Fields Institute Communications Volume 00, 0000

Automorphisms of formal power series rings over a valuation ring

Barry Green

Department of Mathematics University of Stellenbosch 7602 Stellenbosch South Africa

0 Introduction

The aim of this paper is to report on recent work on liftings of groups of automorphisms of a formal power series ring over a field k of characteristic p to characteristic 0, where they are realised as groups of automorphisms of a formal power series ring over a suitable valuation ring R dominating the Witt vectors W (k). We show that the lifting requirement for a group of automorphisms places severe restrictions on the structure of the group as well as implying numerical congruences, which need to be satisfied by the fixed points of these automorphisms. In particular if the centre of the group isn’t cyclic then it must be a p-group (an easy consequence observed by Matignon in [Ma2]), and for p = 2 the dihedral groups D_{2n}, with n odd, can never be lifted. We also draw attention to equivalent formulations of assertions on the existence of prescribed finite order automorphisms of formal power series rings over a valuation ring.

Recently these questions have become important in the study of the arithmetic and geometry of curves over a p-adic field, due to a local-global-principle for lifting galois covers of smooth curves from characteristic p to characteristic 0. More precisely the local-global-principle for liftings gives necessary and sufficient conditions, whereby liftings of the inertia groups acting on the completions of the local rings at the points of a galois cover of smooth curves over k to smooth galois covers of the p-adic open disc over R, ensures a global lifting to a galois covering of smooth relative curves over R. The completed local rings are formal power series rings over kand R respectively, and so provide the setting.

The most important question in this regard is the Oort-Sekiguchi Conjecture, which claims that the lifting problem can be solved whenever the group for the cover is cyclic (See [S-O], [O1] and [O2]). Here the only problem is caused by the p-part of the respective inertia groups, and the conjecture has been proved when these are ap or ap^{2} cyclic groups with (a, p) = 1 (See [O-S-S] and [G-M1]).

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 11G20, 14H30; Secondary 14D15, 14E22.

More precisely, if G ⊂ Aut_{k}(k[[z]]) is cyclic and |G| = ap^{r}, with (a, p) = 1 and r = 1, 2, then it can be lifted provided R contains the p^{r}-th roots of unity. For r > 2, when considering the abstract group, in [G-M2] it is shown that there is a realisation of G in Aut_{k}(k[[z]]) which can be lifted to Aut_{R}(R[[Z]]) using Lubin-Tate formal groups. The conjecture above would imply the result for each realisation.

For G an elementary abelian p-group, in [Ma1] Matignon has shown that there is a realisation G ⊂ Aut_{k}(k[[z]]) which can be lifted, based on a construction of p-cyclic covers of ℙ^{1} with good reduction and special geometry of the branch locus. In general, however, there are realisations of elementary abelian p-groups, G ⊂ Aut_{k}(k[[z]]) which cannot be lifted for any R as above. For G = (ℤ/pℤ)^{2} an example is given [G-M1]. We also refer the reader to the paper by J. Bertin, [B], where various obstructions are discussed.

Motivated by the above investigation the following inverse galois problem is presented in [Ma1]:

Inverse Galois Problem If we let G be a p-group, is it possible to find a realisation G ⊂ Aut_{k}(k[[z]]), which can be lifted to G ⊂ Aut_{R}(R[[Z]]) for some complete discrete valuation ring R dominating W (k)?

This is a tantalizing problem and at present we don’t have any example of an non-abelian p-group which can be lifted as above, so progress in this direction would be very interesting.

Situation Let k be an algebraically closed field of characteristic p > 0, W (k) the ring of Witt vectors and R be a complete discrete valuation ring dominating W (k). We denote the fraction field of R by K and the corresponding algebraic and integral closures by K̃ and R̃. We suppose that K is finite over Quot(W (k)). Let π be a uniformising parameter of R and suppose v denotes the exponential valuation. We denote the rings of formal power series over k and R by k[[z]] and R[[Z]], respectively. Note that R[[Z]] is a two dimensional local ring with maximal ideal generated by Z and π, and its height one prime ideals are all principal. If σ ∈ Aut_{R}(R[[Z]]) then we can write

σ(Z) = a_{0} + a_{1}Z + a_{2}Z^{2} + a_{3}Z^{3} + ⋯,

where a_{0} ∈ πR and a_{1} ∈ R^{×}. We denote the identity morphism on R[[Z]] (respectively k[[z]]) by 1. From the context this will not lead to confusion. Observe that reducing the coefficients of σ modulo (π), we obtain a canonical surjective group homomorphism

Ψ : Aut_{R}(R[[Z]]) ⟶ Aut_{k}(k[[z]]).

We are interested in the following questions:

Lifting Questions

1. If \overline{σ} is a k-automorphism of k[[z]] of finite order, when can we find a lifting of \overline{σ} to an R-automorphism of R[[Z]] of the the same order, i.e., an Rautomorphism σ of R[[Z]] of the the same order such that Ψ(σ) = \overline{σ}? More generally, if \overline{G} is a finite subgroup of Aut_{k}(k[[z]]), when can one find a section of \overline{G} in Aut_{R}(R[[Z]]) with respect to Ψ?

2. If G is a finite group, give criteria depending on the structure of the group, which need to be satisfied in order that a realisation G ⊂ Aut_{k}(k[[z]]) can be lifted to G ⊂ Aut_{R}(R[[Z]]), for some complete discrete valuation ring R dominating W (k)?

1 Automorphisms and their fixed points

In order to investigate the questions mentioned above, particularly the second one, it is useful to study numerical data associated with the fixed points determined by the action of σ ∈ Aut_{R}(R[[Z]]) on the open disc D = Spec R[[Z]].

Given σ(Z) = ∑_{i=0}^{∞} a_{i}Z^{i} ∈ Aut_{R}(R[[Z]]), it induces a Spec R automorphism of

the open disc D = Spec R[[Z]], which we call σ̃. For rational points (Z − α) ∈ D,

(α ∈ πR), one has σ̃((Z −α)) = (Z −α̃), where α̃ = ∑_{i=0}^{∞} a_{i}α^{i}. Such a point is a fixed

point if and only if α = α̃. More generally P ∈ D is a fixed point of σ̃ if and only if P = (π, Z), (π), (0), or a height one prime ideal of R[[Z]] and P ⊃ (σ(Z) − Z). When we speak of fixed points we shall always mean the points this last set defines in the geometric generic fibre and denote this set by F_{σ}. These points correspond to the zeros of the series σ(Z) − Z in R̃, and to simplify our discussion we shall enlarge R suitably so that they are R-rational. Using the Weierstrass Preparation Theorem they can be described as follows.

Let m_{σ} = inf_{l≥0}{l : v(a_{l}) ≤ v(a_{i}) for all i}, then

σ(Z) − Z = a_{m_{σ}}f_{m_{σ}}(Z) u(Z),

where u(Z)−1 ∈ (π, Z) and f_{m_{σ}}(Z) is a unitary distinguished polynomial of degree m_{σ}, so that |F_{σ}| = m_{σ}. Note that if m_{σ} ≠ 0, so that for some α, v(α) > 0, we have (Z − α) ∈ F_{σ}, we can centre σ at Z = α (replace the parameter Z by Z − α)

and so obtain σ(Z) = ∑_{i=1}^{∞} b_{i}Z^{i}, for suitable b_{i} with v(b_{1}) = 0. If σ has finite order,

say o(σ) = n, then ζ = b_{1} is an n-th root of unity which must be primitive by 1.1.1 below, so we can express σ by

σ(Z) = ζZ(1 + a_{1}Z + a_{2}Z^{2} + ⋯),

where the coefficients a_{i} are distinct from those used previously.

We next recall a number of elementary facts on finite order automorphisms needed for our discussion. The first five can be found in [G-M1] and for the sixth we give a proof.

1.1 Elementary Facts: Suppose that σ ∈ Aut_{R}(R[[Z]]) has finite order, say o(σ) = n. Then:

1.1.1 If σ(Z) ≡ Z mod Z^{2} then σ = 1, i.e., n = 1.

1.1.2 If Ψ(σ) ≠ 1 then σ can be represented as

σ(Z) = ζZ(1 + a_{1}Z + a_{2}Z^{2} + ⋯),

where ζ is a primitive n-th root of 1.

1.1.3 If (n, p) = 1, then σ is linearizable and consequently σ̃ has exactly one fixed point.

We first note that by [G-M1], p. 242, σ has a fixed point and so can be represented by σ(Z) = ζZ(1 + a_{1}Z + a_{2}Z^{2} + ⋯), where ζ is a primitive n-th root of 1. Now set Z′ = Z + ζ^{−1}σ(Z) + ⋯ + ζ^{−(n−1)}σ^{n−1}(Z). Then σ(Z′) = ζZ′, Z′ ≡ nZ mod Z^{2} and as n ∈ R^{×}, we have R[[Z]] = R[[Z′]].

Note that this argument works over positive characteristic fields with characteristic prime to n as well, provided they contain the required roots of unity.

1.1.4 If o(σ) = p^{r} and Ψ(σ) ≠ 1, then σ isn’t linearizable.

If it were then σ(Z) = ζZ, which implies Ψ(σ) = 1, a contradiction.

1.1.5 If \overline{σ} ∈ Aut_{k}(k[[z]]) with o(\overline{σ}) = n and (n, p) = 1 then there exists σ ∈ Aut_{R}(R[[Z]]) with Ψ(σ) = \overline{σ} and o(σ) = n.

This follows directly from 1.1.3 above.

1.1.6 If \overline{σ} ∈ Aut_{k}(k[[z]]) with o(\overline{σ}) = n = ep^{r}, (e, p) = 1, and if we are able to lift automorphisms of k[[z]] of order p^{r}, then there exists σ ∈ Aut_{R}(R[[Z]]) with Ψ(σ) = \overline{σ} and o(σ) = n.

Let \overline{σ} ∈ Aut_{k}(k[[z]]) with o(\overline{σ}) = n = ep^{r}, (e, p) = 1, be given and set \overline{τ} = \overline{σ}^{p^{r}}. Then o(\overline{τ}) = e and so for some primitive e-th root of 1, say \overline{γ}, and parameter z′ we have \overline{τ}(z′) = \overline{γ}z′ and [[z′]] = k[[z]]. If we let t = z′\overline{τ}(z′) ⋯ \overline{τ}^{e−1}(z′) = (\overline{γ}^{(e−1)/2}z′)^{e}, then k[[z′]]^{⟨\overline{τ}⟩} = k[[t]]. For convenience we replace \overline{γ}^{(e−1)/2}z′ by z′′ and note that k[[z′′]] = k[[z′]]. Now \overline{σ}|_{k[[t]]} has order p^{r} and power series representation

\overline{σ}(t) = t(1 + \overline{a}_{1}t + ⋯).

Rewriting in terms of z′′ we have

\overline{σ}(z′′^{e}) = z′′^{e}(1 + \overline{a}_{1}z′′^{e} + ⋯)

and so

\overline{σ}(z′′) = z′′(1 + \overline{a}_{1}z′′^{e} + ⋯)^{1/e}.

Now let σ(T) = ζT (1 + a_{1}T + ⋯) be an order p^{r} automorphism lifting \overline{σ} to R[[T]], where ζ is a suitable primitive p^{r}-th root of 1. Next we consider the tame extension E of F = Quot(R[[Z]]) defined by Z′′^{e} = T. Then the integral closure of R[[T]] in E is R[[Z′′]] and as σ(Z′′^{e}) = ζZ′′^{e}(1 + a_{1}Z′′^{e} + ⋯), it follows σ extends to an order n = ep^{r} automorphism of R[[Z′′]] defined by σ(Z′′) = ζ^{1/e}Z′′(1 + a_{1}Z′′^{e} + ⋯)^{1/e}. Furthermore with respect to the morphism Ψ : Aut_{R}(R[[Z′′]]) ⟶ Aut_{k}(k[[z′′]]), we have Ψ(σ) = \overline{σ} as desired.

Hence the first lifting question reduces to:

Given \overline{σ} ∈ Aut_{k}(k[[z]]) with o(\overline{σ}) = p^{r}, find σ ∈ Aut_{R}(R[[Z]]) with Ψ(σ) = \overline{σ} and o(σ) = p^{r}.

As mentioned in the introduction we have:

Theorem 1.2 Let \overline{σ} ∈ Aut_{k}(k[[z]]) with o(\overline{σ}) = p^{r}, with r = 1, 2. Then there exists σ ∈ Aut_{R}(R[[Z]]) with Ψ(σ) = \overline{σ} and o(σ) = p^{r}.

The case r = 1 is treated in [O-S-S], and r = 1, 2 in [G-M1].

2 Observations on finite order automorphism groups of formal power series rings

We begin by recalling certain general structural results by Eaken and Sathaye, [E-S], on subgroups of Aut_{A}(A[[Z]]), where A is an integral domain. By applying these results in our context with A = R, together with elementarty observations concerning the fixed points of an automorphism σ ∈ Aut_{R}(R[[Z]]), we give criteria which will be used in the next section to give obstructions to the lifting questions for certain types of abelian and non-abelian groups.

Theorem 2.1 ([E-S]) Let A be an integral domain and G be a group of Aautomorphisms of the formal power series ring A[[Z]]. Then

A[[Z]]^{G} = \begin{cases}A[[ ∏_{σ∈G} σ(Z)]] & if G is finite \\ A & if G is finitely generated and infinite \end{cases}

Corollary 2.2 In the situation above let G ⊂ Aut_{A}(A[[Z]]) be a finitely generated subgroup. Then the following are equivalent:

i) G is finite. ii) A[[Z]]^{G} ⫌ A.

Next we let C ⊂ Aut_{A}(A[[Z]]) denote the subgroup of all automorphisms defined by Z ↦ ζZ, for ζ ∈ A some root of 1. Clearly C is a torsion subgroup of Aut_{A}(A[[Z]]), and will be referred to as the circle group in Aut_{A}(A[[Z]]).

Theorem 2.3 ([E-S]) If ℚ ⊂ A and G is a torsion subgroup of Aut_{A}(A[[Z]]), then G is conjugate to a subgroup of the circle group in Aut_{A}(A[[Z]]) and so is abelian. If G is finitely generated by elements of finite order, then it is a cyclic group.

Now given an integral domain A (which we do not assume contains ℚ), we let G_{0}(A) be the subgroup of Aut_{A}(A[[Z]]) consisting of those automorphisms having constant term a_{0} = 0 (i.e., by abuse of notation with σ(0) = 0). Note that if K = Quot(A), then the canonical map

Φ : G_{0}(A) ⟶ Aut_{K}(K[[Z]])

is a monomorphism. As a direct application using Φ and the results above we obtain:

Corollary 2.4 In the above situation suppose char(A) = 0. Then every torsion subgroup G of G_{0}(A) is abelian and an Aut_{K}(K[[Z]]) conjugate of a subgroup of the circle group C(K). (Note that if A is integrally closed then C(K) = C(A).) If G is finitely generated by elements of finite order, then it is a cyclic group.

For the rest of the paper we set A = R, the discrete valuation ring dominating W (k) defined previously.

Observation 2.5 Interpreted in terms of the fixed points determined by finite order automorphisms of R[[Z]], it follows that if G = ⟨σ_{1}, σ_{2}, …, σ_{n}⟩ ⊂ Aut_{R}(R[[Z]]) is finite and the morphisms σ̃_{i} ∈ Aut_{R}(D), D = Spec R[[Z]], have a common fixed point then G is cyclic.

This observation is particularly useful when finding necessary conditions for, and obstructions to, the lifting questions for certain types of groups.

3 Lifting Criteria

In this section we apply the observation above to a number of types of groups. We begin with an easy lemma, which will be useful when applied to various special abelian and non-abelian types of groups.

Lemma 3.1 Let σ, τ ∈ Aut_{R}(R[[Z]]) and suppose τ^{−1}στ = σ^{l} for some l ∈ ℤ. i) If P ∈ F_{σ}, then τ̃^{−1}(P) ∈ F_{σ} and the orbits O_{τ,P} := {τ̃^{−k}(P) | k = 0, 1, 2, … } for P ∈ F_{σ} are either disjoint or strictly contained in each other. If m_{σ} := |F_{σ}| is finite, then each P ∈ F_{σ} is preperiodic with respect to τ̃^{−1} and O_{τ,P} contains a unique τ̃^{−1}-period. If o(τ) = m then each orbit O_{τ,P} is a τ̃^{−1} period with |O_{τ,P} | dividing m, and the set of orbits O_{τ,P}, P ∈ F_{σ} forms a partition of F_{σ}.

ii) If o(τ) = m is prime and m_{σ} ≢ 0 mod m, then for some P ∈ F_{σ} we have

P ∈ F_{τ}. If furthermore ⟨σ, τ⟩ is finite, then it is a cyclic group. iii) If o(σ) = n with (n, p) = 1, then P ∈ F_{σ} implies P ∈ F_{τ}. If furthermore

⟨σ, τ⟩ is finite, then it is a cyclic group.

Proof i) Suppose P ∈ F_{σ}. Then as τ̃σ̃τ̃^{−1} = σ̃^{l} we have

σ̃(τ̃^{−1}(P)) = τ̃^{−1}(σ̃^{l}(P)) = τ̃^{−1}(P),

so that τ̃^{−1}(P) ∈ F_{σ}. Each of the other assertions in i) follow directly from this using the fact that τ̃^{−1} is an automorphism of D = Spec R[[Z]].

ii) If m is prime and m_{σ} ≢ 0 mod m, then as the τ̃^{−1}-orbits form a partition of F_{σ}, for some P ∈ F_{σ} we must have |O_{τ,P} | = 1 and hence P ∈ F_{τ}. Now since σ and τ have a common fixed point, if the group they generate is finite, by observation 2.5 it must be cyclic.

iii) Since o(σ) = n with (n, p) = 1, it follows by our earlier remarks that σ̃ has a unique fixed point. By i) above τ̃^{−1}(P) is also a fixed point, so we must have τ̃^{−1}(P) = P and P ∈ F_{τ}. The last assertion follows as in ii).

We next apply 2.4 and 3.1 to the lifting question for various special types of groups. Throughout we suppose that G ⊂ Aut_{k}(k[[z]]) is a finite subgroup which can be lifted to G ⊂ Aut_{R}(R[[Z]]).

Corollary 3.2 If p = 2 and n is odd, then the dihedral groups D_{2n} cannot be lifted from Aut_{k}(k[[z]]) to Aut_{R}(R[[Z]]) for any discrete valuation ring R dominating W (k).

Proof Suppose there was a lifting with D_{2n} ⊂ Aut_{R}(R[[Z]]), with representation D_{2n} = ⟨ σ, τ | τ^{2} = σ^{n} = 1, τ^{−1}στ = σ^{−1} ⟩. Since o(σ) = n, with n odd, we obtain a contradiction using 3.1 iii) above.

For G abelian: In [Ma2] Matignon has observed that if G is abelian and has a realisation that can be lifted, then G is either cyclic or a p-group. As this result follows immediately from Lemma 3.1 we include a proof below.

Propositon 3.3 ([Ma2]) Let G ⊂ Aut_{k}(k[[z]]) be a finite abelian group. Then if there is a lifting to G ⊂ Aut_{R}(R[[Z]]), G must either be cyclic or a p-group.

Proof Suppose G is abelian, but neither cyclic nor a p-group. Then the prime to p part of G must be cyclic, as it determines a tame totally ramified extension k[[z]]/k[[t]] for suitable t. Hence G must contain a subgroup of the form (ℤ/pℤ)^{2} × ℤ/lℤ, for some l prime to p. Let σ ∈ G be a generator for ℤ/lℤ and τ_{1}, τ_{2} ∈ G, generators for (ℤ/pℤ)^{2}. By 3.1 iii) σ̃ has a unique fixed point, which is also a fixed point for τ̃_{i}, i = 1, 2. This implies ⟨σ, τ_{1}, τ_{2}⟩ is cyclic, a contradiction.

Corollary 3.4 Let G ⊂ Aut_{k}(k[[z]]) be a finite group. Then if there is a lifting to G ⊂ Aut_{R}(R[[Z]]), the centre of G must either be cyclic or a p-group.

The theorem below gives numerical criteria on the fixed points, which needs to be satisfied in order that a lifting of a p-group can be realised.

Theorem 3.5 If G ⊂ Aut_{k}(k[[z]]) is a finite, non-cyclic p-group which can be lifted to G ⊂ Aut_{R}(R[[Z]]), then, unless G is a generalized quarterion group, for each σ ∈ G we have m_{σ} ≡ 0 mod p.

Proof Note that by the Hasse-Arf theorem applied in this context (see for example [G-M1], p. 251) it suffices to prove the theorem for σ ∈ G ⊂ Aut_{R}(R[[Z]]) of order p.

We first prove the abelian case. As G isn’t cyclic we can find some τ ∈ G so that ⟨σ, τ⟩ ≅ (ℤ/pℤ)^{2}. We next apply 3.1 ii) noting that if m_{σ} ≢ 0 mod p, then σ̃ and τ̃ must have a common fixed point and so generate a cyclic group, a contradiction.

Now suppose G isn’t abelian and let ζ(G) denote the centre, which is non-trivial. Let σ ∈ G with o(σ) = p. If σ ∉ ζ(G), choosing any element τ ∈ ζ(G) with o(τ) = p (which must exist as the centre is non-trivial) it follows that ⟨ σ, τ ⟩ ≅ (ℤ/pℤ)^{2}. Again we deduce that m_{σ} ≡ 0 mod p.

If σ ∈ ζ(G) we use the property which asserts that a p-group has exactly one subgroup of order p if and only if it cyclic or a generalized quarternion group (see for example [R], p. 143). By assumption G isn’t of the above type, so we can find τ ∈ G with ⟨σ⟩ ∩ ⟨τ⟩ = ⟨1⟩. Hence ⟨σ, τ⟩ ≅ (ℤ/pℤ)^{2} and we are finished as before.

Remark 3.6 For the generalized quarternion group p = 2, and it has representation

G = ⟨ σ, τ | σ^{2^{n−1}} = 1, τ^{2} = σ^{2^{n−2}}, τ^{−1}στ = σ^{−1} ⟩.

The only element of order 2 is τ^{2} and together with any other element of G generates a cyclic group. Hence the argument of the previous theorem doesn’t apply here.

4 Equivalent formulations of the lifting question

As an application of the observations in sections 1 and 2 we obtain the following equivalent formulations of the first lifting question:

Propositon 4.1 Let \overline{σ} ∈ Aut_{k}(k[[z]]) with o(\overline{σ}) = p^{r}. Then the following are equivalent:

i) There exists σ ∈ Aut_{R}(R[[Z]]) such that Ψ(σ) = \overline{σ} and o(σ) is finite.

ii) There exists τ ∈ Aut_{R}(R[[Z]]) with Ψ(τ) = \overline{σ} and o(τ) = p^{r}.

iii) There exists σ ∈ Aut_{R}(R[[Z]]) such that Ψ(σ) = \overline{σ} and R[[Z]]^{⟨σ⟩} ⫌ R.

Proof The equivalence of i) and iii) follows directly from 2.2 and ii) ⇒ i) is obvious.

i) ⇒ ii) Suppose o(σ) = ep^{m}, with (e, p) = 1. We set τ_{1} = σ^{ke}, where ke+lp^{r} = 1. Then \overline{τ_{1}}^{p^{m}} = 1 and \overline{τ_{1}} = \overline{σ^{1−lp^{r}}} = \overline{σ}\overline{σ}^{−lp^{r}} = \overline{σ}. As \overline{τ_{1}} ^{p^{m}} = 1 and o(\overline{σ}) = p^{r} it follows that p^{r}|p^{m}.

Thus we have found an R-automorphism τ_{1} of R[[Z]] of order p^{m}, m > r, and with \overline{τ_{1}} = \overline{σ}. We next show how, using τ_{1}, one constructs an automorphism τ of R[[Z]] having the desired properties, i.e., with o(τ) = p^{r} and \overline{τ} = Ψ(τ) = \overline{σ}.

Let R[[Z]]^{⟨τ_{1}⟩}, respectively R[[Z]]^{⟨τ_{1}^{p^{r}} ⟩}, denote the fixed rings with respect to the

group generated by τ_{1}, respectively τ^{p^{r}}. Then R[[Z]]^{⟨τ_{1}⟩} = R[[T]] with T = ∏_{i=1}^{p^{m}} τ_{1}^{i}(Z)

and R[[Z]]^{⟨τ_{1}^{p^{r}} ⟩} = R[[T_{r}]] with T_{r} = ∏_{l=1}^{p^{m−r}} τ_{1}^{lp^{r}}(Z).

Observe that \overline{T_{r}} = z^{p^{m−r}} as Ψ(τ_{1}^{lp^{r}}) = \overline{σ}^{lp^{r}} = 1 for 1 ≤ l ≤ p^{m−r}. Also

\begin{aligned}\overline{T} & = ∏_{k=1}^{p^{m−r}−1} ( ∏_{i=1}^{p^{r}} \overline{τ_{1}}^{kp^{r}+i}(z)) = ∏_{k=1}^{p^{m−r}−1} ( ∏_{i=1}^{p^{r}} \overline{σ}^{kp^{r}+i}(z)) \\ & = ∏_{k=1}^{p^{m−r}−1} ( ∏_{i=1}^{p^{r}} \overline{σ}^{i}(z)) = ∏_{i=1}^{p^{r}} \overline{σ}^{i}(z^{p^{m−r}}). \end{aligned}

The diagram below gives the tower of formal power series rings when taking the fixed rings with respect to ⟨τ_{1}^{p^{r}}⟩ and ⟨τ_{1}⟩ as well as the respective residue rings modπ :

R[[Z]] ⟶ k[[z]]
↑  ↑
R[[T_{r}]] ⟶ k[[t_{r}]] = k[[z^{p^{m−r}}]]
↑  ↑
R[[T]] ⟶ k[[t]] = k[[ ∏_{i=1}^{p^{r}}\overline{σ}^{i}(z^{p^{m−r}})]]

The extension k[[z]] | (k[[t_{r}]] = k[[z^{p^{m−r}}]]) is purely inseparable of degree p^{m−r} and \overline{σ} restricted to k[[t_{r}]] determines an order p^{r} automorphism of k[[t_{r}]] whose power series representation is identical to that of \overline{σ}, but with t_{r} substituted in place of z. We call this automorphism \overline{τ} and set τ = τ_{1}|_{R[[T_{r}]]} = τ_{1} mod τ_{1}^{p^{r}}. Renaming the variables Z := T_{r} and z := t_{r}, we have found τ satisfying the properties of ii).

More generally we have:

Theorem 4.2 Let \overline{G} ⊂ Aut_{k}(k[[z]]) be a finite p-subgroup. Then the following are equivalent:

i) There exists a finite subgroup H ⊂ Aut_{R}(R[[Z]]) such that Ψ(H) = \overline{G}.

ii) There exists a section G of \overline{G} in Aut_{R}(R[[Z]]).

Proof The implication ii) ⇒ i) is obvious.

i) ⇒ ii) Suppose \overline{G} = ⟨\overline{σ}_{1}, …, \overline{σ}_{n}⟩. Then by Proposition 4.1 we can find σ_{i} ∈ H with Ψ(σ_{i}) = \overline{σ}_{i} and o(σ_{i}) = p^{n_{i}} ≥ p^{r_{i}} = o(\overline{σ}_{i}) say, for each i. Replacing H by a suitable subgroup of ⟨σ_{1}, …, σ_{n}⟩, we obtain a p-group H with Ψ(H) = \overline{G}. Let N = Ker(Ψ|_{H}). Then R[[Z]]^{N} = R[[Z′]], where Z′ = ∏_{τ∈N} τ(Z), and \overline{Z′} = z^{o(N)} =: z′,

a p-power of z as N is a p-group. We obtain the following diagram of rings and residue rings mod π :

R[[Z]] ⟶ k[[z]]
↑  ↑
R[[Z′]] ⟶ k[[z′]] = k[[z^{o(N)}]]

The extension k[[z]] | k[[z′]] is purely inseparable and so for each i, \overline{σ}_{i}|_{k[[z′]]} determines an order p^{r_{i}} automorphism of k[[z′]] whose power series representation is identical to that of \overline{σ}_{i}, but with z′ substituted in place of z. For each i we again denote this automorphism by \overline{σ}_{i}.

Observe that H/N = ⟨σ_{1}N, …, σ_{n}N⟩ ⊂ Aut_{R}(R[[Z′]]) and that for each i, Ψ(σ_{i}N) = \overline{σ}_{i}, where now Ψ : Aut_{R}(R[[Z′]]) ⟶ Aut_{k}(k[[z′]]). This finishes the proof of the theorem.

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