{N})\otimes_{A}Q\bigr{)}/(\Phi_{\mathfrak{M}}^{*}\otimes\operatorname{id}_{Q})^{-1}\bigl{(}v^{N}C^{1}(\mathfrak{N})\otimes_{A}Q\bigr{)}\buildrel\delta\over{\longrightarrow}\bigl{(}C^{1}(\mathfrak{N})\otimes_{A}Q\bigr{)}/\bigl{(}v^{N}C^{1}(\mathfrak{N})\otimes_{A}Q\bigr{)}]}}$${{[C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)/\bigl{(}\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)\bigr{)}\buildrel\delta\over{\rightarrow}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)/v^{N}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)]}}$

is a quasi-isomorphism. In fact, it is even an isomorphism. ∎

######  Proposition 3.1.13.

Let $A$ be a ${\mathcal{O}}/\varpi^{a}$-algebra for some $a\geq 1$, and let
$\mathfrak{M}$, $\mathfrak{N}$ be Breuil–Kisin modules with descent data and
$A$-coefficients. Then
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ and
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}/u^{i}\mathfrak{N})$
for $i\geq 1$ are finitely presented $A$-modules.

If furthermore $A$ is Noetherian, then
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ and
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}/u^{i}\mathfrak{N})$
for $i\geq 1$ are also finitely presented (equivalently, finitely generated)
$A$-modules.

###### Proof.

The statements for $\mathfrak{N}/u^{i}\mathfrak{N}$ follow easily from those
for $\mathfrak{N}$, by considering the short exact sequence $0\to
u^{i}\mathfrak{N}\to\mathfrak{N}\to\mathfrak{N}/u^{i}\mathfrak{N}\to 0$ in
$\mathcal{K}(A)$ and applying Corollary 3.1.7. By Corollary 3.1.8, it is
enough to consider the cohomology of the complex $C^{\bullet}$. By Lemma
3.1.10 with $Q=A$, the cohomology of $C^{\bullet}$ agrees with the cohomology
of the induced complex

$C^{0}/\bigl{(}(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}))\to
C^{1}/v^{N}C^{1},$

for an appropriately chosen value of $N$. It follows that for an appropriately
chosen value of $N$,
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ can be
computed as the cokernel of the induced morphism $C^{0}/v^{N}C^{0}\to
C^{1}/v^{N}C^{1}$.

Under our hypothesis on $\mathfrak{N}$, $C^{0}/v^{N}C^{0}$ and
$C^{1}/v^{N}C^{1}$ are finitely generated projective $A$-modules, and thus
finitely presented. It follows that
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ is
finitely presented.

In the case that $A$ is furthermore assumed to be Noetherian, it is enough to
note that since
$v^{N}C^{0}\subseteq(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1})$, the quotient
$C^{0}/\bigl{(}(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1})\bigr{)}$ is a
finitely generated $A$-module. ∎

######  Proposition 3.1.14.

Let $A$ be a ${\mathcal{O}}/\varpi^{a}$-algebra for some $a\geq 1$, and let
$\mathfrak{M}$ and $\mathfrak{N}$ be Breuil–Kisin modules with descent data
and $A$-coefficients. Let $B$ be an $A$-algebra, and let
$f_{B}:\mathfrak{M}\,\widehat{\otimes}_{A}B\to\mathfrak{N}\,\widehat{\otimes}_{A}B$
be a morphism of Breuil–Kisin modules with $B$-coefficients.

Then there is a finite type $A$-subalgebra $B^{\prime}$ of $B$ and a morphism
of Breuil–Kisin modules
$f_{B^{\prime}}:\mathfrak{M}\,\widehat{\otimes}_{A}B^{\prime}\to\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime}$
such that $f_{B}$ is the base change of $f_{B^{\prime}}$.

###### Proof.

By Lemmas 3.1.3 and 3.1.9 (the latter applied with $Q=B$) we can and do think
of $f_{B}$ as being an element of the kernel of
$\delta:C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}B)\to
C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}B)$, the complex $C^{\bullet}$ here
and throughout this proof denoting $C^{\bullet}_{\mathfrak{M}}$ as usual.

Fix $N$ as in Lemma 3.1.10, and write $\overline{f}_{B}$ for the corresponding
element of
$C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}B)/v^{N}=(C^{0}(\mathfrak{N})/v^{N})\otimes_{A}B$
(this equality following easily from the assumption that $\mathfrak{M}$ and
$\mathfrak{N}$ are projective $\mathfrak{S}_{A}$-modules of finite rank).
Since $C^{0}(\mathfrak{N})/v^{N}$ is a projective $A$-module of finite rank,
it follows that for some finite type $A$-subalgebra $B^{\prime}$ of $B$, there
is an element
$\overline{f}_{B^{\prime}}\in(C^{0}(\mathfrak{N})/v^{N})\otimes_{A}B^{\prime}=C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime})/v^{N}$
such that $\overline{f}_{B^{\prime}}\otimes_{B^{\prime}}B=\overline{f}_{B}$.
Denote also by $\overline{f}_{B^{\prime}}$ the induced element of

$C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime})/\bigl{(}\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime})).$

By Lemma 3.1.10 (and Lemma 3.1.3) we have a commutative diagram with exact
rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(C^{\bullet}(\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime})/\bigl{(}(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime}))\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}B^{\prime})/v^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(C^{\bullet}(\mathfrak{N}\,\widehat{\otimes}_{A}B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}B)/\bigl{(}(\Phi_{\mathfrak{M}}^{*})^{-1}(v^{N}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}B))\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}B)/v^{N}}$

in which the vertical arrows are induced by
$\,\widehat{\otimes}_{B^{\prime}}B$. By a diagram chase we only need to show
that $\delta(\overline{f}_{B^{\prime}})=0$. Since $\delta(f_{B})=0$, it is
enough to show that the right hand vertical arrow is an injection. This arrow
can be rewritten as the tensor product of the injection of $A$-algebras
$B^{\prime}\hookrightarrow B$ with the flat (even projective of finite rank)
$A$-module $C^{1}(\mathfrak{N})/v^{N}$, so the result follows. ∎

We have the following key base-change result for $\operatorname{Ext}^{1}$’s of
Breuil–Kisin modules with descent data.

######  Proposition 3.1.15.

Suppose that $\mathfrak{M}$ and $\mathfrak{N}$ are Breuil–Kisin modules with
descent data and coefficients in a ${\mathcal{O}}/\varpi^{a}$-algebra $A$.
Then for any $A$-algebra $B$, and for any $B$-module $Q$, there are natural
isomorphisms
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}Q\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{1}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B)\otimes_{B}Q\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{1}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}Q).$

###### Proof.

We first prove the lemma in the case of an $A$-module $Q$. It follows from
Lemmas 3.1.5 and 3.1.10 that we may compute
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ as the
cokernel of the morphism

$C^{0}(\mathfrak{N})/v^{N}C^{0}(\mathfrak{N})\buildrel\delta\over{\longrightarrow}C^{1}(\mathfrak{N})/v^{N}C^{1}(\mathfrak{N}),$

for some sufficiently large value of $N$ (not depending on $\mathfrak{N}$),
and hence that we may compute
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}Q$
as the cokernel of the morphism

$\bigl{(}C^{0}(\mathfrak{N})/v^{N}C^{0}(\mathfrak{N})\bigr{)}\otimes_{A}Q\buildrel\delta\over{\longrightarrow}\bigl{(}C^{1}(\mathfrak{N})/v^{N}C^{1}(\mathfrak{N})\bigr{)}\otimes_{A}Q.$

We may similarly compute
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\,\widehat{\otimes}_{A}Q)$
as the cokernel of the morphism

$C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)/v^{N}C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)\buildrel\delta\over{\longrightarrow}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)/v^{N}C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}Q).$

(Remark 2.1.6 (2) shows that $\mathfrak{N}\,\widehat{\otimes}_{A}Q$ satisfies
the necessary hypotheses for Lemma 3.1.10 to apply.) Once we note that the
natural morphism

$\bigl{(}C^{i}(\mathfrak{N})/v^{N}C^{i}(\mathfrak{N})\bigr{)}\otimes_{A}Q\to
C^{i}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)/v^{N}C^{i}(\mathfrak{N}\,\widehat{\otimes}_{A}Q)$

is an isomorphism for $i=0$ and $1$ (because $\mathfrak{M}$ is a finitely
generated projective $\mathfrak{S}_{A}$-module), we obtain the desired
isomorphism

$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}Q\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\,\widehat{\otimes}_{A}Q).$

If $B$ is an $A$-algebra, and $Q$ is a $B$-module, then by Lemma 3.1.9 there
is a natural isomorphism

$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\,\widehat{\otimes}_{A}Q)\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{1}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}Q);$

combined with the preceding base-change result, this yields one of our claimed
isomorphisms, namely

$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}Q\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{1}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}Q).$

Taking $Q$ to be $B$ itself, we then obtain an isomorphism

$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}B\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{1}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B).$

This allows us to identify
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}Q$,
which is naturally isomorphic to
$\bigr{(}\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}B\bigr{)}\otimes_{B}Q$,
with
$\operatorname{Ext}^{1}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B)\otimes_{B}Q$,
yielding the second claimed isomorphism. ∎

In contrast to the situation for extensions (_cf_. Proposition 3.1.15), the
formation of homomorphisms between Breuil–Kisin modules is in general not
compatible with arbitrary base-change, as the following example shows.

######  Example 3.1.16.

Take $A=({\mathbf{Z}}/p{\mathbf{Z}})[x^{\pm 1},y^{\pm 1}]$, and let
$\mathfrak{M}_{x}$ be the free Breuil–Kisin module of rank one and
$A$-coefficients with $\varphi(e)=xe$ for some generator $e$ of
$\mathfrak{M}_{x}$. Similarly define $\mathfrak{M}_{y}$ with
$\varphi(e^{\prime})=ye^{\prime}$ for some generator $e^{\prime}$ of
$\mathfrak{M}_{y}$. Then
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M}_{x},\mathfrak{M}_{y})=0$. On
the other hand, if $B=A/(x-y)$ then $\mathfrak{M}_{x}\,\widehat{\otimes}_{A}B$
and $\mathfrak{M}_{y}\,\widehat{\otimes}_{A}B$ are isomorphic, so that
$\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}_{x}\,\widehat{\otimes}B,\mathfrak{M}_{y}\,\widehat{\otimes}B)\not\cong\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M}_{x},\mathfrak{M}_{y})\otimes_{A}B$.

However, it is possible to establish such a compatibility in some settings.
Corollary 3.1.18, which gives a criterion for the vanishing of
$\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B)$
for any $A$-algebra $B$, is a first example of a result in this direction.
Lemma 3.1.20 deals with flat base change, and Lemma 3.1.21, which will be
important in Section 3.3, proves that formation of homomorphisms is compatible
with base-change over a dense open subscheme of $\operatorname{Spec}A$.

######  Proposition 3.1.17.

Suppose that $A$ is a Noetherian ${\mathcal{O}}/\varpi^{a}$-algebra, and that
$\mathfrak{M}$ and $\mathfrak{N}$ are objects of $\mathcal{K}(A)$ that are
finitely generated over $\mathfrak{S}_{A}$ (or, equivalently, over $A[[u]]$).
Suppose also that $\mathfrak{N}$ is a flat $\mathfrak{S}_{A}$-module. Consider
the following conditions:

  1. (1)

$\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B)=0$
for any finite type $A$-algebra $B$.

  2. (2)

$\operatorname{Hom}_{\mathcal{K}(\kappa(\mathfrak{m}))}\bigl{(}\mathfrak{M}\otimes_{A}\kappa(\mathfrak{m}),\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)}=0$
for each maximal ideal $\mathfrak{m}$ of $A$.

  3. (3)

$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}Q)=0$
for any finitely generated $A$-module $Q$.

Then we have (1)$\implies$(2)$\iff$(3). If $A$ is furthermore Jacobson, then
all three conditions are equivalent.

###### Proof.

If $\mathfrak{m}$ is a maximal ideal of $A$, then $\kappa(\mathfrak{m})$ is
certainly a finite type $A$-algebra, and so evidently (1) implies (2). It is
even a finitely generated $A$-module, and so also (2) follows from (3).

We next prove that (2) implies (3). To this end, recall that if $A$ is any
ring, and $M$ is any $A$-module, then $M$ injects into the product of its
localizations at all maximal ideals. If $A$ is Noetherian, and $M$ is finitely
generated, then, by combining this fact with the Artin–Rees Lemma, we see that
$M$ embeds into the product of its completions at all maximal ideals. Another
way to express this is that, if $I$ runs over all cofinite length ideals in
$A$ (i.e. all ideals for which $A/I$ is finite length), then $M$ embeds into
the projective limit of the quotients $M/IM$ (the point being that this
projective limit is the same as the product over all $\mathfrak{m}$-adic
completions). We are going to apply this observation with $A$ replaced by
$\mathfrak{S}_{A}$, and with $M$ taken to be $\mathfrak{N}\otimes_{A}Q$ for
some finitely generated $A$-module $Q$.

In $A[[u]]$, one sees that $u$ lies in the Jacobson radical (because $1+fu$ is
invertible in $A[[u]]$ for every $f\in A[[u]]$), and thus in every maximal
ideal, and so the maximal ideals of $A[[u]]$ are of the form
$(\mathfrak{m},u)$, where $\mathfrak{m}$ runs over the maximal ideals of $A$.
Thus the ideals of the form $(I,u^{n})$, where $I$ is a cofinite length ideal
in $A$, are cofinal in all cofinite length ideals in $A[[u]]$. Since
$\mathfrak{S}_{A}$ is finite over $A[[u]]$, we see that the ideals $(I,u^{n})$
in $\mathfrak{S}_{A}$ are also cofinal in all cofinite length ideals in
$A[[u]]$. Since $A[[u]]$, and hence $\mathfrak{S}_{A}$, is furthermore
Noetherian when $A$ is, we see that if $Q$ is a finitely generated $A$-module,
and $\mathfrak{N}$ is a finitely generated $\mathfrak{S}_{A}$-module, then
$\mathfrak{N}\otimes_{A}(Q/IQ)$ is $u$-adically complete, for any cofinite
length ideal $I$ in $A$, and hence equal to the limit over $n$ of
$\mathfrak{N}\otimes_{A}Q/(I,u^{n})$. Putting this together with the
observation of the preceding paragraph, we see that the natural morphism

$\mathfrak{N}\otimes_{A}Q\to\varprojlim_{I}\mathfrak{N}\otimes_{A}(Q/IQ)$

(where $I$ runs over all cofinite length ideals of $A$) is an embedding. The
induced morphism

$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}Q)\to\varprojlim_{I}\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}(Q/IQ))$

is then evidently also an embedding.

Thus, to conclude that
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}Q)$
vanishes, it suffices to show that
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}(Q/IQ))$
vanishes for each cofinite length ideal $I$ in $A$. An easy induction (using
the flatness of $\mathfrak{N}$) on the length of $A/I$ reduces this to showing
that
$\operatorname{Hom}_{\mathcal{K}(A)}\bigl{(}\mathfrak{M},\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)},$
or, equivalently,
$\operatorname{Hom}_{\mathcal{K}(\kappa(\mathfrak{m}))}\bigl{(}\mathfrak{M}\otimes_{A}\kappa(\mathfrak{m}),\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)},$
vanishes for each maximal ideal $\mathfrak{m}$. Since this is the hypothesis
of (2), we see that indeed (2) implies (3).

It remains to show that (3) implies (1) when $A$ is Jacobson. Applying the
result “(2) implies (3)” (with $A$ replaced by $B$, and taking $Q$ in (3) to
be $B$ itself as a $B$-module) to $\mathfrak{M}\,\widehat{\otimes}_{A}B$ and
$\mathfrak{N}\,\widehat{\otimes}_{A}B$, we see that it suffices to prove the
vanishing of

$\operatorname{Hom}_{\mathcal{K}(B)}\bigl{(}(\mathfrak{M}\,\widehat{\otimes}_{A}B)\otimes_{B}\kappa(\mathfrak{n}),(\mathfrak{N}\,\widehat{\otimes}_{A}B)\otimes_{B}\kappa(\mathfrak{n})\bigr{)}=\operatorname{Hom}_{\mathcal{K}(A)}\bigl{(}\mathfrak{M},\mathfrak{N}\,\widehat{\otimes}_{A}\kappa(\mathfrak{n})\bigr{)}$

for each maximal ideal $\mathfrak{n}$ of $B$. Since $A$ is Jacobson, the field
$\kappa(\mathfrak{n})$ is in fact a finitely generated $A$-module, hence
$\mathfrak{N}\,\widehat{\otimes}\kappa(\mathfrak{n})=\mathfrak{N}\otimes_{A}\kappa(\mathfrak{n})$,
and so the desired vanishing is a special case of (3). ∎

######  Corollary 3.1.18.

If $A$ is a Noetherian and Jacobson ${\mathcal{O}}/\varpi^{a}$-algebra, and if
$\mathfrak{M}$ and $\mathfrak{N}$ are Breuil–Kisin modules with descent data
and $A$-coefficients, then the following three conditions are equivalent:

  1. (1)

$\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B)=0$
for any $A$-algebra $B$.

  2. (2)

$\operatorname{Hom}_{\mathcal{K}(\kappa(\mathfrak{m}))}\bigl{(}\mathfrak{M}\otimes_{A}\kappa(\mathfrak{m}),\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)}=0$
for each maximal ideal $\mathfrak{m}$ of $A$.

  3. (3)

$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}Q)=0$
for any finitely generated $A$-module $Q$.

###### Proof.

By Proposition 3.1.17, we need only prove that if
$\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B)$
vanishes for all finitely generated $A$-algebras $B$, then it vanishes for all
$A$-algebras $B$. This is immediate from Proposition 3.1.14. ∎

######  Corollary 3.1.19.

Suppose that $\mathfrak{M}$ and $\mathfrak{N}$ are Breuil–Kisin modules with
descent data and coefficients in a Noetherian
${\mathcal{O}}/\varpi^{a}$-algebra $A$, and that furthermore
$\operatorname{Hom}_{\mathcal{K}(A)}\bigl{(}\mathfrak{M}\otimes_{A}\kappa(\mathfrak{m}),\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)}$
vanishes for each maximal ideal $\mathfrak{m}$ of $A$. Then the $A$-module
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ is
projective of finite rank.

###### Proof.

By Proposition 3.1.13, in order to prove that
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ is
projective of finite rank over $A$, it suffices to prove that it is flat over
$A$. For this, it suffices to show that
$Q\mapsto\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}Q$
is exact when applied to finitely generated $A$-modules $Q$. Proposition
3.1.15 (together with Remark 2.1.6 (1)) allows us to identify this functor
with the functor
$Q\mapsto\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}Q).$
Note that the functor $Q\mapsto\mathfrak{N}\otimes_{A}Q$ is an exact functor
of $Q$, since $\mathfrak{S}_{A}$ is a flat $A$-module (as $A$ is Noetherian;
see Remark 2.1.3(3)). Thus, taking into account Corollary 3.1.7, we see that
it suffices to show that
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}\otimes_{A}Q)=0$
for each finitely generated $A$-module $Q$, under the hypothesis that
$\operatorname{Hom}_{\mathcal{K}(A)}\bigl{(}\mathfrak{M}\otimes_{A}\kappa(\mathfrak{m}),\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)}=0$
for each maximal ideal $\mathfrak{m}$ of $A$. This is the implication (2)
$\implies$ (3) of Proposition 3.1.17. ∎

######  Lemma 3.1.20.

Suppose that $\mathfrak{M}$ is a Breuil–Kisin modules with descent data and
coefficients in a Noetherian ${\mathcal{O}}/\varpi^{a}$-algebra $A$. Suppose
that $\mathfrak{N}$ is either a Breuil–Kisin module with $A$-coefficients, or
that $\mathfrak{N}=\mathfrak{N}^{\prime}/u^{N}\mathfrak{N}^{\prime}$, where
$\mathfrak{N}^{\prime}$ a Breuil–Kisin module with $A$-coefficients and $N\geq
1$. Then, if $B$ is a finitely generated flat $A$-algebra, we have a natural
isomorphism

$\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B)\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}B.$

###### Proof.

By Corollary 3.1.8 and the flatness of $B$, we have a left exact sequence

$0\to\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}B\to
C^{0}(\mathfrak{N})\otimes_{A}B\to C^{1}(\mathfrak{N})\otimes_{A}B$

and therefore (applying Corollary 3.1.12 to treat the case that $\mathfrak{N}$
is projective) a left exact sequence

$0\to\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\otimes_{A}B\to
C^{0}(\mathfrak{N}\,\widehat{\otimes}_{A}B)\to
C^{1}(\mathfrak{N}\,\widehat{\otimes}_{A}B).$

The result follows from Corollary 3.1.8 and Lemma 3.1.9. ∎

######  Lemma 3.1.21.

Suppose that $\mathfrak{M}$ is a Breuil–Kisin module with descent data and
coefficients in a Noetherian ${\mathcal{O}}/\varpi^{a}$-algebra $A$ which is
furthermore a domain. Suppose also that $\mathfrak{N}$ is either a
Breuil–Kisin module with $A$-coefficients, or that
$\mathfrak{N}=\mathfrak{N}^{\prime}/u^{N}\mathfrak{N}^{\prime}$, where
$\mathfrak{N}^{\prime}$ is a Breuil–Kisin module with $A$-coefficients and
$N\geq 1$. Then there is some nonzero $f\in A$ with the following property:
writing $\mathfrak{M}_{A_{f}}=\mathfrak{M}\,\widehat{\otimes}_{A}A_{f}$ and
$\mathfrak{N}_{A_{f}}=\mathfrak{N}\,\widehat{\otimes}_{A}A_{f}$, then for any
finitely generated $A_{f}$-algebra $B$, and any finitely generated $B$-module
$Q$, there are natural isomorphisms

$\operatorname{Hom}_{\mathcal{K}(A_{f})}(\mathfrak{M}_{A_{f}},\mathfrak{N}_{A_{f}})\otimes_{A_{f}}Q\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B,\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B)\otimes_{B}Q\\\
\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B,\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}Q).$

###### Proof of Lemma 3.1.21..

Note that since $A$ is Noetherian, by Remark 2.1.3(3) we see that
$\mathfrak{N}$ is $A$-flat. By Corollary 3.1.8 we have an exact sequence

$0\to\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\to
C^{0}(\mathfrak{N})\to
C^{1}(\mathfrak{N})\to\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\to
0.$

Since by assumption $\mathfrak{M}$ is a projective $\mathfrak{S}_{A}$-module,
and $\mathfrak{N}$ is a flat $A$-module, the $C^{i}(\mathfrak{N})$ are also
flat $A$-modules.

By Proposition 3.1.13,
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})$ is a
finitely generated $A$-module, so by the generic freeness theorem [Sta13, Tag
051R] there is some nonzero $f\in A$ such that
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})_{f}$ is
free over $A_{f}$.

Since localisation is exact, we obtain an exact sequence

$0\to\operatorname{Hom}_{\mathcal{K}(A_{f})}(\mathfrak{M},\mathfrak{N})_{f}\to
C^{0}(\mathfrak{N})_{f}\to
C^{1}(\mathfrak{N})_{f}\to\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})_{f}\to
0$

and therefore (applying Corollary 3.1.12 to treat the case that $\mathfrak{N}$
is a Breuil–Kisin module) an exact sequence

$0\to\operatorname{Hom}_{\mathcal{K}(A_{f})}(\mathfrak{M}_{A_{f}},\mathfrak{N}_{A_{f}})\to
C^{0}(\mathfrak{N}_{A_{f}})\to
C^{1}(\mathfrak{N}_{A_{f}})\to\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})_{f}\to
0.$

Since the last three terms are flat over $A_{f}$, this sequence remains exact
upon tensoring over $A_{f}$ with $Q$. Applying Corollary 3.1.12 again to treat
the case that $\mathfrak{N}$ is a Breuil–Kisin module, we see that in
particular we have a left exact sequence

$0\to\operatorname{Hom}_{\mathcal{K}(A_{f})}(\mathfrak{M}_{A_{f}},\mathfrak{N}_{A_{f}})\otimes_{A_{f}}Q\to
C^{0}(\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}Q)\to
C^{1}(\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}Q),$

and Corollary 3.1.8 together with Lemma 3.1.9 yield one of the desired
isomorphisms, namely

$\operatorname{Hom}_{\mathcal{K}(A_{f})}(\mathfrak{M}_{A_{f}},\mathfrak{N}_{A_{f}})\otimes_{A_{f}}Q\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B,\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}Q).$

If we consider the case when $Q=B$, we obtain an isomorphism

$\operatorname{Hom}_{\mathcal{K}(A_{f})}(\mathfrak{M}_{A_{f}},\mathfrak{N}_{A_{f}})\otimes_{A_{f}}B\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B,\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B).$

Rewriting the tensor product $\text{--}\otimes_{A_{f}}Q$ as
$\text{--}\otimes_{A_{f}}B\otimes_{B}Q,$ we then find that

$\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B,\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B)\otimes_{B}Q\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(B)}(\mathfrak{M}_{A_{f}}\,\widehat{\otimes}_{A_{f}}B,\mathfrak{N}_{A_{f}}\,\widehat{\otimes}_{A_{f}}Q),$

which gives the second desired isomorphism. ∎

Variants on the preceding result may be proved using other versions of the
generic freeness theorem.

######  Example 3.1.22.

Returning to the setting of Example 3.1.16, one can check using Corollary
3.1.18 that the conclusion of Lemma 3.1.21 (for
$\mathfrak{M}=\mathfrak{M}_{x}$ and $\mathfrak{N}=\mathfrak{M}_{y})$ holds
with $f=x-y$. In this case all of the resulting $\operatorname{Hom}$ groups
vanish (_cf_. also the proof of Lemma 3.3.7). It then follows from Corollary
3.1.19 that
$\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})_{f}$ is
projective over $A_{f}$, so that the proof of Lemma 3.1.21 even goes through
with this choice of $f$.

As well as considering homomorphisms and extensions of Breuil–Kisin modules,
we need to consider the homomorphisms and extensions of their associated étale
$\varphi$-modules; recall that the passage to associated étale
$\varphi$-modules amounts to inverting $u$, and so we briefly discuss this
process in the general context of the category $\mathcal{K}(A)$.

We let $\mathcal{K}(A)[1/u]$ denote the full subcategory of $\mathcal{K}(A)$
consisting of objects on which multiplication by $u$ is invertible. We may
equally well regard it as the category of left
$\mathfrak{S}_{A}[1/u][F,\operatorname{Gal}(K^{\prime}/K)]$-modules (this
notation being interpreted in the evident manner). There are natural
isomorphisms (of bi-modules)

(3.1.23)
$\mathfrak{S}_{A}[1/u]\otimes_{\mathfrak{S}_{A}}\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]\buildrel\sim\over{\longrightarrow}\mathfrak{S}_{A}[1/u][F,\operatorname{Gal}(K^{\prime}/K)]$

and

(3.1.24)
$\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]\otimes_{\mathfrak{S}_{A}}\mathfrak{S}_{A}[1/u]\buildrel\sim\over{\longrightarrow}\mathfrak{S}_{A}[1/u][F,\operatorname{Gal}(K^{\prime}/K)].$

Thus (since $\mathfrak{S}_{A}\to\mathfrak{S}_{A}[1/u]$ is a flat morphism of
commutative rings) the morphism of rings
$\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]\to\mathfrak{S}_{A}[1/u][F,\operatorname{Gal}(K^{\prime}/K)]$
is both left and right flat.

If $\mathfrak{M}$ is an object of $\mathcal{K}(A)$, then we see from (3.1.23)
that
$\mathfrak{M}[1/u]:=\mathfrak{S}_{A}[1/u]\otimes_{\mathfrak{S}_{A}}\mathfrak{M}\buildrel\sim\over{\longrightarrow}\mathfrak{S}_{A}[1/u][F,\operatorname{Gal}(K^{\prime}/K)]\otimes_{\mathfrak{S}_{A}[F,\operatorname{Gal}(K^{\prime}/K)]}\mathfrak{M}$
is naturally an object of $\mathcal{K}(A)[1/u]$. Our preceding remarks about
flatness show that $\mathfrak{M}\mapsto\mathfrak{M}[1/u]$ is an exact functor
$\mathcal{K}(A)\to\mathcal{K}(A)[1/u]$.

######  Lemma 3.1.25.

  1. (1)

If $M$ and $N$ are objects of $\mathcal{K}(A)[1/u]$, then there is a natural
isomorphism

$\operatorname{Ext}^{i}_{\mathcal{K}(A)[1/u]}(M,N)\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{i}_{\mathcal{K}(A)}(M,N).$

  2. (2)

If $\mathfrak{M}$ is an object of $\mathcal{K}(A)$ and $N$ is an object of
$\mathcal{K}(A)[1/u]$, then there is a natural isomorphism

$\operatorname{Ext}^{i}_{\mathcal{K}(A)}(\mathfrak{M},N)\buildrel\sim\over{\longrightarrow}\operatorname{Ext}^{i}_{\mathcal{K}(A)}(\mathfrak{M}[1/u],N),$

for all $i\geq 0$.

###### Proof.

The morphism of (1) can be understood in various ways; for example, by
thinking in terms of Yoneda Exts, and recalling that $\mathcal{K}(A)[1/u]$ is
a full subcategory of $\mathcal{K}(A).$ If instead we think in terms of
projective resolutions, we can begin with a projective resolution
$\mathfrak{P}^{\bullet}\to M$ in $\mathcal{K}(A)$, and then consider the
induced projective resolution $\mathfrak{P}^{\bullet}[1/u]$ of $M[1/u]$.
Noting that $M[1/u]\buildrel\sim\over{\longrightarrow}M$ for any object $M$ of
$\mathcal{K}(A)[1/u]$, we then find (via tensor adjunction) that
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{P}^{\bullet},N)\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(A)[1/u]}(\mathfrak{P}^{\bullet}[1/u],N)$,
which induces the desired isomorphism of $\operatorname{Ext}$’s by passing to
cohomology.

Taking into account the isomorphism of (1), the claim of (2) is a general fact
about tensoring over a flat ring map (as can again be seen by considering
projective resolutions). ∎

######  Remark 3.1.26.

The preceding lemma is fact an automatic consequence of the abstract
categorical properties of our situation: the functor
$\mathfrak{M}\mapsto\mathfrak{M}[1/u]$ is left adjoint to the inclusion
$\mathcal{K}(A)[1/u]\subset\mathcal{K}(A),$ and restricts to (a functor
naturally equivalent to) the identity functor on $\mathcal{K}(A)[1/u]$.

The following lemma expresses the Hom between étale $\varphi$-modules arising
from Breuil–Kisin modules in terms of a certain direct limit.

######  Lemma 3.1.27.

Suppose that $\mathfrak{M}$ is a Breuil–Kisin module with descent data in a
Noetherian ${\mathcal{O}}/\varpi^{a}$-algebra $A$, and that $\mathfrak{N}$ is
an object of $\mathcal{K}(A)$ which is finitely generated and $u$-torsion free
as an $\mathfrak{S}_{A}$-module. Then there is a natural isomorphism

$\varinjlim_{i}\operatorname{Hom}_{\mathcal{K}(A)}(u^{i}\mathfrak{M},\mathfrak{N})\buildrel\sim\over{\longrightarrow}\operatorname{Hom}_{\mathcal{K}(A)[1/u]}(\mathfrak{M}[1/u],\mathfrak{N}[1/u]),$

where the transition maps are induced by the inclusions
$u^{i+1}\mathfrak{M}\subset u^{i}\mathfrak{M}$.

######  Remark 3.1.28.

Note that since $\mathfrak{N}$ is $u$-torsion free, the transition maps in the
colimit are injections, so the colimit is just an increasing union.

###### Proof.

There are compatible injections
$\operatorname{Hom}_{\mathcal{K}(A)}(u^{i}\mathfrak{M},\mathfrak{N})\to\operatorname{Hom}_{\mathcal{K}(A)[1/u]}(\mathfrak{M}[1/u],\mathfrak{N}[1/u])$,
taking
$f^{\prime}\in\operatorname{Hom}_{\mathcal{K}(A)}(u^{i}\mathfrak{M},\mathfrak{N})$
to $f\in\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}[1/u])$
where $f(m)=u^{-i}f^{\prime}(u^{i}m)$. Conversely, given
$f\in\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}[1/u])$,
there is some $i$ such that $f(\mathfrak{M})\subset u^{-i}\mathfrak{N}$, as
required. ∎

We have the following analogue of Proposition 3.1.17.

######  Corollary 3.1.29.

Suppose that $\mathfrak{M}$ and $\mathfrak{N}$ are Breuil–Kisin modules with
descent data in a Noetherian ${\mathcal{O}}/\varpi^{a}$-algebra $A$. Consider
the following conditions:

  1. (1)

$\operatorname{Hom}_{\mathcal{K}(B)[1/u]}\bigl{(}(\mathfrak{M}\,\widehat{\otimes}_{A}B)[1/u],(\mathfrak{N}\,\widehat{\otimes}_{A}B)[1/u]\bigr{)}=0$
for any finite type $A$-algebra $B$.

  2. (2)

$\operatorname{Hom}_{\mathcal{K}(\kappa(\mathfrak{m}))[1/u]}\bigl{(}(\mathfrak{M}\otimes_{A}\kappa(\mathfrak{m}))[1/u],(\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m}))[1/u]\bigr{)}=0$
for each maximal ideal $\mathfrak{m}$ of $A$.

  3. (3)

$\operatorname{Hom}_{\mathcal{K}(A)[1/u]}\bigl{(}\mathfrak{M}[1/u],(\mathfrak{N}\otimes_{A}Q)[1/u]\bigr{)}=0$
for any finitely generated $A$-module $Q$.

Then we have (1)$\implies$(2)$\iff$(3). If $A$ is furthermore Jacobson, then
all three conditions are equivalent.

###### Proof.

By Lemma 3.1.27, the three conditions are respectively equivalent to the
following conditions.

  1. (1′)

$\operatorname{Hom}_{\mathcal{K}(B)}\bigl{(}u^{i}(\mathfrak{M}\,\widehat{\otimes}_{A}B),\mathfrak{N}\,\widehat{\otimes}_{A}B\bigr{)}=0$
for any finite type $A$-algebra $B$ and all $i\geq 0$.

  2. (2′)

$\operatorname{Hom}_{\mathcal{K}(\kappa(\mathfrak{m}))}\bigl{(}u^{i}(\mathfrak{M}\otimes_{A}\kappa(\mathfrak{m})),\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)}=0$
for each maximal ideal $\mathfrak{m}$ of $A$ and all $i\geq 0$.

  3. (3′)

$\operatorname{Hom}_{\mathcal{K}(A)}\bigl{(}u^{i}\mathfrak{M},\mathfrak{N}\otimes_{A}Q\bigr{)}=0$
for any finitely generated $A$-module $Q$ and all $i\geq 0$.

Since $\mathfrak{M}$ is projective, the first two conditions are in turn
equivalent to

  1. (1′′)

$\operatorname{Hom}_{\mathcal{K}(B)}\bigl{(}(u^{i}\mathfrak{M})\,\widehat{\otimes}_{A}B,\mathfrak{N}\,\widehat{\otimes}_{A}B\bigr{)}=0$
for any finite type $A$-algebra $B$ and all $i\geq 0$.

  2. (2′′)

$\operatorname{Hom}_{\mathcal{K}(\kappa(\mathfrak{m}))}\bigl{(}(u^{i}\mathfrak{M})\otimes_{A}\kappa(\mathfrak{m}),\mathfrak{N}\otimes_{A}\kappa(\mathfrak{m})\bigr{)}=0$
for each maximal ideal $\mathfrak{m}$ of $A$ and all $i\geq 0$.

The result then follows from Proposition 3.1.17. ∎

######  Definition 3.1.30.

If $\mathfrak{M}$ and $\mathfrak{N}$ are objects of $\mathcal{K}(A)$, then we
define

$\operatorname{ker-
Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}):=\ker\bigl{(}\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\to\operatorname{Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M}[1/u],\mathfrak{N}[1/u])\bigr{)}.$

The point of this definition is to capture, in the setting of Lemma 2.2.4, the
non-split extensions of Breuil–Kisin modules whose underlying extension of
Galois representations is split.

Suppose now that $\mathfrak{M}$ is a Breuil–Kisin module. The exact sequence
in $\mathcal{K}(A)$

$0\to\mathfrak{N}\to\mathfrak{N}[1/u]\to\mathfrak{N}[1/u]/\mathfrak{N}\to 0$

gives an exact sequence of complexes

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{0}(\mathfrak{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{0}(\mathfrak{N}[1/u])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{0}(\mathfrak{N}[1/u]/\mathfrak{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{1}(\mathfrak{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{1}(\mathfrak{N}[1/u])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{1}(\mathfrak{N}[1/u]/\mathfrak{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

It follows from Corollary 3.1.8, Lemma 3.1.25(2), and the snake lemma that we
have an exact sequence

(3.1.31)
$\begin{split}0\to\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\to\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}[1/u])\qquad\qquad\\\
\to\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}[1/u]/\mathfrak{N})\to\operatorname{ker-
Ext}^{1}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N})\to 0.\end{split}$

######  Lemma 3.1.32.

If $\mathfrak{M}$, $\mathfrak{N}$ are Breuil–Kisin modules with descent data
and coefficients in a Noetherian ${\mathcal{O}}/\varpi^{a}$-algebra $A$, and
$\mathfrak{N}$ has height at most $h$, then $f(\mathfrak{M})$ is killed by
$u^{i}$ for any
$f\in\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}[1/u]/\mathfrak{N})$
and any $i\geq\lfloor e^{\prime}ah/(p-1)\rfloor$.

###### Proof.

Suppose that $f$ is an element of
$\operatorname{Hom}_{\mathcal{K}(A)}(\mathfrak{M},\mathfrak{N}[1/u]/\mathfrak{N})$.
Then $f(\mathfrak{M})$ is a finitely generated submodule of
$\mathfrak{N}[1/u]/\mathfrak{N}$, and it therefore killed by $u^{i}$ for some
$i\geq 0$. Choosing $i$ to be the exponent of $f(\mathfrak{M})$ (that is,
choosing $i$ to be minimal), it follows that
$(\varphi^{*}f)(\varphi^{*}\mathfrak{M})$ has exponent precisely $ip$. (From
the choice of $i$, we see that $u^{i-1}f(\mathfrak{M})$ is nonzero but killed
by $u$, i.e., it is just a $W(k^{\prime})\otimes A$-module, and so its
pullback by $\varphi:\mathfrak{S}_{A}\to\mathfrak{S}_{A}$ has exponent
precisely $p$. Then by the flatness of
$\varphi:\mathfrak{S}_{A}\to\mathfrak{S}_{A}$ we have
$u^{ip-1}(\varphi^{*}f)(\varphi^{*}\mathfrak{M})=u^{p-1}\varphi^{*}(u^{i-1}f(\mathfrak{M}))\neq
0$.)

We claim that $u^{i+e^{\prime}ah}(\varphi^{*}f)(\varphi^{*}\mathfrak{M})=0$;
admitting this, we deduce that $i+e^{\prime}ah\geq ip$, as required. To see
the claim, take $x\in\varphi^{*}\mathfrak{M}$, so that
$\Phi_{\mathfrak{N}}((u^{i}\varphi^{*}f)(x))=u^{i}f(\Phi_{\mathfrak{M}}(x))=0$.
It is therefore enough to show that the kernel of

$\Phi_{\mathfrak{N}}:\varphi^{*}\mathfrak{N}[1/u]/\varphi^{*}\mathfrak{N}\to\mathfrak{N}[1/u]/\mathfrak{N}$

is killed by $u^{e^{\prime}ah}$; but this follows immediately from an
application of the snake lemma to the commutative diagram