Table Of ContentComputation of framed deformation functors
Pietro Ploner
Abstract
3
1
In this work we compute the framed deformation functor associated
0
to a reducible representation given as direct sum of 2-dimensional repre-
2
sentations associated to elliptic curves with appropriate local conditions.
n Suchconditionsarise in theworks of Schoofand correspond toreduction
a properties of modular elliptic curves.
J
2
2
]
T
N
1 Introduction
.
h
t
a
m
[
2 This work originates from the articles of Schoof about classification of abelian
v varieties [15]. There he examines the case of abelian varieties over Q with
3
semistable reduction in only one prime ℓ and good reduction everywhere else
7
and proves that they do not exist for ℓ=2,3,5,7,13,while for ℓ=11 they are
8
3 given by products of the Jacobian variety J0(11) of the modular curve X0(11).
. In [16] he makes some generalisations of this result when ℓ is not a prime and
1
1 the base field is not Q, but a quadratic field. Some similar results, given in
2 terms of p-divisible groups, were also previously obtained by Abrashkin in [1].
1 Themainpurposeofthisworkistotranslatesomeresultsofthosearticlesin
:
v termsofdeformationtheoryofrepresentationsassociatedtoelliptic curves. We
i examine the following setting: let p 6= ℓ be distinct primes and S = {p,ℓ,∞}.
X
Letρ¯ :G →GL (k) i=1,...,nbeGaloisrepresentations,wherek isafinite
i S 2
r fieldofcharacteristicpandG istheGaloisgroupofthemaximalextensionofQ
a S
unramified outside S. We can suppose that there are exactly r non-isomorphic
representations among them and, up to reordering indexis, suppose that they
are ρ¯ ,...,ρ¯ . Then we can write
1 r
ρ¯=ρ¯ ⊕···⊕ρ¯ =⊕r ρ¯ei :G →GL (k) (1)
1 n i=1 i S 2n
r
where e =n. The main result is the following
i=1 i
TheorPem 1.1. : Suppose that:
1. the k-vector space Ext1 (V ,V ) of killed-by-p extensions is trivial for
D,p ρ¯i ρ¯j
every i,j =1,...,r;
1
1 INTRODUCTION 2
k if i=j
2. Hom (V ,V )= .
G ρ¯i ρ¯j (0 if i6=j
(cid:3)
Let F be the framed deformation functor associated to ρ¯ with the local
D
conditions:
• ρ is p-flat over Z[1/ℓ];
• ρ satisfies (ρ (g)−Id)2 =0 for every g ∈I ;
i ℓ
• ρ is odd.
(cid:3) (cid:3)
Then F is represented by a framed universal ring R which is isomorphic
D D
to W(k)[[x ,...,x ]], where N =4n2− r e 2.
1 N i=1 i
P
The setting works in particular when ρ¯ is the representation associated to
i
the p-torsion points of an elliptic curve E over Q having semistable reduction
i
in ℓ and good supersingular reduction at p, as the varieties described in [15].
Moreover the local condition in ℓ corresponds to the condition of semistable
actiondescribedin[15,Section2],whiletheconditioninpistheclassicalflatness
condition introduced in [13]. The final result want to express that the framed
deformation ring turns out to be the “simplest” possible, giving an analog for
deformation of [15, Th. 8.3].
The work is structured as such: in section 2 we recall the main features of
deformationtheory,followingmainlyMazur’soriginalformulation(see[11,12])
and we introduce framed deformation functors, following Kisin’s approach[9],
but avoidingthe use ofthe formalismof groupoids. Then inthe following three
chapters we describe the local conditions we have used in our theorem: first
we deal with the flatness condition in the residual prime p, which is treated
according the initial results of Ramakrishna [13] and expanded by the work of
Conrad [3, 4, 5]; then we pass to examine the semistability condition, which is
treatedasaparticularcaseofrepresentationofSteinbergtype: thecomputation
of the universal ring in this part is pretty indirect and passes through the use
of formal schemes [10]. Finally, for the archimedean prime, the computation of
the deformation ring is performed directly. Chapters 6 and 7 are dedicated to
local-to-globalarguments, which consent to build up a presentation of a global
deformation ring using the computations on the local ones we did in the pre-
vious chapters. The definition of geometric deformation functor is introduced,
too. It is due to Kisin [10] and the name comes form the “geometric”represen-
tationsdefinedinSerre’sconjecture. Inthe finalchapterTheorem1.1isproved
with the use of the technical instruments introduced and through a final direct
computation of the universal framed deformation, using matrix algebras.
2 BASICS NOTIONS OF DEFORMATION THEORY 3
2 Basics Notions of Deformation theory
Let k be a finite field of characteristic p and let S be a finite set of rational
primes containing p and the archimedean prime. Denoting by Q the maximal
S
extensionofQunramifiedoutsideS andbyG =Gal(Q /Q),consideraGalois
S S
representation ρ¯: G →GL (k) and denote by V the associated G -module.
S N ρ¯ S
Let A be a complete noetherian local W(k)-algebra with residue field k and
denote by Aˆr the category of such algebras. A lift of ρ¯to A is a representation
ρ :G →GL (A) such that the diagram
A S N
GL (A)
vvρvAvvvvvv;; N(cid:15)(cid:15)πA
G ρ¯ //GL (k)
S N
commutes, where π is the natural projection on the residue field. Two lifts
A
ρ ,ρ are said to be equivalent if there exists a matrix M ∈Ker(π ) such that
1 2 A
Mρ (g)M−1 =ρ (g) for every g ∈G .
1 2 S
A deformation of ρ¯to A is an equivalence class of lifts. The starting repre-
sentation ρ¯is called residual.
Deformations can be better understood via a categoricalapproach. Given a
Galois representation ρ¯, we can define the deformation functor
F :Aˆr →Sets, (2)
ρ¯
which associates to an element A∈Aˆr the set of deformation classesof ρ¯to A.
Kisinhasintroducedavariantofthedeformationfunctor. Letβ beak-basis
of the Galois module V . A framed deformation of the couple (V ,β) to a ring
ρ¯ ρ¯
A∈Aˆr is a couple (V ,β ), where V is a free N-dimensional A-module with
A A A
continuous G -action lifting the action of V and β is an A-basis of V lifting
S ρ¯ A A
β. We can then define the framed deformation functor F(cid:3) : Aˆr → Sets which
ρ¯
associatesto analgebraA∈Aˆr the setof frameddeformationclassesof(V ,β)
ρ¯
to A.
Theorem 2.1. 1. Theframeddeformation functorisrepresentablebyaring
R(cid:3) =R(cid:3) ∈Aˆr.
ρ¯
2. If ρ¯satisfies the trivial centralizer condition End (V )≃ k, then F is
k[G] ρ¯ ρ¯
representable by a ring R=R ∈Aˆr.
ρ¯
(cid:3)
The rings R and R are called the universal deformation ring and the uni-
versal framed deformation ringofρ¯respectively. Theyareuniversalinthesense
that any deformation of ρ¯to an element A∈Aˆr can be recovered via a unique
homomorphism R→A.
AmongthealgebrasinAˆrthereisonewithparticularproperties. Letk[ǫ]be
theringofpolynomialsinǫwiththeconditionǫ2 =0andletF beadeformation
functor. The tangent space of F is the set F(k[ǫ]). It has a natural structure of
k-vector space.
2 BASICS NOTIONS OF DEFORMATION THEORY 4
Proposition 2.2. 1. F (k[ǫ]) is a finite dimensional k-vector space;
ρ¯
2. F (k[ǫ])≃H1(G ,Ad(ρ¯))≃Ext1 (V ,V ) as k-vector spaces;
ρ¯ S k[G] ρ¯ ρ¯
3. dim F(cid:3)(k[ǫ])=dim F (k[ǫ])+N2−dim H0(G ,Ad(ρ¯)).
k ρ¯ k ρ¯ k S
Let F be a deformation functor and let P be the category of pairs (A,V )
ρ¯ A
with A ∈ Aˆr and V ∈ F (A). Let D be a full subcategory of P. We say that
A ρ¯
D is a deformation condition if the following conditions hold:
1. if(A,V )→(B,V )isamorphisminP and(A,V )∈D,then(B,V )∈
A B A B
D;
2. if(A,V )→(B,V )isaninjectivemorphisminP and(B,V )∈D,then
A B B
(A,V )∈D;
A
3. if(A× B,V)liesinD,thenalsothe projections(A,V )and(B,V )do.
C A B
Given a deformation condition D, we can consider the functor F :Aˆr →
ρ¯,D
Sets that sends a ring A∈Aˆr to the set of deformations ρ of ρ¯to A such that
(A,V ) ∈ D. If F is representable by a ring R , then F is representable,
ρ ρ¯ ρ¯ ρ¯,D
too, by a quotient of R . Moreover the tangent space F (k[ǫ]) is a k-vector
ρ¯ ρ¯,D
subspace of F (k[ǫ]).
ρ¯
Given a Galois representation ρ¯: G → GL (k) a natural way to attach a
S N
deformation condition is the following: for every ℓ ∈S we consider the restric-
tion ρ¯ : G → GL (k) together with a local deformation condition D . Then
ℓ ℓ N ℓ
wecandefineaglobaldeformationconditionDgivenbytheobjects(A,V )∈P
A
whose local restriction to ℓ lies in D for every ℓ∈S.
ℓ
Another important example of deformation condition is given by the fixed
determinant. Let χ : G → W(k)∗ be a linear character. We say that a
S
representation ρ : G → GL (A) has determinant χ if the G -action induced
S N S
on the wedge product ΛN(V ) is given by the character
ρ
χ :G →χ W(k)∗ →φ A∗, (3)
A S
whereφistherestrictionoftheW(k)-algebrastructuremorphism. Thesubcate-
goryDofpairs(A,V )suchthatρhasdeterminantχisadeformationcondition
ρ
and the corresponing deformation functor is denoted as Fχ. Moreoverwe have
ρ¯
that
Fχ(k[ǫ])≃H1(G ,Ad0(ρ¯)), (4)
ρ¯ S
where Ad0 denotes the vectorspaceofmatriceswith tracezeroprovidedby the
adjoint G -action.
S
3 THE LOCAL FLAT DEFORMATION FUNCTOR 5
3 The local flat deformation functor
Inthissectionwewanttodealwithalocaldeformationconditionwhichrefersto
the prime p, characteristic of the finite base field k. This condition was mainly
studied by Ramakrishna in [13] and then generalised by Conrad in [4],[5] and
Kisin in [9]. From now on, we will only deal with representations of degree 2.
Lemma 3.1. (Ramakrishna) Let C be a full subcategory of Rep (G) closed
k
under passage to subobjects, direct products and quotients and let D be the full
subcategory of P of pairs (A,V ) such that V ∈ C. Then D is a deformation
A A
condition.
Proof. We need to prove that D satisfies the three properties of deformation
conditions. Property 1 comes from the fact that, if (A,V ) → (B,V ) is a
A B
morphisminD,thenV ⊗ B ≃V andthetensorproductcanberecoveredvia
A A B
directsumsandquotients. Property2comesfromtheclosureundersubobjects.
Property 3 comes from the fact that the fiber product and its projections can
be constructed via direct sums and quotients, too.
Let F be a finite extension of Q and ρ¯: G → GL (k) be a Galois repre-
p F 2
sentation. If ρ is a deformation of ρ¯to a coefficient ring A, we say that ρ is flat
if there exists a finite flat group scheme X over the ring of integers O such
F
that V ≃X(F¯), that is, V is the generic fiber of X.
ρ ρ
Proposition 3.2. The subcategory D of flat deformations is a deformation
condition.
Proof. it suffices to show that D satisfies lemma 3.1.
Let 0 → T → U → V → 0 be a sequence of G-modules such that U is the
generic fiber of a finite flat group scheme X over O . Then we can take the
F
schematicclosureX ofT inX (see[13,Lemma2.1]fordetails)andX =X/X
1 2 1
to see that also T and V are generic fibers of finite flat group schemes. This
argument and the fact that a direct sum of finite flat group schemes is still
a finite flat group scheme show that the subcategory of flat deformations is a
deformation condition.
If ρ¯ satisfies the trivial centralizer condition End (V ) = k, then the
k[GF] ρ¯
deformation functor which assigns to a coefficient ring the set of deformations
of ρ¯ which are flat, called the flat deformation functor and denoted as Ffl, is
representable by a noetherian ring Rfl, which is called the local flat universal
p
deformation ring. Wewanttogiveaproofofthemainresultofrepresentability
for this condition, which was proven by Ramakrishna for p6=2 and by Conrad
for all cases. First we need some technical data
Definition 3.3. Let φ denote the absolute Frobenius morphism. A Fontaine-
Lafaille module is a W(k)-module M provided with a decreasing, exhaustive,
separated filtration of W(k)-submodules {M } such that, for every index i, there
i
exists a φ-semilinear map φ :M →M with the property that φ (x)=pφ (x)
i i i i+1
for every x∈M.
3 THE LOCAL FLAT DEFORMATION FUNCTOR 6
We denote by MF the category of Fontaine-Lafaille modules over W(k).
Moreover we denote by MFf the full subcategory of objects such that M has
tor
finitelengthand Im(φ )=M andbyMFf,j the subcategoryofobjectssuch
i tor
that M = M and M = 0. Finally we say that a Fontaine-Lafaille module
0 j
P
is connected if the morphism φ is nilpotent The main result about Fontaine-
0
Lafaille modules is the following
Theorem 3.4 (Fontaine-Lafaille). For every j ≤p there exists a faithful exact
contravariant functor
MFf,j →Repf (G), (5)
tor Zp
which is fully faithful if j < p and becomes fully faithful when restricted to the
subcategory of connected Fontaine-Lafaille modules if j = p. Morevoer MFf,2
tor
is antiequivalent to the category of finite flat group schemes over W(k)
Proof. See [8, Ch.8-9] for a proof and description of the functor.
We say that a representation ρ has weight j if it comes from a Fontaine-
Lafaillemodule lying inMFf,j andwedenote byF the subfunctor ofF given
tor j ρ¯
by deformations of ρ¯which are of weight j. It follows that if ρ¯is flat, then the
functors F and F are the same, therefore we will identify them in the rest of
2 fl
the chapter.
We can now prove the main result for flat deformation functor. The proof
is due to Ramakrishna for the case p>2 (see [13, section 3]); then Conrad has
shown (see [3]) that the proof works also in the case p=2, since the Fontaine-
Lafaille module used is connected.
Theorem 3.5. (Ramakrishna) Let ρ¯:G →GL (k) be a flat residual Galois
Qp 2
representation with trivial centralizer and such that det(ρ¯) = χ, where χ is the
cyclotomic character. Then
Rfl(ρ¯)≃W(k)[[T ,T ]]. (6)
p 1 2
Proof. We split the proof in two parts. Suppose first that k = F and ρ¯is the
p
representation attached to the p-torsion points of an elliptic curve E over Q
p
with good supersingular reduction. We prove the theorem in this particular
case,where computations are relativelyeasy,andthen pass to the generalcase.
In the particular case we have chosen, we know that ρ¯ satisfies the triv-
ial centralizer hypothesis and is of weight 2. We calculate the tangent space
F (F [ǫ]). Viewing F [ǫ]2 as a 4-dimensional F -vector space, we can see an
2 p p p
element ρ∈F (F [ǫ]) as a matrix
2 p
ρ¯(g) 0
ρ(g)= (7)
R ρ¯(g)
g
(cid:18) (cid:19)
and such a representationgives clearly an element of Ext1 (V ,V ), the exten-
2,p ρ¯ ρ¯
sions in the category of weight 2 representations which are killed by p. It is
immediate to check that equivalence of litings correspond to equivalent exten-
sions.
3 THE LOCAL FLAT DEFORMATION FUNCTOR 7
Let M be the Fontaine-Lafaille module associated to V via Theorem 3.4.
ρ¯
By fullfaithfulness ofthe functor, wehavethatExt1 (V ,V )=Ext1 (M,M)
2,p ρ¯ ρ¯ 2,p
and that End (M)=F .
MF p
We want to write the module in a compactified manner in terms of a 2×2-
matrixX . ForthatweusethefactthatM is1-dimensional(itwillbeproved
M 1
shortly) and that φ (M )=0. Then we write
0 1
α 0 ∗ γ α γ
φ = , φ = , X = . (8)
0 β 0 1 ∗ δ M β δ
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
ThematrixX encodesallthe informationsofthe structureofM. Wealso
M
wantto write the elements of Ext1 (M,M)via these matrices. If N is such an
2,p
element, we have
X C
X = M , C ∈M (F ). (9)
N 0 X 2 p
M
(cid:18) (cid:19)
The matrix C corresponds to an element of Hom(M,M). If N′ is another
elementofExt1 (M,M)andD isthe2×2matrixinitsuppertriangularpart,
2,p
then it represents the same extension of N if and only if there exist a matrix
Id R
∈M (F ) such that
0 Id 4 p
(cid:18) (cid:19)
Id R X C X D Id R
M = M (10)
0 Id 0 X 0 X 0 Id
M M
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
and this happens if and only if C −D = [R,X ]. Moreover R must preserve
M
the filtration of M, because the isomorphism between N and N′ does so. Let
H be the set of such matrices R. It follows that
Ext1 (M,M)≃Hom(M,M)/{[R,X ]: R∈H} (11)
2,p M
Now we know that dim M = 2 and dim M = 0. If dim M 6= 1
Fp 0 Fp 2 Fp 1
then any endomorphism of M does not need to respect any filtration structure
and therefore the centralizerof X in M (F ), which has at least dimension 2,
M 2 p
wouldbelongtoEnd (M);thisisimpossiblebecausetheendomorphismring
MF
is 1-dimensional. Therefore dim M =1.
Fp 1
Now we can compute the dimension of the tangent space: observe that
Hom(M,M)hasdimension4,thesetofmatricesRwhichpreservesthefiltration
of M has dimension 3 and the kernel of the map R → [R,X ] has dimension
M
1 (it is isomorphic to End (M)). Therefore the tangent space has dimension
MF
4−(3−1)=2.
Now we have that R (ρ¯) = Z [[T ,T ]]/I. We count the number of Z /pl-
2 p 1 2 p
valued points of the universal ring, which is the number of objects N ∈MFf,2
tor
which are free Z /pl-modules of rank 2. If N denotes the kernel of multiplica-
p p
tion by p in N, then we need N ≃ M, in terms of matrices, since X ≡ X
p N M
(mod p). Since X ∈ M (Z /pl) and we have to consider modulo p, there are
N 2 p
p4(l−1) such matrices. We have to consider them modulo isomorphism. Now
if X ≃ X , then there exists a matrix R ∈ M (Z /pl) which respects the
N1 N2 2 p
3 THE LOCAL FLAT DEFORMATION FUNCTOR 8
filtration of M such that RX = X R; there are p3(l−1) such matrices and
N1 N2
pl−1 lie in the center of M (Z /pl), therefore commute with allthe X . So the
2 p N
number of Z /pl-valued points is p4(l−1)/(p3(l−1)/pl−1)=p2(l−1). Observethat
p
this is the same number of Z /pl-valued points of Z [[T ,T ]].
p p 1 2
Let now f ∈ I and (x,y) ∈ (Z /pl)2, then f(x,y) ≡ 0 (mod pl) for every
p
positiveintegerl. Itfollowsthat,takingliftingstocharacteristiczero,f(x,y)=
0 for all (x,y)∈(pZ )2 and therefore f =0. So I =0 and R (ρ¯)=Z [[T ,T ]].
p 2 p 1 2
Nowwecanpasstothe proofofthe theoreminthe generalcaseandremove
the hypothesis that k = F and that ρ¯ is the representation coming from an
p
elliptic curve. A lemmaofSerre(whose proofcanbe foundin[13])tells us that
ρ¯has restriction to inertia given by
ψ 0
ρ¯| = (12)
I 0 ψp
(cid:18) (cid:19)
whereψ isafundamentalcharacteroflevel2. Forsucharepresentationwewill
compute both the “unrestricted” universal ring and the flat one. First of all
we want to show that H2(G,Ad(ρ¯)) = 0. By Tate local duality we have that
H2(G,Ad(ρ¯)) = H0(G,Ad(ρ¯)∗) = (Ad(ρ¯)∗)G. Let φ ∈ (Ad(ρ¯)∗)G, we want to
show that its kernel is 4-dimensional and therefore φ = 0. Let R ∈ Ad(ρ¯), we
have φ(gR)=gφ(R), where the G-action is given by conjugacy composed with
ρ¯ on the left and by determinant on the right. It follows that, if g ∈ I, then
ρ¯(g)Rρ¯(g)−1−det(ρ¯(g))R∈Ker(φ). Then, if we define the map
T :R→ρ¯(g)Rρ¯(g)−1−det(ρ¯(g))R, (13)
g
it suffices to show that there exists g ∈I such that Ker(T )=0. We choose a
g
g such that ψ(g)=α where α is an element of order p2−1 in k∗. Then, taking
explicit formulas
x y x(1−αp+1) y(α1−p−αp+1)
R= , T (R)= (14)
z w g z(αp−1−αp+1) w(1−αp+1)
(cid:18) (cid:19) (cid:18) (cid:19)
and the last matrix is zero if and only if R=0. Then our claim is proved.
Nowwe usethe formulaforthe Euler-Poincar´echaracteristicforAd(ρ¯). Let
hi =dim(Hi(G,Ad(ρ¯))). We have
c (Ad(ρ¯))=h0−h1+h2 =−dim Ad(ρ¯). (15)
EP k
We have that h2 = 0, h0 = 1 (it is the trivial centralizer condition) and
dim Ad(ρ¯) = 4, therefore h1 = 5. It follows that the unrestricted universal
k
ring for such a representation is isomorphic to W(k)[[T ,T ,T ,T ,T ]].
1 2 3 4 5
The flat deformationring canbe computed by means of calculationssimilar
to the ones performed in the case k = F , except that we have to consider
p
Fontaine-Lafaille modules overW(k) instead of Z and all the dimensions have
p
to be computed over k. We obtain again that R (ρ¯) = W(k)[[T ,T ]]. In
2 1 2
particularR (ρ¯)is aquotientofR(ρ¯)andthe surjectivemapbetweenthemhas
2
a 3-dimensional kernel. The theorem is therefore proved.
Now we give a refinement of this result, which is due to Conrad [4].
4 STEINBERG REPRESENTATIONS AT PRIMES ℓ6=P 9
Theorem 3.6. Letρ¯beas in theprevious theorem andlet Ffl,χ bethesubfunc-
tor of flat deformations of ρ¯which have fixed determinant χ. Then this functor
is representable by the ring
Rfl,χ(ρ¯)≃Z [[T]]. (16)
p p
Proof. For the proof see [4, Ch. 4, Th.4.1.2].
EXAMPLE: Let E be an elliptic curve over Q that has supersingular re-
p
duction in p. Let ρ¯: G → GL (F ) be the representation coming from the
Qp 2 p
Galois action on the p-torsion points of E. Then, by the results of [5], ρ¯ is
absolutelyirreducibleandthereforethefunctorFfl isrepresentable. Therefore,
ρ¯
applying Ramakrishna’s theorem, we have that the flat universal deformation
ring is Z [[T ,T ]].
p 1 2
4 Steinberg representations at primes ℓ 6= p
Now we want to analyse local conditions at finite primes which are different
from p. We continue to assume that the representationspace V has dimension
ρ¯
2.
Definition 4.1. : A 2-dimensional representation ρ¯: G → GL (k) is called
ℓ 2
of Steinberg type if it is a non-split extension of a character λ:G →k∗ by the
ℓ
twist λ(1)=λ⊗χ of λ by the p-adic cyclotomic character χ .
p p
A representation of Steinberg type has the matricial form
λ(1)(g) ∗
ρ¯(g)= ∀g ∈I (17)
0 λ(g) ℓ
(cid:18) (cid:19)
Observethatsince ℓ6=p the modp cyclotomiccharacteris unramifiedand, if p
is not a square mod ℓ, it also happens that χ and its twists are trivial. We do
p
notimposeanyramificationrestrictiononthecharacterλ. Uptotwistingbythe
inversecharacterofλ,wemayassumethatdet(ρ¯)=χp andthatV(ρ¯)(−1)Gℓ 6=
0, which means that there is a subrepresentation of dimension 1 on which G
ℓ
acts via χ .
p
We define a subfunctor
Lχp :Aˆr →Sets (18)
ρ¯
of the deformation functor Fχp as
ρ¯
Lχp(A)=(V ,L ) (19)
ρ¯ A A
where
• V is a deformation of ρ¯to A.
A
4 STEINBERG REPRESENTATIONS AT PRIMES ℓ6=P 10
• L is a submodule of rank 1 of V on which G acts via χ .
A A ℓ p
We define in the same way the framed subfunctor Lχp,(cid:3) :Aˆr →Sets as
ρ¯
Lχp,(cid:3)(A)=(V ,β ,L ) (20)
ρ¯ A A A
where
• (V ,β ) is a framed deformation of ρ¯to A.
A A
• L is a submodule of rank 1 of V on which G acts via χ .
A A ℓ p
This is the subfunctor corresponding to liftings of Steinberg type. In the
following we work with the framed setting to avoid representability problems.
Inorderto dealwithrepresentabilityofdeformationsfunctorsofSteingberg
type, we need to recall the main definitions of formal schemes. Let R be a
noetherian ring and I an ideal and assume that R is I-adically complete, so
that we have
R=limR/In. (21)
←
We define a topologycal space Spf(R) in the following way: given an element
f ∈Randf¯itsreductionmoduloI,wedefineD(f¯)tobethesetofprimeideals
of R/I not containing f¯. Then the set Spec(R/I) with the induced topology
is called the formal spectrum of R, with respect to I, and denoted by Spf(R).
The sets D(f¯) are a basis for the topology of Spf(R).
For each f ∈R we define
Rhf−1i=limR[f−1]/In (22)
←
Then the assignment D(f¯)7→Rhf−1i defines a structure sheaf on Spf(R).
Definition 4.2. The affine formal scheme Spf(R) over R with respect to I
is the locally ringed space (X,O ), where X = Spec(R/I) and O (D(f¯)) =
X X
Rhf−1i for each f ∈R.
A noetherian formal scheme is a locally ringed space (X,O ), where X is a
X
topological space and O is a sheaf of rings over X such that each point x∈X
X
admits a neighborhood U such that (U,O | ) is isomorphic to an affine formal
X U
scheme Spf(R).
A morphism of formal schemes is a pair (f,f∗):(X,O )→(Y,O ), where
X Y
f : X → Y is a continuous map of topologycal spaces and f∗ : O → f O is
Y ∗ X
a morphism of sheaves.
If (X,O ) is a scheme, we can obtain a formal scheme Xˆ by the following
X
construction: let I ⊆ O be an ideal sheaf and consider Xˆ the completion of
X
X along I. Its underlying topological space is given by the subscheme Z of
X defined by I and the structure sheaf is defined as before. A formal scheme
obtained in this way is called algebrizable.
