Table Of ContentModels of group schemes of roots of unity
Ariane M´ezard, Matthieu Romagny, Dajano Tossici
January 15, 2013
3
Abstract
1
0
Let O be a discrete valuation ring of mixed characteristics 0,p , with residue field k. Using work of
2 K p q
n SekiguchiandSuwa,weconstructsomefinite flatOK-modelsofthegroupschemeµpn,K ofpn-throotsof
a unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic
J properties ofthis construction. When k is perfect andO is a complete totally ramifiedextensionof the
K
4 ringofWittvectorsW k ,weprovideaparallelstudyofthe Breuil-Kisinmodulesoffinite flatmodels of
1 p q
µpn,K,insuchawaythattheconstructionofKummergroupsandBreuil-Kisinmodulescanbecompared.
] We compute these objects for n 3. This leads us to conjecture that all finite flat models of µpn,K are
T ď
Kummer group schemes.
N
.
h
t
a
m Contents
[
2 1 Introduction 2
v
2 2 Breuil-Kisin modules and µ-lattices 4
3
2.1 Breuil-Kisin modules of finite flat group schemes . . . . . . . . . . . . . . . . . . . . 5
2
2 2.2 Lattices of W u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
n
. pp qq
4 2.3 Breuil-Kisin modules of models of µpn,K . . . . . . . . . . . . . . . . . . . . . . . . . 9
0
1 3 The loop of µ-matrices 10
1
3.1 p-adic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
:
v
3.2 The loop of µ-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
i
X
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
r
a
4 Relating lattices and matrices 18
4.1 Matrices and lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Matricial description of Breuil-Kisin modules . . . . . . . . . . . . . . . . . . . . . . 21
5 Computation of µ-matrices for n 3 23
“
5.1 Computation of the matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Translation of the conditions of the theorem . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 The tamely ramified case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Sekiguchi-Suwa Theory 27
6.1 Some definitions about Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 Deformed Artin-Hasse exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Main theorems of Sekiguchi-Suwa Theory . . . . . . . . . . . . . . . . . . . . . . . . 33
6.4 Sekiguchi-Suwa Theory from a matricial point of view . . . . . . . . . . . . . . . . . 35
1
7 Kummer group schemes 41
7.1 Finiteness of closures of finite flat subgroups . . . . . . . . . . . . . . . . . . . . . . . 41
7.2 Matricial translation of the integrality conditions . . . . . . . . . . . . . . . . . . . . 45
8 Computation of Kummer group schemes for n 3 46
“
8.1 Two lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.2 Computations for n 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
“
8.3 Computations for n 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
“
8.4 Five lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1 Introduction
1.1. Context. Let k be a perfect field of characteristic p, W k its ring of Witt vectors, K
0
p q
the fraction field of W k , K K a finite totally ramified field extension, and O its ring of in-
0 K
p q {
tegers. The aim of the present paper is the determination of the models over O of the group
K
scheme µpn,K of roots of unity, or what is the same by Cartier duality, of the cyclic group scheme
Z pnZ . Apart from the intrinsic interest of the problem, a first motivation for doing this lies
K
p { q
in the study of the representations of the absolute Galois group of K. Indeed, finite flat group
schemes and p-divisible groups are extremely important examples of crystalline representations.
Work of Fontaine, Breuil and Kisin has culminated into a fairly nice description of these groups us-
ing modules with semilinear Frobenius. This description remains however very abstract and many
arithmetico-geometric properties of the group schemes do not have an easy translation in terms of
modules. Thus, one is in search of concrete examples witnessing the constructions and conjectures
of the general theory, like the filtrations defined by Abbes-Saito [AS] and Fargues [Fa]. We wish to
provide such explicit examples and test these general constructions.
Another importantmotivation istounderstandthereductionof Galois covers of K-varieties. In
the case of covers of curves, it is visible already for isogenies of elliptic curves (Katz-Mazur [KM])
but also in higher genus (Abramovich-Romagny [AR]) that it is necessary to let degenerate, along
with the varieties, also the Galois group of the covers. The existence of such group degenerations
is studied more precisely in [Ro] and [To1]. In the particular case of cyclic covers, this leads to the
question of understanding the models of Z pnZ. Here it is worth emphasizing that whereas in the
{
context of Galois representations one is by choice sticking to the original field K, in the context of
reduction of covers it is natural to allow finite extensions K1 K. This enhances the importance of
{
cyclic K-group schemes, since any finite flat commutative group scheme becomes isomorphic to a
product of such after a finite field extension.
A third motivation comes from the problem of finding an explicit description of the Hopf
algebras of group schemes over a discrete valuation ring with prescribed generic fiber G, in other
words the Hopf orders of the algebra K G . The most studied and best-known case is that of
r s
G µpn,K. Going beyond the Tate-Oort classification [TO], work in this trend is due mainly to
“
Larson [Lar], Greither [Gr], Byott [By], Underwood [Un] and Childs [Ch]. As a result, one has
a complete classification of Hopf orders for n 1,2 and a wealth of examples for n 3. One
“ “
difference between our approach and some of these constructions is that we shall find descriptions
which offer information aboutthe cohomology of theassociated groupschemes. Anotherimportant
feature of our constructions is that they require no assumption on the discretely valued field K,
whereas the results obtained by the above authors are valid for K complete, with perfect residue
field, containing a primitive pn-th root of unity.
2
1.2. Our approach. In this text, building on work of Sekiguchi and Suwa, we present a family of
finite flat models of µpn,K which we call Kummer group schemes. For this, we consider models of
G n constructed by successive extensions of affine, smooth, one-dimensional models of G
m,K m,K
p q
with connected fibres, called filtered group schemes. Kummer group schemes are defined as the
kernels G of some well-chosen isogenies E F between filtered group schemes, and their name
Ñ
comes from the fact that the exact sequence 0 G E F 0 is an integral model of the
Ñ Ñ Ñ Ñ
usual Kummer isogeny. This sequence is especially well-suited for the description of torsors under
the group schemes at hand, which as we said before is one of our motivations. We also point out
that the isogenies are given by explicit equations, and hence so are the kernels. We formulate the
following conjecture:
Conjecture. Any model of µpn,K over OK is a Kummer group scheme.
Our aim is to give strong evidence for this statement. We remark that this conjecture is true,
without assuming the discrete valuation ring complete with perfect residue field, in the case n 2
ď
(forn 1seee.g. [WW], discussionafter Theorem2.5, andforn 2see[To2]). Inordertoexplain
“ “
why we think it is true in general and what we actually do, let us first consider the category of
finiteflatmodels of µpn,K. Usingscheme-theoretic closures, it is nothardto see thatany morphism
G G1 between finite flat O -group schemes factors as the composition G։ G N H ã G1 of
K
Ñ { Ñ Ñ
the quotient by a finite flat subgroup scheme, a morphism which is an isomorphism on the generic
fibre (a so-called model map) and the closed immersion of a finite flat subgroup scheme. Models of
µpn,K are special because they have a unique (finite, flat) subgroup and quotient of a given order.
Thus the category of models of µpn,K may be completely described by the subcategory of groups
with model maps as morphisms, which is just a partially ordered set C , , and the two families
n
p ěq
of functors Q ,S : C C given by the finite flat quotient of degree pn`1´i, and the finite flat
i i n i
Ñ
subgroup of degree pn`1´i, for i 1,...,n 1 .
P t ` u
Now let us describe what we do. As we said, we take up and extend a construction of Sekiguchi
and Suwa, and use it to produce models of µpn, the Kummer group schemes. These group schemes
are parametrized by matrices with coefficients in the ring of Witt vectors W O . The choice of
K
p q
a uniformizer π for O allows to single out a certain set M of upper triangular matrices with an
K n
interesting structure: it is embedded in a bigger set of matrices endowed with a non-associative
product, giving rise to a natural order ą. This set has also operators Ui and Li that take a matrix
to its ”upper left” and ”lower right” square submatrices.
Then,we studyBreuil-Kisin modulesof modelsof µpn,K. They can beidentified with u-integral
lattices of the ring of Laurent series W k u , where k is the residue field of K. The set L of
n n
p qpp qq
lattices is ordered by inclusion and is endowed with functors K ,I : L L given by the kernel
i i n i
Ñ
and image of the endomorphisms pn`1´i and pi´1 of a given lattice. The lattices have unique
distinguished systems of generators whose p-adic coefficients can be put together into an upper
triangular matrix. In this way, we obtain a set G of matrices with coefficients in k u , with a
n
pp qq
non-associative product very similar to that of M and giving rise to a natural order ą. This set
n
also has functors Ui and Li.
Although not quite ”isomorphic”, the partially ordered sets C , , M ,ą and G ,ą with
n n n
their pairs of functors have strong analogies. There are naturalpfunctěoqrspM qC „pG givqen by
n n n
Ñ Ñ
mapping a matrix to the Kummer group scheme it defines, and then to the Breuil-Kisin module
of that group. The second functor is an equivalence, constructed in [Ki1]. The basic idea to prove
the conjecture above is to compute the Breuil-Kisin modules of Kummer groups and check that all
modulescan beobtained in thisway. Unfortunately, thereis nodirectway to computeBreuil-Kisin
modules. However, computations for n 2 (doneby Carusoin [To2] AppendixA) and n 3(done
“ “
3
in the present article) show a surprising phenomenon: it seems that if we replace π by u in the
matrix of a Kummer group, we obtain the matrix of its Breuil-Kisin lattice. In fact, we set up a
precise, nontrivial dictionary that indicates how totranslate thecongruences inadiscretevaluation
ring of characteristic 0 on one side, into congruences in a discrete valuation ring of characteristic p
ontheotherside. Thereadermaybeinspiredbyalookatthetablesin8.2.6(comparisonforn 2)
“
and 8.3.5 (comparison for n 3 under a simplifying assumption). She/he will see for her/himself
“
how striking the correspondence is. However, we wish to say that writing the dictionary already
for n 3 in the general case seems challenging.
“
Finally we observe that in particular we prove that Breuil-Kisin modules of models of µpn are
npn`1q
classifiedby parameters, asconjectured, in[GrCh]and[ChUn],moregenerallyforallmodels
2
of a fixed group scheme of order pn. Moreover the Kummer group schemes we constructed form a
family with exactly this number of parameters.
1.3. Summary of contents. Here is a short overview of the contents of the article; each section
starts with a more detailed introduction. The article is divided in two parts written to be readable
independently (to a reasonable extent). The first part (§§2-5) is devoted to Breuil-Kisin Theory
over a complete discrete valuation ring with perfect residue field. We apply that theory in order to
parametrize themodelsof µpn interms ofBreuil-Kisin modules(§2). Thenweexplain thealgebraic
structure(called aloop) of acertain set ofmatrices (§3)allowing usto rewriteBreuil-Kisin modules
in matricial terms (§4). The main result of this first part is Theorem 4.2.2 which is a computable
interpretation of Breuil-Kisin Theory (§4.2). The second part (§§6-8) is devoted to Sekiguchi-Suwa
Theory over a general discrete valuation ring of unequal characteristics. We recall the construction
of filtered group schemes and formalize it in matricial terms (§6). Then we describe the conditions
for certain modelmaps of filtered group schemes to beisogenies, whosekernels are by definitionthe
Kummer group schemes (§7). Finally we proceed with the explicit computation of models of µp3
(§8) with a comparison of the congruences coming from Breuil-Kisin Theory and Sekiguchi-Suwa
Theory (8.2.6 an 8.3.5).
1.4. Acknowledgements. We thank Xavier Caruso, Marco Garuti, Noriyuki Suwa and Angelo
Vistoliforinterestingconversations related tothisarticle. WearealsogratefultoChristopheBreuil
forvaluablecomments onthegenesis oftheclassification offiniteflatgroupschemes andtoLindsay
Childswhokindlysentusaversionofthepaper[GrCh]. Wethanktherefereeforhiscarefulreading
whichallowed tocorrectseveralinaccuracies. Thefirstandsecondauthorsespecially enjoyedastay
in the Scuola Normale Superiore di Pisa where part of this work was done. The third author had
fruitfulstays at the MPIM in Bonn, at theIHES in Bures-sur-Yvette, and spentsome time in Paris
to work on this project invited by the University Paris 6, the University of Versailles Saint-Quentin
and the IHP, during the Galois Trimester. The three authors also spent a very nice week in the
CIRM in Luminy. We thank all these institutions for their support and hospitality.
2 Breuil-Kisin modules and µ-lattices
In this section, we recall the description due to Breuil and Kisin of the category of finite flat group
schemes (understood commutative, of p-power order) in terms of modules with Frobenius. Then,
we specialize to the subcategory of models of the group scheme µpn of roots of unity.
We fix the following notations. Let k be a perfect field of characteristic p, W W k the
“ p q
ring of Witt vectors with coefficients in k, and S W u . We write W W k the ring
n n
“ rr ss “ p q
of Witt vectors of length n and S W u . The rings S and S are endowed with a ring
n n n
“ rr ss
endomorphism φ which is continuous for the u-adic topology, defined as the usual Frobenius on
4
W k and by φ u up. Let K be the fraction field of W k , let K K be a totally ramified
n 0 0
p q p q “ p q {
extension of degree e and O its ring of integers. We fix a uniformizer π of K and denote by E u
K
p q
its minimal polynomial over K and v the p-adic valuation with v π 1. We always use the
0
p q “
phrase finite flat group scheme as a shortcut for commutative finite flat group scheme of p-power
order. We denote by Gr O the corresponding category.
K
p { q
2.1 Breuil-Kisin modules of finite flat group schemes
2.1.1 The Breuil-Kisin Theorem. In recent papers, Breuil and Kisin have proven a classifi-
cation theorem for finite flat O -group schemes, in terms of the category Mod S described as
K
p { q
follows:
• the objects of Mod S are the finitely generated S-modules M of projective dimension 1,
p { q
killed by some power of p, and endowed with a φ-semilinear map φM : M M such that
Ñ
E u M is contained in the S-module generated by φM M .
p q p q
• the morphisms in Mod S are the S-linear maps compatible with φ.
p { q
For any M Mod S , the map φM is called the Frobenius and most often written simply φ. Note
P p { q
that to φ is associated a linear map φ˚M M, where φ˚M : M S,φ S. The classification of
Ñ “ b
Breuil and Kisin is the following:
2.1.2 Theorem. There is a contravariant exact equivalence of categories Gr O Mod S .
K
p { qÑ p { q
One may compose with Cartier duality to get a covariant equivalence, and in this paper this is
what we will do.
The category Mod S was introduced in [Br1]. To be more precise, Breuil required moreover
p { q
that the underlying S-module of an object Mod S should be a finite direct sum of modules
p { q
S pniS. He conjectured the existence of an equivalence between Mod S and the category of
{ p { q
finite flat group schemes whose pm-kernels are finite flat for all m, and he proved the conjecture for
group schemes killed by p, when p 2. After that, Kisin realized that arbitrary finite flat group
ą
schemes could betaken into account by requiringtheunderlyingmodulesmerely to have projective
dimension 1, and he proceeded to prove the conjecture in general (see [Ki1], Thm 0.5) for p 2.
ą
Later Lau [Lau] and Liu [Li] independently proved that the statement also holds for p 2, which
“
in fact was the original motivation of the note of Breuil [Br1].
For the convenience of the reader, here is a very rough sketch of how the equivalence of the
theorem works. Let S denote the p-adic completion of the divided power envelope of W u with
r s
respect to the ideal generated by E u . There is a natural inclusion S S but one has to
p q Ñ
notice that the ring S is much more complicated than S. Breuil introduces a category Mod S
p { q
whose objects are S-modules with a 1-step filtration and a semi-linear Frobenius. On the syntomic
site of the formal scheme Spf O , all finite flat group schemes define abelian sheaves. Breuil
K
p q
constructs another abelian sheaf Ocris. This sheaf plays the role of a sort of dualizing object:
8,π
Breuil shows that there is a contravariant equivalence Gr O Mod S that takes a group
K
p { q Ñ p { q
scheme G to the module Hom G,Ocris , with a quasi-inverse that takes a module M to the group
p 8,πq
schemethatrepresentsthesyntomicsheafX Hom M,Ocris X . Nowthereisacovariantfunctor
ÞÑ p 8,πp qq
Mod S Mod S given by tensoring with the map φ :S S. Kisin proves that for any M
p { q Ñ p { q Ñ P
Mod S there is a unique sub-S-module M M such that E u M φ M M. Moreover we
p { q Ă p q Ă x p qy Ă
can recover M from this submodulein the sense that M M S,φS so that Mod S Mod S
» b p { q Ñ p { q
is an equivalence.
5
2.1.3 Group schemes killed by pn. The modules killed by pn correspond to the group schemes
killed by pn. We will use a somewhat different description of the full subcategory of Mod S of
p { q
modules killed by pn, based on the following lemma.
2.1.4 Lemma. Let M be an S-module endowed with a φ-semilinear map φ : M M such that
Ñ
coker φ˚M M is killed by E u . Assume that M is killed by pn. Then M is an S-module of
p Ñ q p q
projective dimension 1 if and only if M is a finite S -module without u-torsion.
n
Proof: It follows from [Ki1], Lemma 2.3.2 that M has projective dimension 1 if and only if it is an
iterated extension of finitefreeS pS-modules. By induction, itisimmediate thatthisis equivalent
{
to the fact that M is a finite S -module without u-torsion. l
n
Therefore, the full subcategory of Mod S of modules killed by pn is the category Mod S
n
p { q p { q
defined as follows:
• the objects of Mod S are the finite S -modules M with no u-torsion endowed with a
n n
p { q
φ-semilinear map φ: M M such that coker φ˚M M is killed by E u .
Ñ p Ñ q p q
• the morphisms in Mod S are the S -linear maps compatible with φ.
n n
p { q
We now record some basic facts concerning Mod S .
n
p { q
2.1.5 Lemma. For any object M of Mod S the map φ˚M M is injective.
n
p { q Ñ
Proof: This is [Ki2], Lemma 1.1.9. l
2.1.6 Lemma. The category Mod S has kernels, cokernels, images and coimages. Kernels
n
p { q
and images are given as the kernels and images in the category of S-modules.
Proof: Let us prove first that Mod S has kernels and images. For a morphism f : M N, let
n
p { q Ñ
K and I be the kernel and the image in the category of S -modules. It is easy to see that K and
n
I are finite S -modules, stable under φ, with no u-torsion. Also note that the map f1 : φ˚f :
n
“
φ˚M φ˚N has kernel φ˚K (since φ is flat) and image φ˚I. The main point is to see that E u
Ñ p q
kills the cokernels of the maps φ˚K K and φ˚I I. We start with the kernel. For any x K
Ñ Ñ P
we have x M and since the cokernel of φ˚M M is killed by E u there exists y M1 such
P Ñ p q P
that E u x φ y . Then f1 y maps to 0 in N and hence is 0 in φ˚N. It follows that y φ˚K,
p q “ p q p q P
as desired. We come to the image. Let x I so that x f y for some y N. Then there exists
P “ p q P
z φ˚N such that E u y φ z . Therefore E u x φ f1 z with f1 z φ˚I, as desired.
P p q “ p q p q “ p p qq p q P
By Theorem 2.1.2, there is on Mod S a contravariant exact involutive equivalence given by
n
p { q
Cartier duality. It follows that Mod S has cokernels and coimages. l
n
p { q
2.1.7 Remark. Ingeneral,foramorphismf : M Ntheobjectscoker ker f andker coker f
Ñ p p qq p p qq
are not isomorphic. In the category Mod S this is not so easy to see, because we have not
n
p { q
worked out thedescription of cokernels. Thingsare a little easier in the category of finiteflat group
schemes. There, the kernel of a map u : G H is the scheme-theoretic closure of the kernel of
Ñ
the generic fibre u : G H inside G, and the cokernel is the Cartier dual of the kernel of the
K K K
Ñ
dual of u. For example, if R contains a primitive p-th root of unity and u : Z pZ µ is an
R p,R
p { q Ñ
isomorphism on the generic fibre, then ker u coker u 0 even though u is not an isomorphism.
p q “ p q “
6
2.2 Lattices of W ppuqq
n
We shall see in 2.3 that the Breuil-Kisin modules of models of µpn can be identified with lattices
in the W u -module W u . For this reason, it is useful to collect some basic facts on these
n n
rr ss pp qq
lattices; knowing their generating systems will be particularly important in Section 4. Since the
lattices we are interested in are Breuil-Kisin modules, for simplicity we keep the letters M,N (etc.)
to denote them.
2.2.1 Definition. A lattice M is a finitely generated sub-W u -module of W u such that
n n
rr ss pp qq
M 1 u W u . We denote by L the partially ordered set of lattices with inclusions between
n n
r { s “ pp qq
them. If a lattice M is contained in W u , we say that it is positive and we write M 0.
n
rr ss ě
Note that it is simpler here not to follow british mathematical usage, so we say positive instead
of non-negative.
For any two lattices M,N, there exists α N such that uαN M. We define the volume (or
P Ă
index) of M with respect to N as
vol M,N unα´lgpM{uαNq
p q “
where lg denotes the length as a W u -module. Using the fact that lg N uαN nα, one sees
n
rr ss p { q “
that the definition is indeed independent of α. Although our base ring is not a Dedekind ring, this
is the analogue of the symbol χ M,N of [Fa], d´ef. 5 and [Se], chap. III, no. 1. The volume (or
p q
index) of M is defined by vol M vol M,W u , and we have vol M,N vol M vol N .
n
p q “ p rr ssq p q “ p q{ p q
2.2.2 Kernels and images of p. For any lattice M and any integer i with 1 i n 1, we
ď ď `
define M i ker pn`1´i : M M and M i im pi´1 : M M . We have M i M i and
r s “ p Ñ q p q “ p Ñ q p q Ă r s
these submodules fit into compatible decreasing filtrations:
M M 1 ... M n M n 1 0
“ r s Ľ Ľ r s Ľ r ` s“
Y Y Y
M M 1 ... M n M n 1 0 .
“ p q Ľ Ľ p q Ľ p ` q“
For any two submodules N,N1 of W u , consider the ideal
n
pp qq
N : N1 x W u , xN1 N .
n
p q“ t P rr ss Ă u
Let 1 i j n 1 be integers. One can see easily, by inverting u, that
ď ď ď `
M j :M i M j :M i pj´iW u .
n
p r s r sq “ p p q p qq “ rr ss
Besides, since M has no u-torsion then M i M j 1 u M j and the map
r sX r sr { s “ r s
M i M j M i 1 u M j 1 u
r s{ r s ÝÑ r sr { s{ r sr { s
is injective. Since M i 1 u pi´1W u and pi´1W u pj´1W u W u , this
n n n j´i
r sr { s “ pp qq pp qq{ pp qq “ pp qq
proves that M i M j is canonically a lattice of W u . Exactly the same arguments show
j´i
r s{ r s pp qq
that M i M j is canonically a lattice of W u . In particular, for j n 1 this says that
j´i
p q{ p q pp qq “ `
M i and M i are lattices of W u .
n`1´i
r s p q pp qq
2.2.3 Generating sets. For each x k, let x W k be its Teichmu¨ller representative (see 3.1
P r s P p q
for a reminder on this notion). The map x x is the unique multiplicative section of the
ÞÑ r s
projection onto the residue field. If e ,...,e is a set of generators for M, then we will call T-
1 n
combination a linear combination t e t e where t ,...,t are Teichmu¨ller representatives.
1 1 n n 1 n
`¨¨¨`
In the following result, and in other places of the paper, we use the same letter for the valuation
of a discrete valuation ring and for the induced function on its artinian quotients.
7
2.2.4 Lemma. Let M be a lattice of W u and let e ,...,e be a system of generators. Let v
n 1 n p
pp qq
denote the p-adic valuation on W . Then the following conditions are equivalent:
n
(1) For 1 i n, we have v e i 1 and pe e ,...,e .
p i i i`1 n
ď ď p q “ ´ P x y
(2) For 1 i n, we have M i e ,...,e .
i n
ď ď r s “ x y
(3) For 1 i n, we have v e i 1 and each element x M can be written in a unique way
p i
ď ď p q“ ´ P
as a T-combination x x e x e with x k u .
1 1 n n i
“ r s `¨¨¨`r s P rr ss
Proof: (1) (2). Set N e ,...,e . It is obvious that N M i , so we only prove the
i i n i
ñ “ x y Ă r s
opposite inclusion. Since v e i 1, we have N 1 u pi´1W u . Let x M i and write
p i i n
p q “ ´ r { s “ pp qq P r s
x x1e x1 e
1 1 n n
“ `¨¨¨`
for some coefficients x1 W u . The fact that pe N implies that this linear combination
i P nrr ss i P i`1
may be transformed into a T-combination x x e x e . If x 0 there exists ν minimal
1 1 n n
“ r s `¨¨¨`r s ‰
such that x 0. Then the assumption that x M i gives x e M i N . After tensoring
ν ν ν ν`1
‰ P r s r s P r s`
with W u we obtain
n
pp qq
pν´1W u pi´1W u pνW u pminpi´1,νqW u
n n n n
pp qq Ă pp qq` pp qq “ pp qq
hence ν i, so that x N .
i
ě P
(2) (3). From M i 1 u pi´1W u we deduce by decreasing induction on i that v e
n p i
ñ r sr { s “ pp qq p q “
i 1. Now fix x M. Since pM i M i 1 , we have pe e ,...,e for all i. Using this, we
i i`1 n
´ P r s Ă r ` s P x y
may as above write x as a T-combination x x e x e with x k u . Moreover, if
1 1 n n i
“ r s `¨¨¨`r s P rr ss
x e x e x1 e x1 e are two expressions for x, then x x1 e M 2 .
r 1s 1`¨¨¨`r ns n “ r 1s 1`¨¨¨`r ns n pr 1s´r 1sq 1 P r s
From the fact that M 2 : M 1 pW u it follows that x x1 pW u and hence
p r s r sq “ nrr ss r 1s´r 1s P n`1rr ss
x x1 0. By induction we get similarly x x1 for all i.
1´ 1 “ i “ i
(3) (1). Since v e i 1, the p-valuation of a nonzero element x e x e is equal
p i 1 1 n n
ñ p q “ ´ r s `¨¨¨`r s
to ν 1 where ν is the least integer such that x 0. For x pe we find ν i 1, so that
ν i
´ ‰ “ “ `
pe e ,...,e . l
i i`1 n
P x y
2.2.5 Definition. A set of generators e ,...,e of a lattice M satisfying the equivalent conditions
1 n
of Lemma 2.2.4 is called a Teichmu¨ller basis, or a T-basis for short.
2.2.6 Remark. Let e ,...,e be a T-basis of M and for each i, let l be the u-adic valuation of
1 n i
the class of e in M i M i 1 which is a lattice of k u . Then, we have l l ... l .
i 1 2 n
r s{ r ` s pp qq ě ě ě
Indeed, by the definition of l , we have e α pi´1 mod pi with val α l . Thereforepe α pi
i i i u i i i i
“ p q“ “
mod pi`1 and e α pi mod pi`1. Since pe e ,...,e , it follows at once that l l .
i`1 i`1 i i`1 n i i`1
“ P x y ě
2.2.7 Proposition. Let M be a lattice of W u . Then there exists a unique T-basis e ,...,e
n 1 n
pp qq
of the form:
e ulipi´1 a pi a pi`1 a pn´1
i i,i`1 i,i`2 in
“ `r s `r s `¨¨¨`r s
where a k u,u´1 is such that deg a l for all i,j. Moreover, we have l l ... l .
ij u ij j 1 2 n
P r s p q ă ě ě ě
Finally M is positive if and only if l 0 and a k u for all i,j.
n ij
ě P r s
8
Proof: Existence: we construct the e by decreasing induction on i, starting from i n. The
i
“
module M n is isomorphic via a canonical isomorphism to a lattice of W u k u , hence
1
r s pp qq “ pp qq
generated by uln for a unique l Z. The preimage via this isomorphism of this generator is
n
P
e ulnpn´1. For i n, assume by induction that e ,...,e have been constructed. The
n i`1 n
“ ă
module M i M i 1 is again canonically a lattice of k u , generated by uli for a unique li Z.
r s{ r ` s pp qq P
Since M i piW u , a lift in M i of this generator may be written in the form
n
r s Ă pp qq r s
e ulipi´1 a pi a pi`1 a pn´1
i i,i`1 i,i`2 in
“ `r s `r s `¨¨¨`r s
for some Laurent series aij P kppuqq. Now write ai,i`1 “ a1i,i`1`uli`1a2i,i`1 where a1i,i`1 P kru,u´1s
is the truncation of a in degrees l . Replacing e by e a2 e , and rewriting the
i,i`1 ě i`1 i i ´r i,i`1s i`1
p-adic expansion of the tail e a2 pi, we can fulfillthe condition deg a l . Applying
i´r i,i`1s up i,i`1qă i`1
the same process to a for s 1,...,n i we can fulfill the conditions deg a l for all j.
i,i`s u ij j
“ ´ p q ă
This finishes the construction of e , and by induction, of e ,...,e . The elements e are such that
i 1 n i
M i e ,...,e by construction.
i n
r s “ x y
Uniqueness: thechoiceofthegenerator ofM i M i 1 inthepreviousinductionisnormalized
r s{ r ` s
by the fact that we are looking for generators ei with leading coefficients ulipi´1. The choice of
the remaining coefficients of e is imposed by the condition on the degrees. This proves that the
i
system e ,...,e is unique. Finally the inequalities between the l are given by Remark 2.2.6 and
1 n i
the statement about positivity is obvious. l
2.2.8 Definition. The T-basis of Lemma 2.2.7 is called the distinguished basis of M.
2.2.9 Remark. Let M be a lattice with distinguished basis e ,...,e . Then there exist series
1 n
b k u and a set of equalities
ij
P rr ss
R : pe b e b e
i i ii i`1 i,n´1 n
“ r s `¨¨¨`r s
for 1 i n. It can be proven that in fact
ď ď
e ,...,e R ,...,R
1 n 1 n
x | y
is a presentation by generators and relations of M as a W u -module. We will not need this.
n
rr ss
2.3 Breuil-Kisin modules of models of µ
pn,K
We finally specialize to our main object of interest, namely, the finite flat models of µpn,K.
2.3.1 Models and µ-lattices. The natural morphisms between models are the model maps,
which are by definition morphisms of R-group schemes inducing an isomorphism on the generic
fibre. Let us see how the category of models of µpn,K with model maps can be described concretely
in terms of Breuil-Kisin modules.
Let K be an algebraic closure of K. For any two finite flat group schemes G,G1 with associated
Breuil-Kisin modules M,M1, we have:
G G1 G K G1 K M 1 u M1 1 u
K K
» ðñ p q» p q ðñ r { s » r { s
whereG K and G1 K are viewed as representations of the absolute Galois group Gal K K . The
p q p q p { q
first equivalence is clear, let us explain briefly the second. If we introduce the Kummer extension
K K p?n π , then a result of Fontaine says that the module M 1 u determines the
8 ně0
“ Y p q r { s
9
Gal K K -representation associated toG (see [Fo], Remark A.3.4.1). By aresultof Breuil([Br2],
8
p { q
Theorem 3.4.3), this representation in turn determines the crystalline Gal K K -representation
p { q
G K .
p q
Recall that we are using the covariant equivalence Gr O Mod S given by 2.1.2 and
K
p { q Ñ p { q
Cartier duality. ThustheS-moduleassociated to thegroupscheme µpn,R is M Sn with its usual
“
Frobenius. From this, we deduce that M is the module associated to a model of µpn,K if and only
if M 1 u is isomorphic to S 1 u W k u with its Frobenius. Since M has no u-torsion, we
n n
r { s r { s “ p qpp qq
may then see it as a submodule of W k u . As far as the morphisms are concerned, the model
n
p qpp qq
maps correspond to inclusions between submodules of W k u . We are lead to the following
n
p qpp qq
notions.
2.3.2 Definitions. A µ-lattice is a lattice M W u such that E u M φ M M,
n
where φ is the Frobenius of W u . We denote bĂy Lµptpheqqpartially orderpedqsetĂof xµ-lpattqicyesĂwith
n n
pp qq
inclusions between them.
The letter ’µ’ reminds us of µpn. Note that since a µ-lattice M is stable under Frobenius, it
is positive, for otherwise there would exist an element x M with negative u-valuation and then
P
the valuation of φn x would tend to , in contradiction with the finite generation of M. What
p q ´8
has been said before means that the Breuil-Kisin classification gives an equivalence of categories
between Lnµ and the category of models of µpn with model maps.
2.3.3 Kernels and images of p. Let G be a model of µpn,K. For 1 i n 1, define:
ď ď `
• G i the scheme-theoretic closure of ker pn`1´i :G G in G,
K K
r s p Ñ q
• G i the scheme-theoretic closure of im pi´1 :G G in G.
K K
p q p Ñ q
These are finite flat models of µpn`1´i. By definition, there are exact sequences:
0 // G n 2 i // G // G i //0
r ` ´ s p q
pi´1
0 // G i tt // G(cid:15)(cid:15) // G n 2 i //0
r s p ` ´ q
On the generic fibre, the vertical map pi´1 : G G vanishes on G n 2 i and its image is a
Ñ r ` ´ s
subscheme of G i . By taking closures, the same is true everywhere. Therefore, this map induces
r s
a morphism of R-group schemes G i G i which is a model map.
p q Ñ r s
Let M be the µ-lattice associated to G. Starting from the exact sequences above and using the
fact that the Breuil-Kisin equivalence is exact, we see that M i is the µ-lattice of G i and M i
r s r s p q
is the µ-lattice of G i . Moreover, the inclusion M i M i and the model map G i G i
p q p q Ă r s p q Ñ r s
correspond to each other.
3 The loop of µ-matrices
In 2.2, we have seen that lattices have ”nice” systems of generators. The p-adic coefficients of such
systems of generators may be put together into ”nice” matrices, called µ-matrices. We will come
back to this in more detail in Section 4. In the present section, we focus on the abstract algebra
of the set of µ-matrices. This set has a natural operation A,B A B whose meaning is that if
p q ÞÑ ˚
i :M N is an inclusion of lattices, if A is a matrix associated with a generating system of M and
Ñ
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