Galois representations and (phi, Gamma)-modules [Lecture notes]

123 Pages·2016·0.956 MB·English

by  Peter Schneider

Galois representations and (ϕ,Γ)-modules
Peter Schneider
Course at Mu¨nster in 2015, Version 5.10.16
Contents
Overview 1
1 Relevant constructions 5
1.1 Ramified Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Unramified extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Lubin-Tate formal group laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Tilts and the field of norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5 The weak topology on Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . 53
1.6 The isomorphism between H and H . . . . . . . . . . . . . . . . . . . . . 56
L EL
1.7 A two dimensional local field . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2 (ϕ ,Γ )-modules 69
L L
2.1 The coefficient ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.2 The modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3 An equivalence of categories 91
3.1 The functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 The case of characteristic p coefficients . . . . . . . . . . . . . . . . . . . . . . 101
3.3 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 Further topics 112
4.1 Iwasawa cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 Wach modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 (ϕ ,Γ )-modules over the Robba ring . . . . . . . . . . . . . . . . . . . . . . 114
L L
4.4 (ϕ ,Γ )-modules and character varieties . . . . . . . . . . . . . . . . . . . . . 115
L L
4.5 Multivariable (ϕ ,Γ )-modules . . . . . . . . . . . . . . . . . . . . . . . . . . 116
L L
4.6 Variation of (ϕ ,Γ )-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
L L
4.7 (ϕ ,Γ )-modules and p-adic local Langlands . . . . . . . . . . . . . . . . . . 117
L L
References 119
Notations 121
Index 123
1
Overview
Let L be a finite extension of the field of p-adic numbers Q with ring of integers o and
p
residueclassfieldF .Itisanimportantprobleminnumbertheorytounderstandtheabsolute
q
Galois group G := Gal(L/L) of this field. Local class field theory establishes a canonical
L
homomorphism rec : L× −→ Gab – the reciprocity map – into the maximal abelian quotient
L
Gab of G , which is injective and ”nearly” surjective. It seems extremely difficult to extend
L L
this reciprocity map to a natural homomorphism whose target is the full group G . Instead
L
the Langlands philosophy proposes to understand the group G through its representations.
L
Of course, there are various types of representations one can look at. In this book we will
concentrate on
Rep (G ) := category of all finitely generated o-modules
o L
with a continuous linear G -action.
L
One method to study a category is to establish an equivalence to another category which
arises in a different way. This opens up completely new perspectives on the original category.
In this book we will describe such a category equivalence, which in its original form for the
field Q is due to Fontaine. It will also be a category of modules equipped with a certain
p
group action. But, whereas in Rep (G ) the coefficient ring o is very simple and the group
o L
G is very complicated, in the new category the coefficient ring will be more complicated and
L
the relevant group is of a rather simple nature.
The new coefficient ring is a rather big complete discrete valuation ring A . It is (isomor-
L
phic to) the ring
{(cid:88)a Xi : a ∈ o, a −i−→−−−∞→ 0}
i i i
i∈Z
of all infinite Laurent series in a variable X and with coefficients in o, which tend to zero in
the negative direction. If ν denotes the normalized discrete valuation of L then the discrete
L
valuation ν of A is given by
AL L
(cid:88)
ν ( a Xi) := minν (a ) .
AL i i L i
i∈Z
Any prime element π of o also is a prime element of A . The field of fractions L⊗ A of
L o L
A then is a complete nonarchimedean field. Its residue class field is the usual field F ((X))
L q
of Laurent series in the variable X over the finite field F .
q
In order to equip this field with a group action we pick once and for all a Lubin-Tate
formal group law F (X,Y) ∈ o[[X,Y]] for the prime element π ∈ o. This formal group law
π
has two important features:
– Adjoining its torsion points to L gives an infinite Galois extension L /L. Furthermore,
∞
there is a natural isomorphism of groups χ : Γ := Gal(L /L) −→∼= o×. In particular,
L L ∞
the structure of the group Γ is easy to describe.
L
– The ring o embeds into the endomorphism ring End(F ) of the formal group law via a
π
natural map which sends a ∈ o to a power series [a](X) : aX +... ∈ o[[X]].
2
The latter allows us to let the multiplicative monoid o\{0} act on the ring A through
L
o\{0}×A −→ A
L L
(a,f(X)) (cid:55)−→ f([a](X)) .
Since o\{0} = o×πN0 this action splits into the action of the group o× and a single endomor-
phism
ϕ : A −→ A
L L L
f(X) (cid:55)−→ f([π](X)) .
By using the isomorphism χ we obtain the action
L
Γ ×A −→ A
L L L
(γ,f(X)) (cid:55)−→ f([χ (γ)](X))
L
of the Galois group Γ on the ring A . Note that the endomorphism ϕ commutes with the
L L L
Γ -action.
L
The new category now is defined as follows. A (ϕ ,Γ )-module M is a finitely generated
L L
A -module M which is equipped with
L
– a semilinear action of the group Γ and
L
– a ϕ -linear endomorphism ϕ : M −→ M which commutes with the Γ -action.
L M L
(In fact, there is an additional continuity requirement which we do not formulate here.) Such
a (ϕ ,Γ )-module M is called etale if the linearized map
L L
ϕlin : A ⊗ M −→ M
M L ϕL,AL
(f,m) (cid:55)−→ fϕ (m)
M
is an isomorphism. The main goal in this book is to explain the existence of an equivalence
of categories
Rep (G ) (cid:39) category of etale (ϕ ,Γ )-modules.
o L L L
This equivalence will be given by an explicit functor which arises in the following way. Let
H := Gal(L/L ), so that G /H = Γ . Furthermore, let A denote the ring of integers
L ∞ L L L
of the completion B of the maximal unramified extension Bnr of the nonarchimedean field
L
B := L⊗ A . The main technical point will be to show that the Galois group G acts on
L o L L
B in such a way that
– H = Gal(Bnr/B ) and
L L L
– the residual Γ -action on B coincides with the Γ -action defined above.
L L L
The residue class field of B is a separable algebraic closure F ((X))sep of the Laurent series
q
field F ((X)). The q-Frobenius α (cid:55)→ αq on this residue class field lifts to an endomorphism
q
φ of B which preserves A and such that φ |A = ϕ . These facts allow us to introduce the
q q L L
functor
Rep (G ) −→ etale (ϕ ,Γ )-modules
o L L L
V (cid:55)−→ M := (A⊗ V)HL with ϕ := φ ⊗id.
o M q
3
The H -fixed elements on the right hand side are formed with respect to the diagonal G -
L L
action on A⊗ V. Hence M carries a residual Γ -action. It is this functor which will be shown
o L
to be an equivalence of categories.
As already said this approach to p-adic Galois representations was initiated by Fontaine
who established the case L = Q of the above category equivalence in [Fon] (also [FO], [Sc]).
p
Later Kisin and Ren sketched in [KR] how this extends to arbitrary L by using the theory of
Lubin-Tateformalgroups.FortheconstructionoftheringA theyreliedonresultsbyColmez
L
in [Co1] which there were given only under the assumption that the power series [π](X) is a
polynomial. We fill in the necessary details to allow for a general [π](X). Moreover, we begin
with a detailed account of the theory of ramified Witt vectors, which is not well covered in
the literature. In the construction of the ring A we replace Fontaine’s original approach
L
with Scholze’s recent theory of tilts in [Sch]. But instead of his use of almost ring theory we
take the more explicit route via ramified Witt vectors. This has been studied in a general
framework by Kedlaya in [Ked].
In Chapter 1 we will set up the necessary background theories. In Chapter 2 the various
relevant rings together with the action of G (or Γ ) upon them and their “Frobenius en-
L L
domorphisms” will be introduced. Chapter 3 then contains the proof of the equivalence of
categories. For the interested reader we list in Chapter 4 various more advanced topics in
which (ϕ ,Γ )-modules play an essential role.
L L
4
1 Relevant constructions
Thepurposeofthisfirstchapteristodevelopallthetechniqueswhichlaterinthethirdchapter
will be used to prove the equivalence of categories between p-adic Galois representations and
etale(ϕ ,Γ )-modules.ThesourcecategoryreferstotheabsoluteGaloisgroupofalocalfield
L L
L of characteristic zero with (finite) residue field of characteristic p. On the other hand the
coefficient ring of the (ϕ ,Γ )-modules in the target category is a complete discrete valuation
L L
ring whose residue field is a local field of characteristic p. Therefore it should not come as a
surprise that much of this chapter will be devoted to setting up formalisms which allow to
pass between fields (or even rings) of characteristic zero and of characteristic p.
Historically the first such formalism is Witt’s functorial construction of the ring of Witt
vectors W(B) for any (commutative) ring B (cf. [B-AC] §9.1). If B = k is a perfect field of
characteristic p then W(k) is a complete discrete valuation ring with maximal ideal pW(k),
residue field k, and field of fractions of characteristic zero. For example, we have W(F ) = Z .
p p
Moreover, the pth power map on k lifts naturally to a “Frobenius” endomorphism F of W(k).
Since we will work over a finite, possibly ramified extension L of Q we need a generalization
p
of Witt’s original construction. Suppose that o is the ring of integers in L and F its residue
q
field. The rings of ramified Witt vectors W(B) , for any o-algebra B, have all the features of
L
the usual Witt vectors but are designed in such a way that W(F ) = o. This generalization
q
is not well covered in the literature, although most details can be extracted from the rather
technical treatment in [Haz]. In section 1.1 we therefore give a complete and detailed, but
nevertheless streamlined discussion of ramified Witt vectors.
In section 1.2 we recall the theory of unramified extensions of a complete discretely valued
field K. Its importance lies in the fact that, if Knr/K denotes the maximal unramified exten-
sion and ksep/k the separable algebraic closure of the residue field k of K, then one has a
K K K
natural isomorphism of Galois groups Gal(Knr/K) −−∼=→ Gal(ksep/k ). For us this means that
K K
the absolute Galois group of k can be identified naturally with a quotient of the absolute
K
Galois group of K.
The coefficient ring of (ϕ ,Γ )-modules carries an endomorphism ϕ as well as a group
L L L
of operators Γ ∼= o×. Their ultimate origin lies in the theory of Lubin-Tate formal group
L
laws which we explain in detail in section 1.3. If we fix a prime element π of o then there is an
up to isomorphism unique Lubin-Tate formal group law in two variables F(X,Y) ∈ o[[X,Y]]
which contains the ring o in its ring of endomorphisms. In particular, the prime element π
corresponds to an endomorphism which later on will give rise to the ϕ . By adjoining to
L
L the torsion points of this formal group law we obtain an abelian extension L /L. The
∞
action of the Galois group Γ := Gal(L /L) on these torsion points is given by a character
L ∞
χ : Γ := Gal(L /L) −→∼= o×. In an appendix to this section we determine explicitly the
L L ∞
higher ramification theory of the extension L /L.
∞
The section 1.4 is the technical heart of the matter. Let C be the completion of the
p
algebraic closure of Q . Already Fontaine introduced a very simple recipe how to construct
p
out of C an algebraically closed complete field C(cid:91) of characteristic p. But it was Scholze who
p p
saw the general principle behind this recipe. He introduced the notion of a perfectoid field K
in characteristic zero and used Fontaine’s recipe to associate with it a complete perfect field
5
K(cid:91) in characteristic p naming it its tilt. For us a very important example of a perfectoid field
will be the completion Lˆ of the extension L /L. The tilting procedure is natural. Hence
∞ ∞
the absolute Galois group G of L acts on the tilt C(cid:91). The absolute Galois group H of
L p L
L fixes Lˆ and consequently also the tilt Lˆ(cid:91) . In this way we obtain a residual action of
∞ ∞ ∞
Γ = G /H on Lˆ(cid:91) . On the other hand, using the torsion points of our Lubin-Tate formal
L L L ∞
group law we will exhibit an explicit element ω in Lˆ(cid:91) . We then embed the Laurent series
∞
field k((X)) in one variable over the residue field k of L into Lˆ(cid:91) by sending the variable
∞
X to ω. Its image is a local field E of characteristic p. The Γ -action on Lˆ(cid:91) preserves
L L ∞
E . This field E is called the field of norms of L because it also can be constructed via a
L L
projective limit with respect to the norm maps in the tower L /L. In this form it was used
∞
in Fontaine’s original approach to the theory of (ϕ ,Γ )-modules. The route we will take
L L
instead is via Scholze’s tilting correspondence. The main result which we prove in this section
will be Theorem 1.4.24 which says that the tilting K (cid:55)→ K(cid:91) induces a bijection between the
perfectoid intermediate fields Lˆ ⊆ K ⊆ C and the complete perfect intermediate fields
∞ p
Lˆ(cid:91) ⊆ F ⊆ C(cid:91). The proof will be given through the construction of an inverse map F (cid:55)→ F(cid:93).
∞ p
If o is the ring of integers of F, then F(cid:93) is the field of fractions of the quotient of W(o )
F F L
by an explicit element c ∈ W(o ) , which only depends on L.
Lˆ(cid:91) L
∞
A ring like o has its own valuation topology. This leads to a natural topology on the
F
ramified Witt vectors W(o ) . It is called the weak topology since it is coarser than the
F L
p-adic topology on W(o ) . It plays an important technical role in proofs, and it induces the
F L
relevant topology on the coefficient ring of (ϕ ,Γ )-modules. In section 1.5 we will introduce
L L
this weak topology in a slightly more general setting and will provide the tools to work with
it.
In section 1.6 we will deduce from the tilting correspondence that the G -action on C
L p
induces a topological isomorphism of profinite groups between the absolute Galois group H
L
of L and the absolute Galois group H of the local field E . This is the crucial fact which,
EL L
for our purposes, governs the passage between characteristic zero and characteristic p.
In preparation for the coefficient ring of (ϕ ,Γ )-modules we finally consider in section
L L
1.7 the p-adic completion A of the ring of Laurent series o((X)) in one variable X over o. Its
L
elements are “infinite” Laurent series of a certain kind. We will show that the endomorphisms
of our Lubin-Tate formal group law, which correspond to elements in o, extend to operators
on A . In this way we obtain an endomorphism ϕ corresponding to the prime element π as
L L
well as an action of Γ ∼= o× on A . We also will see that, on the one hand, A carries a
L L L
weak topology of its own and, on the other hand, it is a complete discrete valuation ring with
prime element π and residue field k((X)).
ThroughoutwefixaprimenumberpandafinitefieldextensionL/Q ofthefieldofp-adic
p
numbers. Let o ⊆ L be the ring of integers with residue class field k of cardinality q = pf. We
also fix, once and for all, a prime element π of the discrete valuation ring o.
By Alg we denote the category of (commutative unital) o-algebras.
1.1 Ramified Witt vectors
For any integer n ≥ 0 we call
Φ (X ,...,X ) := Xqn +πXqn−1 +...+πnX
n 0 n 0 1 n
6
the nth Witt polynomial. These Witt polynomials satisfy the recursion Φ (X ) = X and
0 0 0
(1) Φ (X ,...,X ) = Φ (Xq,...,Xq)+πn+1X
n+1 0 n+1 n 0 n n+1
= Xqn+1 +πΦ (X ,...,X ) .
0 n 1 n+1
Let B be in Alg.
Lemma 1.1.1. For any m,n ≥ 1 and b ,b ∈ B we have:
1 2
b ≡ b mod πmB =⇒ bqn ≡ bqn mod πm+nB .
1 2 1 2
Proof. By induction it suffices to consider the case n = 1. The polynomial P(X,Y) :=
(cid:80)q−1XiYq−1−i satisfies(X−Y)P(X,Y) = Xq−Yq.HenceitsufficestoshowthatP(b ,b ) ∈
i=0 1 2
πB. But our assumption implies P(b ,b ) ≡ P(b ,b ) = qbq−1 mod πmB.
1 2 1 1 1
Lemma 1.1.2. For m ≥ 1, n ≥ 0, and b ,...,b ,c ,...,c ∈ B we have:
0 n 0 n
i. If b ≡ c mod πmB for 0 ≤ i ≤ n then
i i
Φ (b ,...,b ) ≡ Φ (c ,...,c ) mod πm+iB for 0 ≤ i ≤ n;
i 0 i i 0 i
ii. if π1 is not a zero divisor in B then in i. the reverse implication holds as well.
B
Proof. Both assertions will be proved by induction with respect to n. The case n = 0 being
trivial we assume that n ≥ 1.
i. By assumption and the Lemma 1.1.1 we have bq ≡ cq mod πm+1 for 0 ≤ i ≤ n−1. The
i i L
induction hypothesis then implies that
Φ (bq,...,bq ) ≡ Φ (cq,...,cq ) mod πm+nB .
n−1 0 n−1 n−1 0 n−1
Inserting this into the recursion formula (1) gives
Φ (b ,...,b )−πnb ≡ Φ (c ,...,c )−πnc mod πm+nB .
n 0 n n n 0 n n
But as a consequence of the assumption we have πnb ≡ πnc mod πm+nB. It follows that
n n
Φ (b ,...,b ) ≡ Φ (c ,...,c ) mod πm+nB.
n 0 n n 0 n
ii. By the induction hypothesis we have b ≡ c mod πmB for 0 ≤ i ≤ n−1. As above we
i i
deduce that
Φ (b ,...,b )−πnb ≡ Φ (c ,...,c )−πnc mod πm+nB .
n 0 n n n 0 n n
Butbyassumptionwehavethecorrespondingcongruencefortheleftsummandsalone.Hence
we obtain πn(b −c ) ∈ πm+nB and therefore b −c ∈ πmB by the additional assumption
n n n n
that π1 is not a zero divisor.
B
Let
N
B 0 := {(b ,b ,...) : b ∈ B}
0 1 n
be the countably infinite direct product of the algebra B with itself (so that addition and
multiplication are componentwise). We introduce the following maps:
N N
f : B 0 −→ B 0
B
(b ,b ,...) (cid:55)−→ (b ,b ,...) ,
0 1 1 2
7
which is an endomorphism of o-algebras,
N N
v : B 0 −→ B 0
B
(b ,b ,...) (cid:55)−→ (0,πb ,πb ,...) ,
0 1 0 1
which respects the o-module structure but neither multiplication nor the unit element, and
N
Φ : B 0 −→ B
n
(b ,b ,...) (cid:55)−→ Φ (b ,...,b ) ,
0 1 n 0 n
for n ≥ 0, and
N N
Φ : B 0 −→ B 0
B
b (cid:55)−→ (Φ (b),Φ (b),Φ (b),...) .
0 1 2
Lemma 1.1.3. i. If π1 is not a zero divisor in B, then Φ is injective.
B B
ii. If π1 ∈ B×, then Φ is bijective.
B B
N
Proof. Let b = (bn)n, u = (un)n ∈ B 0. As a consequence of the recursion relations (1) the
relation Φ (b) = u is equivalent to the system of equations
B
u = b ,
0 0
(2)
u = Φ (bq,...,bq )+πnb for n ≥ 1.
n n−1 0 n−1 n
Undertheassumptionini.,resp.inii.,theelementbthereforeis,inaninductiveway,uniquely
determined by u, resp. can recursively be computed from u.
Remark 1.1.4. The system of equations (2) in fact shows the following: Let b = (b ) ,u =
n n
N
(un)n ∈ B 0 such that ΦB(b) = u. Let C ⊆ B be a subalgebra with the property that the
π·
additive map B/C −−→ B/C is injective. Then we have for any m ≥ 0:
u ,...,u ∈ C ⇐⇒ b ,...,b ∈ C .
0 m 0 m
Proposition 1.1.5. Suppose that B has an endomorphism of o-algebras σ such that
σ(b) ≡ bq mod πB for any b ∈ B.
We then have:
i. Let b ,...,b ∈ B for some n ≥ 1 and put u := Φ (b ,...,b ); an element
0 n−1 n−1 n−1 0 n−1
u ∈ B then satisfies:
n
u = Φ (b ,...,b ) for some b ∈ B ⇐⇒ σ(u ) ≡ u mod πnB .
n n 0 n n n−1 n
ii. B(cid:48) := im(ΦB) is an o-subalgebra of BN0 which satisfies:
- B(cid:48) = {(un)n ∈ BN0 : σ(un) ≡ un+1 mod πn+1B for any n ≥ 0},
- f (B(cid:48)) ⊆ B(cid:48), v (B(cid:48)) ⊆ B(cid:48).
B B
8
Proof. i. By assumption on σ we have σ(b ) ≡ bq mod πB for any 0 ≤ i ≤ n−1. Applying
i i
Lemma 1.1.2.i with m = 1 gives
σ(u ) = Φ (σ(b ),...,σ(b )) ≡ Φ (bq,...,bq ) mod πnB .
n−1 n−1 0 n−1 n−1 0 n−1
Theexistenceofanelementb ∈ B suchthatu = Φ (b ,...,b ) = Φ (bq,...,bq )+πnb
n n n 0 n n−1 0 n−1 n
is equivalent to u −Φ (bq,...,bq ) ∈ πnB, hence to u −σ(u ) ∈ πnB.
n n−1 0 n−1 n n−1
ii. By i., the image B(cid:48), as a subset of BN0, has the asserted description. The other claims
are easily derived from this.
First of all we apply this last result to the ring o with its identity endomorphism. We
N
obtain, for any λ ∈ o, an element Ω(λ) = (Ω0(λ),Ω1(λ),...) ∈ o 0 such that
Φ (Ω(λ)) = (λ,...,λ,...) .
o
By Lemma 1.1.3.i this Ω(λ) is uniquely determined by λ. For any o-algebra B we use the
N
canonical homomorphism o → B to also view Ω(λ) as an element in B 0.
Example. Ω (λ) = λ, Ω (λ) = π−1(λ−λq), Ω (λ) = π−2(λ−λq2)−π−1−q(λ−λq)q.
0 1 2
Next we consider the polynomial o-algebra
A := o[X ,X ,...,Y ,Y ,...]
0 1 0 1
intwosetsofcountablymanyvariables.Obviouslyπ1 isnotazerodivisorinA.Weconsider
A
on A the o-algebra endomorphism θ defined by
θ(X ) := Xq and θ(Y ) := Yq for any i ≥ 0.
i i i i
Remark 1.1.6. θ(a) ≡ aq mod πA for any a ∈ A.
Proof. The subset {a ∈ A : θ(a) ≡ aq mod πA} is a subring of A which, since k× has order
q−1, contains o as well as, by the definition of θ, all the variables X and Y . Hence it must
i i
be equal to A.
N
Let X := (X0,X1,...) and Y := (Y0,Y1,...) in A 0. Because of Lemma 1.1.3.i and Prop.
1.1.5.ii there exist uniquely determined elements S = (S ) , P = (P ) , I = (I ) , and
n n n n n n
N
F = (Fn)n in A 0 such that
Φ (S) = Φ (X)+Φ (Y),
A A A
Φ (P) = Φ (X)Φ (Y),
A A A
Φ (I) = −Φ (X),
A A
Φ (F) = f (Φ (X)) ,
A A A
resp. such that
Φ (S ,...,S ) = Φ (X ,...,X )+Φ (Y ,...,Y ),
n 0 n n 0 n n 0 n
Φ (P ,...,P ) = Φ (X ,...,X )Φ (Y ,...,Y ),
n 0 n n 0 n n 0 n
(3)
Φ (I ,...,I ) = −Φ (X ,...,X ),
n 0 n n 0 n
Φ (F ,...,F ) = Φ (X ,...,X )
n 0 n n+1 0 n+1
9
for any n ≥ 0. The Remark 1.1.4 implies that
S ,P ∈ o[X ,...,X ,Y ,...,Y ],
n n 0 n 0 n
I ∈ o[X ,...,X ],
n 0 n
F ∈ o[X ,...,X ].
n 0 n+1
Lemma 1.1.7. F ≡ Xq mod πA for any n ≥ 0.
n n
Proof. We have
Φ (F ,...,F ) = Φ (X ,...,X ) = Φ (Xq,...,Xq)+πn+1X
n 0 n n+1 0 n+1 n 0 n n+1
≡ Φ (Xq,...,Xq) mod πn+1A .
n 0 n
Hence the assertion follows from Lemma 1.1.2.ii.
The S ,P ,I ,F can be computed inductively from the system of equations (3).
n n n n
Example. 1) S = X +Y , S = X +Y −(cid:80)q−1π−1(cid:0)q(cid:1)XiYq−i.
0 0 0 1 1 1 i=1 i 0 0
2) P = X Y , P = πX Y +XqY +X Yq.
0 0 0 1 1 1 0 1 1 0
3) F = Xq +πX , F = Xq +πX −(cid:80)q−1(cid:0)q(cid:1)πq−i−1XqiXq−i.
0 0 1 1 1 2 i=0 i 0 1
Exercise. 1) S −X −Y ∈ o[X ,...,X ,Y ,...,Y ].
n n n 0 n−1 0 n−1
2) If p (cid:54)= 2 then I = −X for any n ≥ 0.
n n
N
Let B again be an arbitrary o-algebra. On the one hand we have the o-algebra (B 0,+,·)
definedasadirectproduct.Anyo-algebrahomomorphismρ : B −→ B inducestheo-algebra
1 2
homomorphism
N N N
ρ 0 : B 0 −→ B 0
1 2
(b ) (cid:55)−→ (ρ(b )) .
n n n n
N
On the other hand we define on the set W(B)L := B 0 a new “addition”
(a ) (cid:1)(b ) := (S (a ,...,a ,b ,...,b ))
n n n n n 0 n 0 n n
and a new “multiplication”
(a ) (cid:0)(b ) := (P (a ,...,a ,b ,...,b )) .
n n n n n 0 n 0 n n
Moreover we put
0 := (0,0,...) and 1 := (1,0,0,...) .
Because of (3) the map
N
Φ : W(B) −→ B 0
B L
satisfies the identities
Φ (a(cid:1)b) = Φ (a)+Φ (b),
B B B
(4)
Φ (a(cid:0)b) = Φ (a)·Φ (b) .
B B B
10