Some additive galois cohomology rings

M. Künzer, H. Weber

June 17, 2004

Abstract

Let p ≥ 3 be a prime. We consider the cyclotomic extension Z_{(p)}[ζ_{p^{2}}] | Z_{(p)}, with galois group G = (Z/p^{2})^{∗}. Since this extension is wildly ramified, the Z_{(p)}G-module Z_{(p)}[ζ_{p^{2}}] is not projective. We calculate its cohomology ring H^{∗}(G, Z_{(p)}[ζ_{p^{2}}]; Z_{(p)}), carrying the cup product induced by the ring structure of Z_{(p)}[ζ_{p^{2}}]. Proceeding in a somewhat greater generality, our results also apply to certain Lubin-Tate extensions.

Contents

0 Introduction 2

0.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.1.1 A cohomology ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.1.2 A reduction isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.1.3 A cyclotomic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.3 Known results and some historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.3.1 Galois module structure, wild case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3.2 Wedderburn embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.5 Leitfaden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.6 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 A twisted cyclic group ring 7 1.1 Subrings defined by derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 A Wedderburn embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 The intermediate ring Λ^{D} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 A description of Λ^{D} via ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 The cyclic case of order p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Nebe decomposition 17 2.1 A block decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 A reduction isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Cup product and Yoneda product coincide 22

4 Cohomology 25 4.1 A projective resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 The Yoneda ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

AMS subject classification: 11R33, 16S35.

5 Applications 33 5.1 Lubin-Tate extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Cyclotomic number field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

A The case C_{p^{2}}: a conjecture and an experiment 36 A.1 The conjectural situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.2 Simplifying u̇, v̇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.2.1 The case p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A.2.2 The case p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A.3 A spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

0 Introduction

0.1 Results

0.1.1 A cohomology ring

Suppose given a purely ramified extension T |S of discrete valuation rings of residue characteristic p ≥ 3, with maximal ideals generated by s ∈ S and t ∈ T, respectively. Let K = frac S, L = frac T, and assume L|K to be galois with Gal(L|K) = C_{p} = ⟨σ⟩. In particular, T |S is wildly ramified, and so, as Speiser remarked [29, §6], T is not projective as a module over SC_{p}. We are interested in its cohomology ring H^{∗}(C_{p}, T; S) =^{def} Ext_{SC_{p}}^{∗}(S, T), carrying the cup product induced by multiplication on T.

We make the assumption that v_{s}(p) ≥ b − \underline{b}, where b =^{def} −1 + v_{t}(t^{σ} − t) and b = p\underline{b} + \overline{b} with \overline{b} ∈ [0, p − 1]. This assumption is void if K is of characteristic p. We obtain

H^{∗}(C_{p}, T; S)
=^{def} Ext_{SC_{p}}^{∗}(S, T)
= χ_{0}^{(0)}S
⊕ (χ_{0}^{(1)}(S/s^{\underline{b}+1}S) ⊕ ⋯ ⊕ χ_{\overline{b}−1}^{(1)}(S/s^{\underline{b}+1}S)) ⊕ (χ_{\overline{b}+1}^{(1)}(S/s\underline{b}S) ⊕ ⋯ ⊕ χ_{p−1}^{(1)}(S/s\underline{b}S))
⊕ χ_{0}^{(2)}(S/s^{b−\underline{b}}S)
⊕ (χ_{0}^{(3)}(S/s^{\underline{b}+1}S) ⊕ ⋯ ⊕ χ_{\overline{b}−1}^{(3)}(S/s^{\underline{b}+1}S)) ⊕ (χ_{\overline{b}+1}^{(3)}(S/s\underline{b}S) ⊕ ⋯ ⊕ χ_{p−1}^{(3)}(S/s\underline{b}S))
⊕ χ_{0}^{(4)}(S/s^{b−\underline{b}}S)
⊕ ⋯,

with S-linear generators χ_{i}^{(k)} ∈ H^{k}(C_{p}, T; S); where the multiplication χ_{0}^{(2)} ∪ − acts as a degree shift by 2, and where multiplication of odd degree elements is given by

χ_{j}^{(2l+1)} ∪ χ_{k}^{(2m+1)} = ∂_{\overline{j+k},\overline{2b}} s^{b+\underline{j+k−2b}} (\overline{b − j})^{−1}χ_{0}^{(2l+2m+2)}

for l, m ≥ 0 and j, k ∈ [0, p − 1] ∖ \{\overline{b}\} (4.9).

0.1.2 A reduction isomorphism

Let U|T be a further purely ramified extension of discrete valuation rings, and denote E = frac U. Suppose E|K to be finite galois, write H = Gal(E|K), and suppose that

[E : L] is coprime to p. So we are given K ⊆^{p} L ⊆^{p′} E, containing S ⊆ T ⊆ U, respectively. We have the reduction isomorphism

H^{∗}(H, U; S) ≃ H^{∗}(C_{p}, T; S)

of graded S-algebras (2.10, 3.1), thus cutting the upper p′-part.

0.1.3 A cyclotomic example

The results apply in particular to the following situation.

\begin{aligned}S & = & Z_{(p)}[π_{n−1}] & s & = & π_{n−1} \\ T & = & Z_{(p)}[π_{n}] & t & = & π_{n} \\ U & = & Z_{(p)}[ζ_{p^{n}}] & u & = & ζ_{p^{n}} − 1 \\ H & = & C_{p−1} × C_{p} \\ b & = & (p^{n−1} − 1)/(p − 1) \\ \underline{b} & = & (p^{n−2} − 1)/(p − 1) & \overline{b} & = & 1 , \end{aligned}  [figure: L = Q(π_{n}); p; K = Q(π_{n−1}); p^{n−2}; Q; p−1; p−1; p−1; Q(ζ_{p^{n}}) = E; p; Q(ζ_{p^{n−1}}); p^{n−2}; Q(ζ_{p})]

where n ≥ 2 and

π_{n} =^{def} ∏_{j∈[1, p−1]}(ζ_{p^{n}}^{j^{p^{n−1}}} − 1) = N_{E|L}(ζ_{p^{n}} − 1) ,

cf. (5.4). In the case n = 2, we note that π_{1} = p.

0.2 Method

We reinterpret H^{∗}(C_{p}, T; S) =^{def} Ext_{SC_{p}}^{∗}(S, T) ≃ Ext_{T ≀C_{p}}^{∗}(T, T) by adjunction, invoking the twisted group ring T ≀ C_{p} ⊇ SC_{p} which carries the multiplication (ρy)(τz) = (ρτ)(y^{τ}z), where ρ, τ ∈ C_{p} and y, z ∈ T. In this way, we have gained freedom in our choice of a projective resolution — when resolving T over T ≀ C_{p}, we are not bound to take a projective resolution of S over SC_{p} and to tensor it with T ≀ C_{p} over SC_{p}. The cup product on Ext_{SC_{p}}^{∗}(S, T) corresponds to the Yoneda product on Ext_{T ≀C_{p}}^{∗}(T, T) (3.1).

Still, we need an interpretation of T ≀ C_{p} that facilitates the Yoneda product calculation. Rationally, there is the Wedderburn isomorphism L ≀ C_{p} \overline{}^{ω}_{∼} End_{K} L = K^{p×p}, sending ρy ^{\overline{}^{ω}} (z \overline{}z^{ρ}y). Restricting to the locally integral situation and using the resulting Wedderburn embedding T ≀ C_{p} [figure: ω] End_{S} T = S^{p×p}, we get a workable isomorphic copy (T ≀ C_{p})ω ⊆ S^{p×p} of T ≀ C_{p}.

Namely, a matrix in S^{p×p} is contained in (T ≀ C_{p})ω if and only if it satisfies a set of congruences between its entries that can be deduced from the single congruence

t^{σ} ≡_{t^{1+b}} t ;

see (1.19, 1.17). With this tie description of (T ≀ C_{p})ω inside S^{p×p} at our disposal, it is easy to calculate the cohomology ring.

In particular, we believe that it is easier to use this tie description to resolve projectively than to work with the classical projective resolution of T over T ≀ C_{p}. Using the latter, it seems that at some point the operation of σ on T as an element of End_{S} T, i.e. the matrix (σ)ω, enters the picture; for instance when solving equations occurring in the resolution of cocycles (cf. 4.1). Using our tie description, we circumvent this problem by choosing an S-linear basis of (T ≀ G)ω without specifying the coefficients of (σ)ω therein. So it is no longer necessary to determine the matrix (σ)ω, which indeed, we have not been able to control.

0.3 Known results and some historical remarks

0.3.1 Galois module structure, wild case

Let L|K be a finite galois extension with G = Gal(L|K), and let S ⊆ K be a Dedekind domain with field of fractions K and integral closure T in L. Since T is not isomorphic to SG as soon as a prime ideal of S wildly ramifies in T, Leopoldt split the galois module structure problem in two parts. If K = Q, S = Z and G is abelian, he determined, firstly, generators for the associated order

A_{L|K} = {ξ ∈ KG : T ξ ⊆ T } ;

secondly, he showed that T ≃ A_{L|K} as a module over A_{L|K} by construction of an isomorphism [16, Satz 6].

We recall some of the recent extensions of this result.

(1) If K = Q(ζ_{n}), K ⊆ L ⊆ Q(ζ_{mn}) and S = Z[ζ_{n}], then Byott and Lettl gave a description of A_{L|K} and showed that T ≃ A_{L|K} [7].

(2) Aiba showed that the analogue of (1) holds in the Carlitz-Hayes function field case if L equals the analogue of Q(ζ_{mn}) and if the analogue of m divides n. On the other hand, if this divisibility condition fails, then T ≄ A_{L|K} [1, th. 4].

(3) If L|K is an extension of finite extensions of Q_{p} with L|Q_{p} abelian, and S the valuation ring of K, Lettl gave a description of A_{L|K} and showed that T ≃ A_{L|K} (4) As Byott observed, similar phenomena as in (2) occur for Lubin-Tate extensions

over Q_{p} [6, th. 5.1].

(5) If L|K is an extension of finite extensions of Q_{p}, and if G = C_{p^{n}}, Elder calculated T as a Z_{p}G-module [13].

It is quite possible that such a description of T ≃ A_{L|K} as a module over SG might be used to calculate cohomology, and also to calculate the cup product on H^{∗}(G, T; S). Nonetheless, generally speaking, Yoneda products are somewhat easier to calculate than cup products.

In the case of an extension L|K of number fields, with S the ring of algebraic integers in K and T |S at most tamely ramified, Fröhlich conjectured a connection between the class of T in Cl(ZG) and the Artin root numbers, which has been proven by Taylor; see e.g. [30, th. 3]. The extensions of this result from the tame to the wild case start by replacing T by an object better suited for class group considerations, thus putting the emphasis on the Artin root numbers; see e.g. [10, sec. 4.2].

0.3.2 Wedderburn embedding

Suppose given a Dedekind domain R and an R-order Ω which is rationally semisimple, that is, for which Ω̄ =^{def} (frac R) ⊗_{R} Ω is semisimple.

The Wedderburn embedding method consists of restricting the Wedderburn isomorphism ω from Ω̄ to Ω, and to describe the image Ωω inside a product of rationally simple maximal orders via congruences of matrix entries, called ties.

For the first time, this method surfaced in the proof of the Brauer-Nesbitt theorem, in which the assumption of the existence of a certain non-maximal overorder of a quasiblock is led to a contradiction [5, eq. (36)]. Here, the quasiblocks of Ω are the R-orders Ωε for rational primitive central idempotents ε ∈ Ω̄.

Around the same time, Higman calculated the ties for Z(C_{p} ⋊ C_{p−1}) with C_{p} ⋊ C_{p−1} = ⟨a, b : a^{p}, b^{p−1}, a^{b} = a^{r}⟩, where r is a generator of F_{p}^{∗} [15]. In particular, he calculated the hereditary quasiblock of size (p − 1) × (p − 1), which is isomorphic to the twisted group ring Z[ζ_{p}] ≀ C_{p−1}. See [26, sec. 6].

In the commutative case, descriptions of suborders via congruences have also been given by Leopoldt [16, p. 125, p. 134, p. 140].

Milnor gave a pullback description of ZC_{p^{n}}, the iteration of which yields the ties describing this ring [3, p. 601 f.].

The first systematic study to describe Wedderburn embeddings via ties has been undertaken by Plesken [24], [25], following hints of Zassenhaus. This has been particularly successful when the endomorphism rings of the indecomposable projective modules of the quasiblocks under consideration are isomorphic to the discrete valuation ring R.

We list some subsequent work related to the twisted group ring T ≀ G, where T |S is a finite galois extension of discrete valuation rings with G = Gal(T |S).

(1) Auslander and Rim showed that T |S is tamely ramified if and only if T ≀ G is hereditary [2, prop. 3.5], which in turn corresponds to an upper triangular shape of its single quasiblock.

(2) Benz and Zassenhaus showed that in the wildly ramified case the radical idealisator process, started with T ≀G, ends up with a hereditary overorder that is unique with respect to certain conditions [8, Satz (a)].

(3) Cliff and Weiss showed that the radical idealisator process of (2) ends up with the unique minimal hereditary overorder of T ≀ G, they determined its shape, and they calculated the number of steps of this process in terms of the different and the ramification index of T |S [12]. For sake of an example, they figured out the ties for T ≀ G in the case of a wildly ramified quadratic extension [11, p. 98, 97].

(4) Wingen calculated the ties of the associated order for G cyclic [32, p. 82].

0.4 Acknowledgements

We would like to thank G. Nebe for the method of reduction by blockwise decomposition. We would like to thank B. Keller for the argument that shows that Yoneda product and cup product coincide. We would like to thank several colleagues for comments and suggestions.

The first author would like to thank P. Littelmann for kind hospitality in Strasbourg, where the first part of this work was done, and where he received support from the EU TMR-network ‘Algebraic Lie Representations’, grant no. ERB FMRX-CT97-0100. Another part of the work was financed by a DFG research grant.

0.5 Leitfaden

[figure: 1 A twisted cyclic group ring; 3 Cup product and Yoneda product coincide; 2 Nebe decomposition; 4 Cohomology; 5 Applications]

0.6 Notations and conventions

(i) Composition of maps is written on the right, ^{\overline{}^{a}} ^{\overline{}^{b}} = ^{\overline{}^{ab}} . Exception is made for ‘standard’ maps, such as traces, characters, derivations …

(ii) For a, b ∈ Z, we denote by [a, b] := {c ∈ Z | a ≤ c ≤ b} the integral interval.

(iii) Given elements x, y of some set X, we let ∂_{x,y} = 1 in case x = y and ∂_{x,y} = 0 in case x ≠ y.

(iv) Given a, b ∈ Z, the binomial coefficient (\begin{matrix}a \\ b \end{matrix}) is defined to be zero unless 0 ≤ b ≤ a.

(v) If R is a discrete valuation ring with maximal ideal generated by r, we write v_{r}(x) for the valuation of x ∈ R ∖ {0} at r, i.e. x/r^{v_{r}(x)} is a unit in R. Moreover, we let v_{r}(0) = +∞.

(vi) If R is a discrete valuation ring and M [figure: f] N an injective R-linear map between R-modules M and N with cokernel of finite length in the sense of Jordan-Hölder, we refer to this length ℓ_{R}(N/M) as the R-linear colength of M in N.

(vii) Let n ≥ 1, let A be a ring. The ring of n × n-matrices over A is denoted by A^{n×n}.

(viii) Given a ring A, by an A-module we mean a finitely generated right A-module, unless specified otherwise.

(ix) Given a ring A, we denote by K^{−}(A) the homotopy category of complexes of A-modules bounded to the right.

(x) Given a category C, and objects X, Y in C, we denote the set of morphisms from X to Y by C(X, Y). If C = mod-A for a ring A, we abbreviate _{A}(X, Y) := _{mod-A}(X, Y).

1 A twisted cyclic group ring

We give a description of a certain twisted group ring of a cyclic group as a subring of a matrix ring. We proceed in descending generality, ending up with complete description in the case of a cyclic group of prime order as the galois group of a purely ramified extension of discrete valuation rings. For an attempt in the next larger case C_{p^{2}}, see appendix A.

For sake of illustration, a continuing example is included, indicated by (cont.).

1.1 Subrings defined by derivations

Let A be a ring, let I ⊆ A be an ideal. Let k ≥ 1, let x = (x_{j})_{j∈[1,k]} be a tuple of elements of A and let the corresponding inner derivations be denoted by

[figure: A; D_{x_{j}}; A; y; yx_{j} − x_{j}y .]

We note that if x_{i}x_{j} = x_{j}x_{i}, then D_{x_{i}} ∘ D_{x_{j}} = D_{x_{j}} ∘ D_{x_{i}}.

Lemma 1.1 Let h = (h_{j})_{j∈[1,k]} be a tuple of positive integers (the height). Let l = (l_{j})_{j∈[1,k]} be a tuple of positive integers (the length). Then the abelian subgroup

A(x, h, l)_{I} := {y ∈ A : D_{x_{1}}^{i_{1}} ∘⋯∘D_{x_{k}}^{i_{k}}(y) ∈ I^{i_{1}l_{1}+···+i_{k}l_{k}} for (i_{j})_{j∈[1,k]} ∈ ∏_{j∈[1,k]}[0, h_{j}]} ⊆ A

is a subring of A.

Given y and z in A(x, h, l)_{I}, we need to show that the product yz is contained in A(x, h, l)_{I}. So suppose given (i_{j})_{j∈[1,k]} ∈ ∏_{j∈[1,k]}[0, h_{j}]. The term

D_{x_{1}}^{i_{1}} ∘ ⋯ ∘ D_{x_{k}}^{i_{k}}(yz)

equals a sum over terms of type

(D_{x_{1}}^{i′_{1}} ∘ ⋯ ∘ D_{x_{k}}^{i′_{k}}(y)) · (D_{x_{1}}^{i′′_{1}} ∘ ⋯ ∘ D_{x_{k}}^{i′′_{k}}(z))

where (i′_{j})_{j}, (i′′_{j})_{j} ∈ ∏_{j∈[1,k]}[0, h_{j}] such that (i′_{j})_{j} + (i′′_{j})_{j} = (i_{j})_{j}; each such summand is contained in I^{i_{1}l_{1}+···+i_{k}l_{k}}.

1.2 A Wedderburn embedding

Let T |S be a finite and purely ramified extension of discrete valuation rings of residue characteristic p ≥ 3. Let L|K be the corresponding extension of fields of fractions, assumed to be galois with galois group G of order g. Let t generate the maximal ideal of T, let s := (−1)^{g+1} N_{L|K}(t) generate the maximal ideal of S. Let

𝔇_{T |S}^{−1} := {x ∈ L : Tr_{L|K}(xT) ⊆ S} ⊆ L

define the different ideal 𝔇_{T |S} ⊆ T, and let Δ_{T |S} := N_{L|K}(𝔇_{T |S}) ⊆ S denote the discriminant ideal of T |S. Write the minimal polynomial of t over K as

μ_{t,K}(X) = X^{g} + ( ∑_{j∈[1,g−1]} e_{j}X^{j}) − s ∈ S[X] .

We recall that T = S[t], that T t^{g} = T s, that S/Ss ^{\overline{}^{∼}} T/T t, that μ_{t,K}(X) ≡_{s} X^{g} and that 𝔇_{T |S} = (μ′_{t,K}(t)) ⊆ T [28, III.§6, cor. 2].

By T ≀ G = {∑_{ρ∈G} ρy_{ρ} : y_{ρ} ∈ T } we denote the twisted group ring carrying the multiplication induced by (ρy)(τz) = (ρτ)(y^{τ}z), where ρ, τ ∈ G and y, z ∈ T. Let Ξ denote the image of the Wedderburn embedding (of S-algebras)

\begin{matrix}T ≀ G & ^{\overline{}^{ω}} & End_{S} T =: Γ \\ ρy & \overline{} & (ρ̇ẏ : x \overline{}x^{ρ}y) . \end{matrix}

We consider the subring

Λ := {f ∈ Γ : (T t^{i})f ⊆ T t^{i} for all i ≥ 0} ⊆ Γ

which contains Ξ, i.e.

T ≀ G \overline{}^{ω}_{∼} Ξ ⊆ Λ ⊆ Γ ,

by a slight abuse of the notation ω. Since dim_{L} L ⊗_{T} Λ = g, we have ℓ_{T} (Λ/ṫΛ) = ℓ_{T} (Λ/Λṫ) = g, and so we obtain

ṫΛ = Λṫ = {f ∈ Γ : (T t^{i})f ⊆ T t^{i+1} for all i ≥ 0} ,

which is thus an ideal of Λ. Moreover, note that we may write

Λ = {f ∈ Γ : (T t^{i})f ⊆ T t^{i} for all i ∈ [0, g − 1]} .

Remark 1.2 Using matrices with respect to the S-linear basis (t^{0}, t^{1}, …, t^{g−1}) of T to represent elements of Γ = End_{S} T, the subring Λ of the full matrix ring S^{g×g} is given by the set of matrices with strictly lower triangular entries contained in Ss. The ideal ṫΛ is given by the set of matrices with lower triangular entries contained in Ss, including the main diagonal. In this interpretation, we shall refer to matrix positions using the coordinates [0, g − 1] × [0, g − 1].

Remark 1.3 (cf. [2, prop. 3.5]) We have g ≢_{p} 0 if and only if

T ≀ G \overline{}^{ω}_{∼} Ξ = Λ .

In particular, in this case T is projective as a module over T ≀ G (cf. [20, prop. 2.4]).

By [28, IV.§2, cor. 1, cor. 3], g ≢_{p} 0 is equivalent to v_{t}(t^{ρ} − t) = 1 for all ρ ∈ G ∖ {1}.

Now by [20, Cor 2.17], the S-linear colength of Ξ in Γ is given by ℓ_{S}(Γ/Ξ) = g·v_{s}(Δ_{T |S})/2. On the other hand, ℓ_{S}(Γ/Λ) = g(g − 1)/2. Therefore, the inclusion Ξ ⊆ Λ is an equality if and only if v_{s}(Δ_{T |S}) = g − 1. But

v_{s}(Δ_{T |S}) = g^{−1} ∑_{\begin{matrix}ρ, τ ∈ G, \\ ρ ≠ τ \end{matrix}} v_{t}(t^{ρ} − t^{τ}) = ∑_{ρ ∈ G∖\{1\}}v_{t}(t^{ρ} − t) .

Example 1.4 (cont.) Let S = Z_{(3)}, s = 3, t = π_{2} := (ζ_{9} − 1)(ζ_{9}^{−1} − 1), and T = Z_{(3)}[π_{2}] (cf. sec. 5.2). Let σ : ζ_{9} \overline{}ζ_{9}^{4}, restricted from Q(ζ_{9}) to T. We have G = \{1, σ, σ^{2}\} ≃ C_{3}. We shall use the matrix interpretation explained in (1.2).

The Wedderburn embedding sends t to ṫ = [\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 3 & −9 & 6 \end{matrix}], the last row resulting from

t^{3} = 3 − 9t + 6t^{2}. Furthermore, it sends σ to σ̇ = [\begin{matrix}1 & 0 & 0 \\ 6 & −5 & 1 \\ 24 & −21 & 4 \end{matrix}], the second row resulting from t^{σ} = 6 − 5t + t^{2}. In particular, t^{σ} − t = t^{2} − 6t + 6 has valuation 2 at t, so Ξ ⊂ Λ is a proper subring, i.e. the Wedderburn embedding does not induce an isomorphism of T ≀ G with Λ.

We have Λ = [\begin{matrix}S & S & S \\ 3S & S & S \\ 3S & 3S & S \end{matrix} ], containing σ̇, and ṫΛ = Λṫ = [\begin{matrix}3S & S & S \\ 3S & 3S & S \\ 3S & 3S & 3S \end{matrix} ].

1.3 The intermediate ring Λ^{D}

We assume that

b := −1 + min_{ρ ∈ G} v_{t}(t^{ρ} − t) ≥ 1 .

For example, if G = ⟨σ⟩ ≃ C_{g}, then b = −1 + v_{t}(t^{σ} − t) since v_{t}(t^{σ^{i}} − t) ≥ v_{t}(t^{σ} − t) for i ∈ [0, g − 1].

Let

Λ^{D} := Λ ((ṫ), (g − 1), (1 + b))_{ṫΛ} ⊆ Λ,

where k = 1 in the notation of (1.1). Explicitly, we have

Λ^{D} = \{u ∈ Λ : ∑_{j∈[0,i]}(−1)^{j} (\begin{matrix}i \\ j \end{matrix}) ṫ^{j}uṫ^{i−j} ≡_{ṫ^{(1+b)i}Λ} 0 for all i ∈ [0, g − 1]\} .

Lemma 1.5 We have

Ξ ⊆ Λ^{D} ( ⊆ Λ ⊆ Γ ) .

Since Λ^{D} is a subring of Λ, it suffices to show that ṫ ∈ Λ^{D}, which follows from Di ; ṫ(ṫ) = 0 for i ≥ 1, and that ρ̇ ∈ Λ^{D} for ρ ∈ G, i.e. that Di ; ṫ(ρ̇) lies in ṫ^{i(b+1)}Λ for ρ ∈ G and i ∈ [0, g −1]. But

\begin{aligned}0 & ≡_{t^{(1+b)i}} & (t^{ρ} − t)^{i} \\ & = & ∑_{j∈[0,i]}(−1)^{i−j} (\begin{matrix}i \\ j \end{matrix}) (t^{j})^{ρ}t^{i−j} \end{aligned}

implies

0 ≡_{ṫ^{(1+b)i}Λ} ∑_{j∈[0,i]}(−1)^{i−j} (\begin{matrix}i \\ j \end{matrix}) (ṫ^{j})ρ̇ṫ^{i−j}

=  ρ̇^{−1} ( ∑_{j∈[0,i]}(−1)^{i−j} (\begin{matrix}i \\ j \end{matrix}) ṫ^{j}ρ̇ṫ^{i−j})
=  ρ̇^{−1} Di ; ṫ(ρ̇) .

1.4 A description of Λ^{D} via ties

For i ∈ Z, we write

i =: g\underline{i} + \overline{i} with \overline{i} ∈ [0, g − 1].

We note that \underline{−i} = −1 − \underline{i − 1}, and that \overline{−i} = g − 1 − \overline{i − 1}.

For i ∈ [0, g − 1] and j ≥ 0, we consider the element ε_{i,j} ∈ Λ ⊆ Γ defined on the S-linear basis (t^{0}, …, t^{g−1}) of T by

(t^{l})ε_{i,j} := ∂_{i,l}t\overline{l+j}s\underline{l+j}

for l ∈ [0, g − 1]. In the interpretation of (1.2), ε_{i,j} is the matrix with a single non-zero entry s\underline{i+j} at position (i, \overline{i + j}). The tuple (ε_{i,j})_{i,j∈[0,g−1]} forms an S-linear basis of Λ. For instance,

ṫ = ( ∑_{i∈[0,g−1]} ε_{i,1}) − s^{−1} ( ∑_{j∈[1,g−2]} e_{j}ε_{g−1,j+1}) − e_{g−1}ε_{g−1,0} ∈ Λ .

The elements ρ̇ seem to be harder to describe in this manner (cf. 1.4). In case G = C_{p}, we shall circumvent this problem by giving a S-linear basis of Ξ without giving the transition matrix from the ‘initial’ S-linear basis (ṫ^{i}σ̇^{j})_{i,j∈[0,g−1]} of Ξ to it. To do so, we use a description of Ξ by congruences, or ties, between the coefficients with respect to the (ε_{i,j})_{i,j}- basis of Λ.

Lemma 1.6 For i, i′ ∈ [0, g − 1] and j, j′ ≥ 0, we obtain

ε_{i,j}ε_{i′,j′} = ∂_{i′,\overline{i+j}}ε_{i,j+j′} .

This follows by evaluation on t^{l} for l ∈ [0, g − 1].

Let

ẗ := ∑_{i∈[0,g−1]} ε_{i,1} ∈ Λ .

Lemma 1.7 For j ≥ 0, we obtain

ẗ^{j} = ∑_{i∈[0,g−1]}ε_{i,j} .

This follows using induction on j together with (1.6).

Lemma 1.8 We have ẗΛ = Λẗ = ṫΛ = Λṫ . An S-linear basis of ẗ^{k}Λ for k ≥ 0 is given by

(s^{−\underline{j−k}}ε_{i,j})_{i,j∈[0,g−1]} .

To see this, we may use the matrix interpretation of (1.2).

Assumption 1.9 Assume that v_{s}(e_{j}) ≥ 1 + b − \underline{b + j} for j ∈ [1, g − 1].

For example, if b = 1, then the assumption (1.9) reads e_{j} ∈ Ss^{2} for j ∈ [1, g − 2] and e_{g−1} ∈ Ss.

Lemma 1.10 Assuming (1.9), we get ẗ− ṫ ∈ ẗ^{1+b(g−1)}Λ .

We have ẗ− ṫ = e_{g−1}ε_{g−1,0} + s^{−1} ∑_{j∈[1,g−2]} e_{j}ε_{g−1,j+1}. Thus, by (1.8), we need to prove the inequalities

\{ \begin{matrix}v_{s}(e_{j}) & ≥ & − \underline{(j + 1) − (1 + b(g − 1))} + 1 & = & 1 + b − \underline{j + b} & for j ∈ [1, g − 2] \\ v_{s}(e_{g−1}) & ≥ & − \underline{0 − (1 + b(g − 1))} & = & 1 + b − \underline{(g − 1) + b} , \end{matrix}

which in turn have been assumed in (1.9).

Using (1.10), we are in position to substitute ṫ by ẗ in the construction of the subring Λ^{D}.

Lemma 1.11 Assume (1.9). Given γ ≥ 0, we obtain

ṫ^{γ}Λ^{D} =^{(1)} ṫ^{γ}Λ((ṫ), (g − 1), (1 + b))_{ṫΛ}
=^{(2)} \{u ∈ Λ : ∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) ṫ^{h}uṫ^{l−h} ≡_{ṫ^{(1+b)l+γ}Λ} 0 for all l ∈ [0, g − 1]\}

=^{(3)} \{u ∈ Λ : ∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) ẗ^{h}uẗ^{l−h} ≡_{ẗ^{(1+b)l+γ}Λ} 0 for all l ∈ [0, g − 1]\}
=^{(4)} ẗ^{γ}Λ((ẗ), (g − 1), (1 + b))_{ẗΛ} .

Equation (1) follows by definition of Λ^{D}. Let us prove (3). First, we remark that for i ≥ 0, (1.10, 1.8) give

ṫ^{i} ≡_{ẗ^{i+b(g−1)}Λ} ṫ^{i−1}ẗ ≡_{ẗ^{i+b(g−1)}Λ} ⋯ ≡_{ẗ^{i+b(g−1)}Λ} ẗ^{i} .

Now, both sets are subsets of ṫ^{γ}Λ = ẗ^{γ}Λ, as we see by putting l = 0. So suppose given u ∈ ṫ^{γ}Λ = ẗ^{γ}Λ. Using (1.10, 1.8) we obtain

ṫ^{h}uṫ^{l−h} ≡_{ẗ^{l+b(g−1)+γ}Λ} ẗ^{h}uṫ^{l−h} ≡_{ẗ^{l+b(g−1)+γ}Λ} ẗ^{h}uẗ^{l−h} .

for 0 ≤ h ≤ l ≤ g − 1. Since l + b(g − 1) + γ ≥ (1 + b)l + γ, this shows (3). Let us prove (4). Again, both sides are contained in ẗ^{γ}Λ. So if v ∈ Λ, then ẗ^{γ}v is in the right hand side of (4) if and only if v ∈ Λ ((ẗ), (g − 1), (1 + b))_{ẗΛ}, for multiplication with ẗ is injective. This in turn is the case if and only if ẗ^{γ}v is contained in the left hand side of (4), again by injectivity of the multiplication with ẗ. This proves (4). The argument for (2) is analogous.

Lemma 1.12 For x, y ∈ Z with y coprime to g, we obtain

∑_{i∈[0,g−1]}\underline{x + iy} = g^{−1} ∑_{i∈[0,g−1]} ((x + iy) − (\overline{x + iy})) = x + (g − 1)(y − 1)/2 .

Now we shall give a description of Λ^{D} via ties; that is, we give certain congruences between the coefficients with respect to the S-linear ε-basis of Λ that are necessary, and, taken together, also sufficient for an element of Λ to lie in Λ^{D}. Necessity will follow from (1.5), whereas sufficiency needs a comparison of colengths.

Lemma 1.13 Assume (1.9). Given γ ≥ 0, we get

ṫ^{γ}Λ^{D} = { ∑_{i,j∈[0,g−1]}a_{i,j}ε_{i,j} : a_{i,j} ∈ S , and for all i, j, l ∈ [0, g − 1], we have

v_{s}( ∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) a_{\overline{i+h},j}) ≥ 1 + \underline{bl − j − 1 + γ} \}

⊆ Λ .

The lth condition of (1.11) (l ∈ [0, g − 1]) for an element u = ∑_{i,j∈[0,g−1]} a_{i,j}ε_{i,j} ∈ Λ (a_{i,j} ∈ S) to lie in ṫ^{γ}Λ^{D} reads

0 ≡_{ẗ^{(1+b)l+γ}Λ} ∑_{h∈[0,l]} ∑_{i,j∈[0,g−1]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) ẗ^{h}a_{i,j}ε_{i,j}ẗ^{l−h}
=^{(1.7)}  ∑_{h∈[0,l]} ∑_{i,j,i′,i′′∈[0,g−1]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) a_{i,j}ε_{i′,h}ε_{i,j}ε_{i′′,l−h}
=^{(1.6)}  ∑_{h∈[0,l]} ∑_{i,j,i′,i′′∈[0,g−1]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) a_{i,j}∂_{i,\overline{i′+h}}∂_{i′′,\overline{i′+h+j}}ε_{i′,l+j}
=  ∑_{j,i′∈[0,g−1]} (∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) a_{\overline{i′+h},j}) ε_{i′,l+j} .

For a ∈ S and k ≥ 0, the element aε_{i,j} is contained in ẗ^{k}Λ if and only if v_{s}(a) ≥ −\underline{j − k} (cf. 1.8). Thus we obtain the set of conditions

(∗_{i,j,l})  v_{s}( ∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) a_{\overline{i+h},j}) ≥ −(\underline{−bl + j − γ}) = 1 + \underline{bl − j − 1 + γ}

for i, j, l ∈ [0, g − 1], as claimed.

1.5 The cyclic case of order p

As we have seen, in general we have an inclusion T ≀ G ^{\overline{}^{∼}} Ξ ⊆ Λ^{D}, and we dispose of a workable description of Λ^{D}. Now we shall consider a case in which this inclusion will turn out to be an equality.

Consider the case g = p, i.e. assume G = ⟨σ⟩ = C_{p} to be of order equal to the residue characteristic p of S. Note that then b = −1+v_{t}(t^{σ} −t), and recall that we have stipulated b ≥ 1.

Remark 1.14 We have

v_{t}(𝔇_{T |S}) = v_{s}(Δ_{T |S}) = p^{−1} v_{t}( ∏_{\begin{matrix}ρ, τ ∈ G , \\ ρ ≠ τ \end{matrix}}(t^{ρ} − t^{τ})) = v_{t}( ∏_{ρ∈G∖\{1\}}(t^{ρ} − t)) = (p−1)(1+b)

(cf. [28, V.§3, lem. 3]). Remark 1.15 In the present case G = C_{p}, assumption (1.9) is fulfilled. Since 𝔇_{T |S} = (μ′_{t,K}(t)), (1.14) implies in particular that

t^{(p−1)(1+b)} | μ′_{L|K}(t) = pt^{p−1} + ∑_{j∈[1,p−1]} je_{j}t^{j−1} ,

i.e. that v_{t}(e_{j}) ≥ (p − 1)(1 + b) − (j − 1) for j ∈ [1, p − 1], for the valuations of the summands are pairwise different. But since e_{j} ∈ S, this implies

v_{s}(e_{j}) ≥ \underline{(p − 1)(1 + b) − (j − 1) + (p − 1)} = 1 + b − \underline{b + j}

for j ∈ [1, p − 1], and thus, assumption (1.9) is fulfilled. Lemma 1.16 Suppose given α_{0}, …, α_{p−1} ≥ 0. Consider the S-linear submodule

M := \{(x_{h})_{h∈[0,p−1]} ∈ S^{p} : for all l ∈ [0, p − 1], we have v_{s}(∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) x_{h}) ≥ α_{l}\}
⊆ S^{p} .

An S-linear basis of M is given by the tuple ((s^{α_{l}} (\begin{matrix}i \\ l \end{matrix}))_{i∈[0,p−1]})_{l∈[0,p−1]}. Let M̃ be the S-linear submodule of S^{p} spanned by ((s^{α_{l′}} (\begin{matrix}i \\ l′ \end{matrix}))_{i∈[0,p−1]})_{l′∈[0,p−1]}. Since

∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) (s^{α_{l′}} (\begin{matrix}h \\ l′ \end{matrix})) = (−1)^{l′}∂_{l,l′}s^{α_{l′}}

for l, l′ ∈ [0, p − 1], we have M̃ ⊆ M. The S-linear colength of M in S^{p} is given by ∑_{l∈[0,p−1]} α_{l}, and so is the colength of M̃. Hence M̃ = M. Proposition 1.17 Suppose given γ ≥ 0 such that v_{s}(p) ≥ b − \underline{b − γ} . Then

ṫ^{γ}Λ^{D} = { ∑_{i,j∈[0,p−1]}a_{i,j}ε_{i,j} : a_{i,j} ∈ S , and for all j, l ∈ [0, p − 1], we have

v_{s}( ∑_{h∈[0,l]}(−1)^{h} (\begin{matrix}l \\ h \end{matrix}) a_{h,j}) ≥ 1 + \underline{bl − j − 1 + γ} \}

⊆ Λ .

An S-linear basis of t^{γ}Λ^{D} is given by

(μ_{l,j}^{(γ)})_{l, j ∈ [0,p−1]} := (s^{1+\underline{bl−j−1+γ}} ∑_{i∈[0,p−1]} (\begin{matrix}i \\ l \end{matrix}) ε_{i,j})_{l, j ∈ [0,p−1]} .

The according basis of Λ^{D} will also be written (μ_{l,j})_{l,j} := (μ_{l,j}^{(0)})_{l,j}. The S-linear colength of Λ^{D} in Λ is given by bp(p − 1)/2.

Comparing with (1.13), we have to show the redundancy of the conditions (∗_{i,j,l}) for

i ∈ [1, p − 1], j, l ∈ [0, p − 1]. First of all, we have (−1)^{h} (\begin{matrix}p−1 \\ h \end{matrix}) ≡_{s^{1+\underline{b(p−1)−j−1+γ}}} 1 for

h ∈ [0, p − 1], since this holds modulo p. Hence the validity of (∗_{i,j,p−1}) is independent of i ∈ [0, p−1]. By downwards induction on l ∈ [0, p−2], forming the difference, equivalence of (∗_{i,j,l}) and (∗_{i+1,j,l}) ensues from (∗_{i,j,l+1}), where i ∈ [0, p − 2].

Now (1.16), applied to a fixed j ∈ [0, p − 1], yields the S-linear basis as claimed. The colength of Λ^{D} in Λ is given by

\begin{aligned}∑_{l∈[0,p−1]} ∑_{j∈[0,p−1]}(1 + \underline{bl − j − 1}) & =^{(1.12)} & p^{2} + ∑_{l∈[0,p−1]}(bl − p) \\ & = & bp(p − 1)/2. \end{aligned}

Remark 1.18 In (1.17), we can as well fix any m ∈ [0, p − 1] and impose (∗_{m,j,l}) on elements of Λ for j, l ∈ [0, p − 1] to describe ṫ^{γ}Λ^{D} inside. Accordingly, we obtain an

S-linear basis (s^{1+\underline{bl−j−1+γ}} ∑_{i∈[0,p−1]} (\begin{matrix}i \\ l \end{matrix}) ε_{\overline{i+m},j})_{l,j} of ṫ^{γ}Λ^{D}.

Theorem 1.19 Recall that we assume b = v_{t}(t^{σ} − t) ≥ 1, where C_{p} = ⟨σ⟩. Suppose in addition that v_{s}(p) ≥ b − \underline{b}.

The Wedderburn embedding factors over the S-algebra isomorphism

T ≀ C_{p} \overline{}^{ω}_{∼} Λ^{D} ⊆ Λ ⊆ Γ .

By (1.5), it suffices to show that the S-linear colengths of (T ≀ G)ω = Ξ and of Λ^{D} in Λ are equal. The colength of Λ in Γ equals p(p − 1)/2, so that, by (1.17), the colength of Λ^{D} in Γ equals (1 + b)p(p − 1)/2. On the other hand, by [20, cor. 2.17], the colength of

Ξ in Γ equals p v_{s}(Δ_{T |S})/2 =^{(1.14)} p(1 + b)(p − 1)/2.

Example 1.20 (cont.) We have ṫ = [\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 3 & −9 & 6 \end{matrix}] and ẗ= [\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 3 & 0 & 0 \end{matrix}].
We have g = p = 3 and b = 1. Since μ_{t,Q}(X) = X^{3} − 6X^{2} + 9X − 3, assumption (1.9) is
satisfied, in accordance with (1.15). Consequently, we have ṫ − ẗ= [\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & −9 & 6 \end{matrix}] ∈ 3Λ (cf. 1.10).

If γ ≤ 1, then 1 = v_{s}(p) ≥ b − \underline{b − γ} = 1, and so (1.17) can be applied to give

Z_{(3)}[π_{2}] ≀ C_{3} = T ≀ C_{3}
\overline{}^{ω}_{∼} Λ^{D}
= \{ [\begin{matrix}a_{0,0} & a_{0,1} & a_{0,2} \\ 3a_{1,2} & a_{1,0} & a_{1,1} \\ 3a_{2,1} & 3a_{2,2} & a_{2,0} \end{matrix} ] : a_{0,0} ∈ 3^{0}S, a_{0,0} − a_{1,0} ∈ 3^{1}S, a_{0,0} − 2a_{1,0} + a_{2,0} ∈ 3^{1}S;

a_{0,1} ∈ 3^{0}S, a_{0,1} − a_{1,1} ∈ 3^{0}S, a_{0,1} − 2a_{1,1} + a_{2,1} ∈ 3^{1}S; a_{0,2} ∈ 3^{0}S, a_{0,2} − a_{1,2} ∈ 3^{0}S, a_{0,2} − 2a_{1,2} + a_{2,2} ∈ 3^{0}S; }

= \{ [\begin{matrix}a_{0,0} & a_{0,1} & a_{0,2} \\ 3a_{1,2} & a_{1,0} & a_{1,1} \\ 3a_{2,1} & 3a_{2,2} & a_{2,0} \end{matrix} ] : a_{0,0} ≡_{3} a_{1,0} ≡_{3} a_{2,0}; a_{0,1} + a_{1,1} + a_{2,1} ≡_{3} 0 \}
⊆ S^{3×3} ,

where S = Z_{(3)}. In particular, we have σ̇ = [\begin{matrix}1 & 0 & 0 \\ 6 & −5 & 1 \\ 24 & −21 & 4 \end{matrix}] ∈ Λ^{D}, as expected. Similarly,

ṫΛ^{D} = \{ [\begin{matrix}a_{0,0} & a_{0,1} & a_{0,2} \\ 3a_{1,2} & a_{1,0} & a_{1,1} \\ 3a_{2,1} & 3a_{2,2} & a_{2,0} \end{matrix} ] : a_{0,0} ∈ 3^{1}S, a_{0,0} − a_{1,0} ∈ 3^{1}S, a_{0,0} − 2a_{1,0} + a_{2,0} ∈ 3^{1}S;

a_{0,1} ∈ 3^{0}S, a_{0,1} − a_{1,1} ∈ 3^{1}S, a_{0,1} − 2a_{1,1} + a_{2,1} ∈ 3^{1}S; a_{0,2} ∈ 3^{0}S, a_{0,2} − a_{1,2} ∈ 3^{0}S, a_{0,2} − 2a_{1,2} + a_{2,2} ∈ 3^{1}S; }

= \{ [\begin{matrix}3a_{0,0} & a_{0,1} & a_{0,2} \\ 3a_{1,2} & 3a_{1,0} & a_{1,1} \\ 3a_{2,1} & 3a_{2,2} & 3a_{2,0} \end{matrix} ] : a_{0,1} ≡_{3} a_{1,1} ≡_{3} a_{2,1}; a_{0,2} + a_{1,2} + a_{2,2} ≡_{3} 0 \}
⊆ S^{3×3} .

Example 1.21 Suppose that S contains a primitive p-th root of unity ζ_{p} (so in particular, char K = 0). Let T := S[\sqrt[p]{s}] and t := \sqrt[p]{s}, i.e. let μ_{t,K}(X) = X^{p} − s. Then L|K is galois with galois group C_{p}, and we have

b = −1 + v_{t}(tζ_{p} − t) = p v_{s}(1 − ζ_{p}) = \frac{p}{p − 1} v_{s}(p) ,

i.e. v_{s}(p) = b − \underline{b}, so that (1.19) may be applied.

Corollary 1.22 Suppose given two discrete valuation rings T and T′ over S with T |S and T′|S both purely ramified and galois with galois group C_{p}. Suppose that

v_{s}(Δ_{T |S}) = v_{s}(Δ_{T′|S}) ≤ p v_{s}(p) + p − 1 .

Then

T ≀ C_{p} ≃ T′ ≀ C_{p}

as S-algebras.

Since v_{s}(Δ_{T |S}) = (p − 1)(b + 1) by (1.14), similarly for T′, and since Λ^{D} depends only on S and b (cf. 1.17), we may conclude

T ≀ C_{p} ≃^{(1.19)} Λ^{D} ≃^{(1.19)} T′ ≀ C_{p} .

Example 1.23 (cont.) Let S = Z_{(3)}, T = Z_{(3)}[π_{2}] and let T′ = S[t′] with μ_{t′,K}(X) = X^{3} + 3X^{2} − 18X + 48 [19]. Then Δ_{T |S} = Δ_{T′|S} = Ss^{4}, whence T ≀ C_{3} ≃ T′ ≀ C_{3} by (1.22). The fields of fractions of T and T′, however, are not isomorphic.

2 Nebe decomposition

The purpose of this section is, translated to the basic example Z_{(p^{2})}[ζ_{p^{2}}]|Z_{(p)}, the reduction of the cohomology calculation from the galois group C_{p} × C_{p−1} to its quotient C_{p}, being the galois group of a subextension.

2.1 A block decomposition

Let T |S and U|T be finite purely ramified extensions of discrete valuation rings with maximal ideals generated by s ∈ S, t ∈ T and u ∈ U, respectively. Let K = frac S, L = frac T and E = frac U, and suppose that E|K is galois with galois group H, and that L|K is galois with galois group G. In particular, if

1 \overline{}N \overline{}H \overline{}G \overline{}1

is short exact, then E/L is galois with galois group N. Denote h := |H|, g := |G| and n := |N|, so h = gn.

The situation can be depicted as follows.

[figure: s ∈ S; t ∈ T; u ∈ U; K; G; L; N; E; H]

Suppose given an S-linear submodule V of U spanned by a T -linear basis of U, i.e. such that T ⊗_{S} V \overline{}^{φ}_{∼} U, x ⊗ y \overline{}xy. In general, V is not a subring of U. We write

Γ := End_{S} U
Γ′ := End_{S} T
Γ′′ := End_{T} U
Γ′′_{0} := End_{S} V .

Here we change our notation slightly in that the ring End_{S} T, which has previously been denoted by Γ, is now denoted by Γ′. Similarly, we denote the Wedderburn embeddings by

[figure: U ≀ H; ω; Γ; T ≀ G; ω′; Γ′; U ≀ N; ω′′; Γ′′ ,]

again switching notation from what has been denoted by ω to ω′. Accordingly, we denote the images by

\begin{aligned}Ξ & := & (U ≀ H)ω \\ Ξ′ & := & (T ≀ G)ω′ \\ Ξ′′ & := & (U ≀ N)ω′′ \end{aligned}

Usage of the notation Γ′, ω′, Ξ′ pertains only to the present section 2, in which we consider the passage from U ≀ H to T ≀ G.

Note that we have an isomorphism of S-algebras

[figure: Γ′; ⊗_{S}; Γ′′_{0}; ∼; End_{S}(T ⊗_{S} V); ∼; Γ]
α ⊗ β_{0}  \overline{}(x ⊗ y \overline{}xα ⊗ yβ_{0})
η  \overline{}φ^{−1}ηφ ,

denoted by Γ′ ⊗_{S} Γ′′_{0} \overline{}^{θ}_{∼} Γ. Moreover, we have an isomorphism of T -algebras

[figure: T; ⊗_{S}; Γ′′_{0}; ∼; End_{T} (T; ⊗_{S}; V); ∼; Γ′′; 1; ⊗; β_{0}; 1; ⊗; β_{0}; φ^{−1}(1 ⊗ β_{0})φ ,]

which shall be denoted by T ⊗_{S} Γ′′_{0} \overline{}^{ψ}_{∼} Γ′′ . Thus, we have an S-linear isomorphism

Γ′ ⊗_{T} Γ′′ \overline{}^{1⊗ψ}_{∼} Γ′ ⊗_{T} T ⊗_{S} Γ′′_{0} ^{\overline{}^{∼}} Γ′ ⊗_{S} Γ′′_{0} \overline{}^{θ}_{∼} Γ ,

denoted by

Γ′ ⊗_{T} Γ′′ \overline{}^{ϑ}_{∼} Γ .

Lemma 2.1 If (x_{1}, …, x_{g}) is an S-linear basis of T, and (y_{1}, …, y_{n}) is a T -linear basis of U lying in V, and given α ∈ Γ′ and β ∈ Γ′′, then (α ⊗ β)ϑ sends x_{i}y_{j} to (x_{i}α)(y_{j}β), where i ∈ [1, g], j ∈ [1, n].

In fact, by T -bilinearity we may assume that β = (1 ⊗ β_{0})ψ with β_{0} ∈ Γ′′_{0}, in which case the assertion follows by construction.

We use ϑ for a transport of the S-algebra-structure from Γ to Γ′ ⊗_{T} Γ′′; i.e. given γ_{1} , γ_{2} ∈ Γ′ ⊗_{T} Γ′′, we let

γ_{1} · γ_{2} := ((γ_{1}ϑ) · (γ_{2}ϑ))ϑ^{−1} ,

so that ϑ becomes an isomorphism of S-algebras.

Now we choose V to be the S-linear span of (u^{0}, …, u^{n−1}) in U. Let

Λ′′ := {φ′′ ∈ Γ′′ : u^{k}φ′′ ∈ Uu^{k} for k ≥ 0 } ⊆ Γ′′ ,

which is a sub-T -algebra.

Lemma 2.2 The S-linear submodule Ξ′ ⊗_{T} Λ′′ ⊆ Γ′ ⊗_{T} Γ′′ is a sub-S-algebra.

Given k, l ∈ [0, n − 1], we let e′′_{k,l} ∈ Γ′′ be defined by u^{j}e′′_{k,l} = ∂_{j,k}u^{l} for j ∈ [0, n − 1]. Then

(∗) Ξ′ ⊗_{T} Λ′′ = { ∑_{k,l∈[0,n−1]} ξ′_{k,l} ⊗ e′′_{k,l} : ξ′_{k,l} ∈ Ξ′ for k ≤ l, ξ′_{k,l} ∈ ṫΞ′ for k > l} .

Since we can write e′′_{k,l} = (1 ⊗ e′′_{k,l;0})ψ with e′′_{k,l;0} ∈ Γ′′_{0}, i.e. since e′′_{k,l} restricts to an S-linear endomorphism of V, multiplication of elements written in the form as in (∗) is given by

(ξ′ ⊗ e′′_{k,l})(ξ̃′ ⊗ e′′_{k̃,l̃}) = (ξ′ξ̃′) ⊗ (∂_{l,k̃}e′′_{k,l̃}) ,

where ξ̃, ξ̃′ ∈ Ξ′ and k, l, k̃, l̃ ∈ [0, n − 1]. Now if k > l̃ and l = k̃, then k > l or k̃ > l̃, hence ξ′ξ̃′ ∈ ṫΞ′. Therefore, the S-linear submodule Ξ′ ⊗_{T} Λ′′ is closed under multiplication.

Lemma 2.3 We have Ξϑ^{−1} ⊆ Ξ′ ⊗_{T} Λ′′ .

Since Ξ′⊗_{T} Λ′′ is a sub-S-algebra of Γ′⊗_{T} Γ′′ (2.2), it suffices to show that uωϑ^{−1} ∈ Ξ′⊗_{T} Λ′′ and that ρωϑ^{−1} ∈ Ξ′ ⊗_{T} Λ′′ for ρ ∈ H.

We have (1 ⊗ uω′′)ϑ = uω by (2.1), hence uωϑ^{−1} ∈ Ξ′ ⊗_{T} Λ′′.

Suppose given ρ ∈ H. Let (x_{1}, …, x_{g}) be an S-linear basis of T. The element x_{i}u^{j} is sent to x_{i}^{ρ}(u^{ρ})^{j} by ρω, where i ∈ [1, g], j ∈ [0, n − 1] (2.1). The element of Γ′ determined by x_{i} \overline{}x_{i}^{ρ} is just ρω′, which is in Ξ′. The element of Γ′′ determined by u^{j} \overline{}(u^{ρ})^{j} is contained in Λ′′. The tensor product of these elements is thus contained in Ξ′ ⊗_{T} Λ′′, as was to be shown.

Remark 2.4 In general, the element u^{j} \overline{}(u^{ρ})^{j}, where j ∈ [0, n − 1], is not contained in Ξ′′ since ρ need not be in N.

By (2.2, 2.3), we have a commutative diagram of S-algebras

[figure: Γ′ ⊗_{T} Γ′′; ϑ; ∼; Γ; Ξ′ ⊗_{T} Λ′′; ϑ^{−1}; Ξ .]

Proposition 2.5 If n ≢_{p} 0, then

Ξϑ^{−1} = Ξ′ ⊗_{T} Λ′′ .

In other words, the Wedderburn embedding U ≀ H [figure: ω] Γ factors as

U ≀ H \overline{}^{ωϑ^{−1}}_{∼} Ξ′ ⊗_{T} Λ′′ [figure] Γ′ ⊗_{T} Γ′′ \overline{}^{ϑ}_{∼} Γ .

The tensor product Ξ′ ⊗_{T} Λ′′ is called the Nebe decomposition of U ≀ H.

First, we remark that if n ≢_{p} 0, then Ξ′′ = Λ′′ by (1.3).

We need to show that the S-linear colengths coincide. By [20, cor. 2.17], the colength of Ξ ⊆ Γ equals

h v_{s}(Δ_{U|S})/2 = h v_{u}(𝔇_{U|S})/2 = h v_{u}(𝔇_{U|T} 𝔇_{T |S})/2 = h(n − 1)/2 + hn v_{s}(Δ_{T |S})/2 .

On the other hand, by [20, cor. 2.17], the colength of the composite embedding Ξ′ ⊗_{T} Λ′′ ⊆ Γ′ ⊗_{T} Λ′′ ⊆ Γ′ ⊗_{T} Γ′′ equals

(g v_{s}(Δ_{T |S})/2) · n^{2} + g · n(n − 1)/2 = hn v_{s}(Δ_{T |S})/2 + h(n − 1)/2 .

Example 2.6 (cont.) Let S = Z_{(3)}, s = 3, T = Z_{(3)}[π_{2}], t = π_{2}, U = Z_{(3)}[ζ_{3^{2}}], u = ζ_{3^{2}} − 1, so that H = (Z/3^{2})^{∗} ≃ C_{3} × C_{2}, G = C_{3} and N = C_{2}. We use the Slinear basis (u^{0}, u^{1}) of V and the S-linear basis (t^{0}, t^{1}, t^{2}) of T, and thus the S-linear basis (u^{0}t^{0}, u^{0}t^{1}, u^{0}t^{2}, u^{1}t^{0}, u^{1}t^{1}, u^{1}t^{2}) of U. By (2.5) and using (∗), the Wedderburn descriptions of Ξ′ and ṫΞ′ obtained in (1.20) can be inserted as blocks into

U ≀ H \overline{}^{ω}_{∼} Ξ =

\{[\begin{matrix}a_{0,0;0,0} & a_{0,0;0,1} & a_{0,0;0,2} & a_{0,1;0,0} & a_{0,1;0,1} & a_{0,1;0,2} \\ 3a_{0,0;1,2} & a_{0,0;1,0} & a_{0,0;1,1} & 3a_{0,1;1,2} & a_{0,1;1,0} & a_{0,1;1,1} \\ 3a_{0,0;2,1} & 3a_{0,0;2,2} & a_{0,0;2,0} & 3a_{0,1;2,1} & 3a_{0,1;2,2} & a_{0,1;2,0} \\ 3a_{1,1;0,0} & a_{1,1;0,1} & a_{1,1;0,2} & a_{1,0;0,0} & a_{1,0;0,1} & a_{1,0;0,2} \\ 3a_{1,1;1,2} & 3a_{1,1;1,0} & a_{1,1;1,1} & 3a_{1,0;1,2} & a_{1,0;1,0} & a_{1,0;1,1} \\ 3a_{1,1;2,1} & 3a_{1,1;2,2} & 3a_{1,1;2,0} & 3a_{1,0;2,1} & 3a_{1,0;2,2} & a_{1,0;2,0} \end{matrix} ] | \begin{matrix}\{ & \begin{matrix}a_{0,0;0,0} ≡_{3} a_{0,0;1,0} ≡_{3} a_{0,0;2,0}, \\ a_{0,0;0,1} + a_{0,0;1,1} + a_{0,0;2,1} ≡_{3} 0; \end{matrix} \\ \{ & \begin{matrix}a_{0,1;0,0} ≡_{3} a_{0,1;1,0} ≡_{3} a_{0,1;2,0}, \\ a_{0,1;0,1} + a_{0,1;1,1} + a_{0,1;2,1} ≡_{3} 0; \end{matrix} \\ \{ & \begin{matrix}a_{1,0;0,0} ≡_{3} a_{1,0;1,0} ≡_{3} a_{1,0;2,0}, \\ a_{1,0;0,1} + a_{1,0;1,1} + a_{1,0;2,1} ≡_{3} 0; \end{matrix} \\ \{ & \begin{matrix}a_{1,1;0,1} ≡_{3} a_{1,1;1,1} ≡_{3} a_{1,1;2,1}, \\ a_{1,1;0,2} + a_{1,1;1,2} + a_{1,1;2,2} ≡_{3} 0 \end{matrix} \end{matrix} \}

⊆ (S^{3×3})^{2×2} .

2.2 A reduction isomorphism

We maintain the notation of section 2.1, having chosen V = S⟨u^{0}, …, u^{n−1}⟩ ⊆ U.

Definition 2.7 Given a Ξ′-module Y, the S-module Y ⊗_{T} U decomposes into a direct sum

Y ⊗_{T} U = ⨁_{j∈[0,n−1]} Y ⊗ u^{j},

so that we may endow it with the structure of a Ξ′ ⊗_{T} Λ′′-module by means of

(y ⊗ u^{j})(ξ′ ⊗ e′′_{k,l}) := yξ′ ⊗ ∂_{j,k}u^{l},

where y ∈ Y, ξ′ ∈ Ξ′, j, k, l ∈ [0, n − 1], ξ′ ∈ ṫΞ if k > l, and where e′′_{k,l} is as in (∗) above.

More naturally explained, we use the isomorphism Y ⊗_{T} U ≃ Y ⊗_{S} V to transport the structure of an Ξ′ ⊗_{S} Γ′′_{0}-module from Y ⊗_{S} V to Y ⊗_{T} U. Restricting the S-algebra isomorphism Γ′ ⊗_{S} Γ′′_{0} ^{\overline{}^{∼}} Γ′ ⊗_{T} Γ′′ to Ξ′ ⊗_{S} Γ′′_{0} ^{\overline{}^{∼}} Ξ′ ⊗_{T} Γ′′, we obtain on Y ⊗_{T} U the structure of a Ξ′ ⊗_{T} Γ′′-module, which we restrict to Ξ′ ⊗_{T} Λ′′. This also shows how the Ξ′ ⊗_{T} Λ′′-module structure on Y ⊗_{T} U depends on the choice of V.

We obtain an exact functor

mod- Ξ′ ^{\overline{}^{F:=(−⊗_{T} U)}} mod-(Ξ′ ⊗_{T} Λ′′)
(Y ^{\overline{}^{α}} Y′)  \overline{}(Y ⊗_{T} U ^{\overline{}^{α⊗1}} Y′ ⊗_{T} U) .

Lemma 2.8 The functor F maps projective Ξ-modules to projective Ξ′ ⊗_{T} Λ′′-modules. Moreover, F is full and faithful.

To show that F Ξ′ is projective over Ξ′ ⊗_{T} Λ′′, we remark that

[figure: F Ξ′ = Ξ′ ⊗_{T} U; ∼; (1 ⊗ e′′_{0,0})(Ξ′ ⊗_{T} Λ′′); ξ′ ⊗ u^{j}; ξ′ ⊗ e′′_{0,j} ,]

where j ∈ [0, n − 1], is an isomorphism of Ξ′ ⊗_{T} Λ′′-modules.

We shall prove that F is full and faithful. Given Y_{1}, Y_{2} ∈ mod- Ξ′, we claim that

Ξ′(Y_{1}, Y_{2}) ^{\overline{}^{F}} _{Ξ′⊗_{T} Λ′′}(F Y_{1}, F Y_{2})

is an isomorphism. Using a two-step free resolution of Y_{1} and exactness of F, we may assume that Y_{1} = Ξ′. Likewise, using projectivity of F Ξ′ and a two-step free resolution of Y_{2}, we may assume Y_{2} = Ξ′. But in this case, we have an isomorphism

[figure: Ξ′; ∼; _{Ξ′}(Ξ′, Ξ′); F; ∼; _{Ξ′⊗_{T} Λ′′}(F Ξ′, F Ξ′); ∼; (1 ⊗ e′′_{0,0})(Ξ′ ⊗_{T} Λ′′)(1 ⊗ e′′_{0,0})]
[figure: ξ′; ξ′ · (−); (ξ′ ⊗ 1) · (−); ξ′ ⊗ e′′_{0,0}.]

Lemma 2.9 There is an isomorphism

Ext_{Ξ′}^{∗}(T, T)  \overline{}^{Ext^{∗}(F)}_{∼}  Ext_{Ξ′⊗_{T} Λ′′}^{∗}(U, U)

of graded S-algebras with respect to the Yoneda product, where U is a Ξ′ ⊗_{T} Λ′′-module via U ≃ T ⊗_{T} U = F T.

Since F preserves projectivity (2.8), we may apply F to a projective resolution of T to obtain a projective resolution of U, and use this resolution to calculate Ext_{U≀H}^{∗}(U, U). Application of F to morphisms of complexes modulo homotopy now yields the morphism of graded S-algebras Ext^{∗}(F). But since F is full and faithful (2.8), this map is an isomorphism.

Proposition 2.10 If n ≢_{p} 0, then

Ext_{T ≀G}^{∗}(T, T) ≃ Ext_{U≀H}^{∗}(U, U)

as graded S-algebras.

By (2.5), we have Ξ′ ⊗_{T} Λ′′ ≃ Ξ ≃ U ≀H. So (2.9) gives the assertion, provided the module structures of Ξ′ ⊗_{T} Λ′′ and of U ≀ H on U coincide. Suppose given ξ′ ⊗ e′′_{k,l} ∈ Ξ′ ⊗_{T} Λ′′, ξ′ ∈ Ξ′, k, l ∈ [0, n − 1]. Let (x_{1}, …, x_{g}) be a S-linear basis of T.

In the interpretation U ≃ T ⊗_{T} U = F T, we have (x_{i} ⊗ u^{j})(ξ′ ⊗ e′′_{k,l}) = (x_{i}ξ′) ⊗ (∂_{j,k}u^{l}) = 1 ⊗ (x_{i}ξ′)(∂_{j,k}u^{l}) by definition of F.

On the other hand, by (2.1), the element ξ′ ⊗e′′_{k,l} is mapped via ϑ to the element in Γ that maps x_{i}u^{j} to (x_{i}ξ′)(∂_{j,k}u^{l}). So after identifying T ⊗_{T} U ≃ U, the operations coincide.

Example 2.11 (cont.) We have

Ext_{Z_{(3)}[ζ_{3^{2}} ]≀(C_{3}×C_{2})}^{∗}(Z_{(3)}[ζ_{3^{2}}], Z_{(3)}[ζ_{3^{2}}]) ≃ Ext_{Z_{(3)}[π_{2}]≀C_{3}}^{∗}(Z_{(3)}[π_{2}], Z_{(3)}[π_{2}])

as graded Z_{(3)}-algebras.

3 Cup product and Yoneda product coincide

In this section we let T be a commutative ring, we let G be a group acting on T via a group morphism G \overline{}Aut_{ring} T, and define S := Fix_{G} T to be the fixed ring of this operation. We assume that T is a free module over S.

We consider the twisted group ring T ≀ G with respect to this operation, which is an S-algebra.

We write ⊗ := ⊗_{S}. Let

P := (⋯ ^{\overline{}^{d_{3}}} SG^{⊗3} ^{\overline{}^{d_{2}}} SG^{⊗2} ^{\overline{}^{d_{1}}} \underbrace{SG^{⊗1}}_{degree 0} \overline{}0 \overline{}⋯)

denote the bar resolution, which is a projective resolution of the trivial module S over SG. The differential is given by

[figure: SG^{⊗i+1}; d_{i}; SG^{⊗i}; g_{[0,i]}; ∑_{l∈[0,i]}(−1)^{i−l}g_{[0,i]∖{l}}]

where i ≥ 1, and where we write shorthand

g_{A} := g_{a_{1}} ⊗ g_{a_{2}} ⊗ ⋯ ⊗ g_{a_{k}},

for A ⊆ Z, where k := #A < ∞, where a_{i} ∈ A and g_{a_{i}} ∈ G for i ∈ [1, k], and where a_{1} < a_{2} < ⋯ < a_{k}. Let SG ^{\overline{}^{d_{0}}} S denote the augmentation map, sending each element of G to 1 ∈ S. We have a quasiisomorphism P ^{\overline{}^{d_{0}}} S, where S is regarded as a complex concentrated in degree 0 and where d_{0} also denotes the morphism of complexes given by d_{0} in degree 0.

Let

Mod- SG ^{\overline{}^{L}} Mod-(T ≀ G)
M \overline{}LM := M ⊗_{SG} T ≀ G
Mod- SG ^{\overline{}^{R}} Mod-(T ≀ G)
N|_{SG} =: RN \overline{}N .

Then L is left adjoint to R, with adjunction morphisms denoted by

LRN ^{\overline{}^{ηN}} N  counit
n ⊗ x \overline{}nx
RLM ^{\overline{}^{εM}} M  unit
m ⊗ 1 \overline{}m

We note that T ≃ LS, by means of x \overline{}1 ⊗ x, x^{g} \overline{}1 ⊗ xg, x ∈ T, g ∈ G, which we use as identification. Since T ≀ G is free as a module over SG, both R and L are exact. To complexes, R and L are applied entrywise. So for instance, RLS is the complex of SG-modules having entry T in degree 0 and entry 0 elsewhere.

We use the description

Ext_{T ≀G}^{i}(T, T) = _{K^{−}(T ≀G)}(LP, LP [i]) ,

where [i] denotes the shift of complexes by i steps to the left, i.e. (X[i])_{j} := X_{j−i}, where i ≥ 0, j ∈ Z. Suppose given u ∈ _{K^{−}(T ≀G)}(LP, LP [i]), v ∈ _{K^{−}(T ≀G)}(LP, LP [j]), for some degrees i, j ≥ 0. The Yoneda product u · v is given as the composite

(LP ^{\overline{}^{u·v}} LP [i + j]) := (LP ^{\overline{}^{u}} LP [i] ^{\overline{}^{v[i]}} LP [i + j]) ∈ _{K^{−}(T ≀G)}(LP, LP [i + j]) .

It turns Ext_{T ≀G}^{∗}(T, T) into a graded S-algebra.

Moreover, we use the description of the cohomology of G with coefficients in T over the ground ring S

H^{i}(G, T; S) := Ext_{SG}^{i}(S, T) = _{K^{−}(SG)}(P, RLS[i]) = H^{i}(CP) ,

where i ≥ 0, and where C : Mod- SG \overline{}Mod- S : X \overline{}_{SG}(X, T) is applied entrywise to complexes. For a morphism f, we denote f^{∗} := Cf. Let Z^{i}(CP) := Kern(CP_{i} ^{\overline{}^{d_{i+1}^{∗}}} CP_{i+1}) denote the S-module of i-cocycles.

We will also make use of the alternative interpretation

[figure: H^{i}(G, T; S) = _{K^{−}(SG)}(P, RLS[i]); λ; ∼; (P, RLP [i])^{K^{−}(SG)}; u(RLd_{0}[i]); u .]

Given a ∈ CP_{i} = _{SG}(SG^{⊗i+1}, T), b ∈ CP_{j} = _{SG}(SG^{⊗j+1}, T), for some degrees i, j ≥ 0, their cup product a ∪ b ∈ CP_{i+j} = _{SG}(SG^{⊗i+j+1}, T) is defined by

(g_{[0,i+j]})(a ∪ b) := (g_{[0,i]})a · (g_{[i,i+j]})b ,

where g_{l} ∈ G for l ∈ [0, i + j]. Because of the Leibniz rule

(a ∪ b)d_{i+j+1}^{∗} = (−1)^{j}(ad_{i+1}^{∗} ∪ b) + (a ∪ bd_{j+1}^{∗}) ,

the cup product restricts to Z^{i}(CP) × Z^{j}(CP) ^{\overline{}^{∪}} Z^{i+j}(CP) and induces a map

H^{i}(G, T; S) × H^{j}(G, T; S) ^{\overline{}^{∪}} H^{i+j}(G, T; S) .

Given cocycles a ∈ Z^{i}(CP), b ∈ Z^{j}(CP), where i, j ≥ 0, we let c_{a,b} ∈ CP_{i+j−1} be defined by

h_{[0,i+j−1]}c_{a,b} := ∑_{m∈[0,j−1]}(−1)^{m(i+j−1)}(h_{[m,m+i]})a · (h_{[m+i,i+j−1]} ⊗ h_{[0,m]})b ,

where h_{l} ∈ G for l ∈ [0, i + j − 1]. We obtain

(g_{[0,i+j]})(c_{a,b}d_{i+j}^{∗}) = g_{[0,i]}a · g_{[i,i+j]}b − (−1)^{ij}g_{[0,j]}b · g_{[j,i+j]}a ,

whence a ∪ b = (−1)^{ij} b ∪ a as elements of H^{i+j}(G, T; S). Thus the cup product turns H^{∗}(G, T; S) into a graded commutative ring.

The following proposition we owe to B. Keller.

Proposition 3.1 The isomorphism λ^{−1}κ of graded S-modules, given by

\begin{matrix}H^{∗}(G, T; S) \overline{}^{λ}_{∼} _{K^{−}(SG)}(P, RLP [∗]) & \overline{}^{κ}_{∼} & _{K^{−}(T ≀G)}(LP, LP [∗]) = Ext_{T ≀G}^{∗}(T, T) \\ u & \overline{} & (Lu)(ηLP [∗]) \\ (εP)(Ru′) & \overline{} & u′ , \end{matrix}

is in fact an isomorphism of graded S-algebras, with respect to the cup product on H^{∗}(G, T; S) and with respect to the Yoneda product on Ext_{T ≀G}^{∗}(T, T).

In particular, Ext_{T ≀G}^{∗}(T, T) is a graded commutative S-algebra.

Note that H^{0}(G, T; S) ≃ Hom_{T ≀G}(T, T) ≃ S as S-algebras.

Given u ∈ _{K^{−}(SG)}(P, RLP [i]), v ∈ _{K^{−}(SG)}(P, RLP [j]), where i, j ≥ 0, we calculate

((uκ) · (vκ))κ^{−1} = ((Lu)(ηLP [i])(Lv[i])(ηLP [i + j]))κ^{−1}
= (εP)(RLu)(RηLP [i])(RLv[i])(RηLP [i + j])
= u(RLv[i])(RηLP [i + j]) .

Now suppose given a ∈ Z^{i}(CP), b ∈ Z^{j}(CP). Letting P_{i+l} ^{\overline{}^{ã_{i+l}}} RLP_{l} be defined by

(g_{[0,i+l]})ã_{i+l} := (−1)^{il}g_{[0,l]} ⊗ (g_{[l,i+l]}a) ,

where l ≥ 0 and g_{k} ∈ G for k ∈ [0, i+l], we obtain a morphism of complexes P ^{\overline{}^{ã}} RLP [i] that is mapped by λ to P ^{\overline{}^{a}} RLS[i]. Hence

(g_{[0,i+j]})(((aλ^{−1}κ) · (bλ^{−1}κ))κ^{−1}λ)
= (g_{[0,i+j]})(((ãκ) · (b̃κ))κ^{−1}λ)
= (g_{[0,i+j]})ã_{i+j}(RLb̃_{j})(RηL(SG))(RLd_{0}[i + j])
= (−1)^{ij}(g_{[0,j]} ⊗ (g_{[j,i+j]}a))(RLb̃_{j})(RηL(SG))(RLd_{0}[i + j])
= (−1)^{ij}(g_{0} ⊗ (g_{[0,j]}b) ⊗ (g_{[j,i+j]}a))(RηL(SG))(RLd_{0}[i + j])
= (−1)^{ij}(g_{0} ⊗ (g_{[0,j]}b) · (g_{[j,i+j]}a))(RLd_{0}[i + j])
= (−1)^{ij}(g_{[0,j]}b) · (g_{[j,i+j]}a)
= (−1)^{ij}(g_{[0,i+j]})(b ∪ a) ,

i.e.

(aλ^{−1}κ) · (bλ^{−1}κ) = (a ∪ b)λ^{−1}κ .

4 Cohomology

4.1 A projective resolution

Remark 4.1 We attempt to explain why in the case of a cyclic galois group, and in presence of a classical periodic resolution, the description in (1.19) is actually useful to calculate products in cohomology.

For the purpose of this remark, let G = ⟨σ | σ^{g} = 1⟩ be a cyclic group of order g ≥ 1 acting on the discrete valuation ring T, let S = Fix_{G}T be the fixed ring in T under G, and let L = frac T, K = frac S. Let

[figure: T ≀ G; α_{0}; T ≀ G; :; 1; (∑_{i∈Z/g} σ^{i}); T ≀ G; β_{0}; T ≀ G; :; 1; (σ − 1); T ≀ G; ε_{0}; T; :; 1; 1]

be T ≀ G-linear maps. We obtain a periodic projective resolution of period 2

P_{0} := (⋯ \overline{}T ≀ G ^{\overline{}^{α_{0}}} T ≀ G ^{\overline{}^{β_{0}}} T ≀ G ^{\overline{}^{α_{0}}} T ≀ G ^{\overline{}^{β_{0}}} \underbrace{T ≀ G}_{degree 0} \overline{}0 \overline{}⋯) ,

mapping quasiisomorphically to T via ε_{0} in degree 0.

where the image of α_{0} is isomorphic to T via T ^{\overline{}^{∼}} Im α_{0}, x \overline{}∑_{i∈Z/g} σ^{i}x. So we have

RHom_{T ≀G}(T, T) =

(⋯ \overline{}T ^{\overline{}^{Tr^{L|K}}} T ^{\overline{}^{x^{σ}−x \overline{}x}} T ^{\overline{}^{Tr^{L|K}}} T ^{\overline{}^{x^{σ}−x \overline{}x}} \underbrace{T}_{degree 0}\overline{}0 \overline{}⋯) ,

whence for instance Ext_{T ≀G}^{1}(T, T) ≃ {x ∈ T : Tr_{L|K}(x) = 0}/{y^{σ} − y : y ∈ T }, or Ext_{T ≀G}^{2}(T, T) ≃ S/ Tr_{L/K}(T). Suppose given an element of, say, Ext_{T ≀G}^{1}(T, T), represented by an element e ∈ T with trace 0. To apply Yoneda multiplication, we need to represent it as an element of _{K^{−}(T ≀G)}(P_{0}, P_{0}[1]). Let us consider the necessary construction of a morphism of complexes. We attempt to construct a periodic resolution of e with period 2. If in odd degrees, the morphism of complexes is given by 1 \overline{}∑_{i∈Z/g} σ^{i}x_{i}, and in even degrees ≥ 2 by 1 \overline{}∑_{i∈Z/g} σ^{i}y_{i}, where x_{i}, y_{i} ∈ T, then these coefficients are subject to the following conditions.

\begin{aligned}∑_{i} x_{i} & = & e \\ (∑_{i} σ^{i}x_{i})(∑_{j} σ^{j}) & = & (σ − 1)(∑_{i} σ^{i}y_{i}) \\ (∑_{j} σ^{j})(∑_{i} σ^{i}x_{i}) & = & (∑_{i} σ^{i}y_{i})(σ − 1) , \end{aligned}

i.e.

∑_{i} x_{i} = e
∑_{i+j≡_{g}k} x_{i}^{σ^{j}} = y_{k−1} − y_{k} for all k ∈ Z/g
e = y_{k−1}^{σ} − y_{k} for all k ∈ Z/g .

As far as we can see, solving this system requires knowledge of the operation of σ on T in terms of an S-linear basis of T. Cf. e.g. (1.4) or appendix A.2.

We maintain the notation and assumptions of section 1.5. In particular, G = C_{p} = ⟨σ⟩ is cyclic of order equal to the residue characteristic p of S, and b = −1 + v_{t}(t^{σ} − t) ≥ 1. To dispose of the equality Ξ = Λ^{D}, we assume that v_{s}(p) ≥ b − \underline{b} (1.19).

Let

[figure: Ξ; α; Ξ; :; 1; μ̃_{p−1,p−\overline{b}}; :=; s^{b−\underline{b}−1} ε_{\overline{b},p−\overline{b}}]
[figure: Ξ; β; Ξ; :; 1; μ_{1,\overline{b}}; =; s\underline{b} ∑_{i∈[0,p−1]} iε_{i,\overline{b}}]
[figure: Ξ; χ_{k}; T; :; 1; t^{k}; for k ∈ [0, p − 1]]

be Ξ-linear maps (cf. 1.17). Here μ̃_{p−1,p−\overline{b}} is a ‘shifted version’ of μ_{p−1,p−\overline{b}} (cf. 1.18).

Proposition 4.2 The complex of Ξ-linear maps

⋯ ^{\overline{}^{β}} Ξ ^{\overline{}^{α}} Ξ ^{\overline{}^{β}} Ξ ^{\overline{}^{α}} Ξ ^{\overline{}^{β}} Ξ ^{\overline{}^{α}} ⋯,

periodic of period 2, is acyclic. The image of α is isomorphic to T as a module over Ξ, more precisely, we have a factorization

(Ξ ^{\overline{}^{α}} Ξ) = (Ξ ^{\overline{}^{χ_{0}}} T \overline{}Ξ),

with χ_{0} surjective and T \overline{}Ξ injective. Denote by

P := (⋯ \overline{}Ξ ^{\overline{}^{α}} Ξ ^{\overline{}^{β}} \underbrace{Ξ}_{degree 0} \overline{}0 \overline{}⋯) ∈ K^{−}(Ξ)

the resulting projective resolution of T.

We claim exactness of Ξ ^{\overline{}^{α}} Ξ ^{\overline{}^{β}} Ξ. To prove that αβ = 0, we calculate

μ_{1,\overline{b}} μ̃_{p−1,p−\overline{b}} = s^{b−1} ∑_{i∈[0,p−1]} iε_{i,\overline{b}} ε_{\overline{b},p−\overline{b}}
=^{(1.6)} s^{b−1} ∑_{i∈[0,p−1]} i∂_{\overline{b},\overline{i+b}} ε_{i,p}
= 0 .

Now, let B ⊆ Λ be the kernel of the Λ-linear map Λ \overline{}Λ, 1 \overline{}μ_{1,\overline{b}}. To prove that Im α = Kern β, we shall show that the S-linear colengths of both submodules in B coincide. For j, k ∈ [0, p − 1], we have

μ_{1,\overline{b}} ε_{j,k} = s\underline{b} ∑_{i∈[0,p−1]} iε_{i,\overline{b}} ε_{j,k}
=^{(1.6)} s\underline{b} ∑_{i∈[0,p−1]} i∂_{j,\overline{i+b}} ε_{i,k+\overline{b}}
= s\underline{b} \overline{j − b} ε_{\overline{j−b},k+\overline{b}} ,

and thus B has the S-linear basis (ε_{\overline{b},k})_{k∈[0,p−1]}. By (1.17), an element ∑_{k∈[0,p−1]} x_{k}ε_{\overline{b},k}, where x_{k} ∈ S, is in Ξ if and only if

v_{s}(x_{k}) ≥ 1 + \underline{b(p − 1) − k − 1}

for k ∈ [0, p − 1]. By (1.12), we obtain the colength of Kern β in B to be equal to ∑_{k∈[0,p−1]}(1 + \underline{b(p − 1) − k − 1}) = b(p − 1). On the other hand we obtain

μ_{l,j}α = μ̃_{p−1,p−\overline{b}} μ_{l,j}
= s^{b−\underline{b}+\underline{bl−j−1}} ∑_{i∈[0,p−1]} (\begin{matrix}i \\ l \end{matrix}) ε_{\overline{b},p−\overline{b}} ε_{i,j}
=^{(1.6)} s^{b−\underline{b}+\underline{bl−j−1}} ∑_{i∈[0,p−1]} (\begin{matrix}i \\ l \end{matrix}) ∂_{i,0} ε_{\overline{b},j+p−\overline{b}}
= ∂_{l,0}s^{b−\underline{b}−1}ε_{\overline{b},j+p−\overline{b}}
= \{ \begin{matrix}s^{b−\underline{b}−1}ε_{\overline{b},j+p−\overline{b}} & for l = 0 and j ∈ [0, \overline{b} − 1] \\ s^{b−\underline{b}} ε_{\overline{b},j−\overline{b}} & for l = 0 and j ∈ [\overline{b}, p − 1] \\ 0 & for l ∈ [1, p − 1] , \end{matrix}

where l, j ∈ [0, p−1], whence the colength of Im α in B equals (b−\underline{b}−1)p+(p−\overline{b}) = b(p−1), too. Thus Im α = Kern β.

Moreover, since

\begin{aligned}μ_{l,j}χ_{0} & = & s^{1+\underline{bl−j−1}} ∑_{i∈[0,p−1]} (\begin{matrix}i \\ l \end{matrix}) t^{0}ε_{i,j} \\ & = & ∂_{l,0}t^{j} , \end{aligned}

where l, j ∈ [0, p − 1], the Ξ-linear isomorphism

[figure: T; ∼; Ξα; t^{j}; s^{b−\underline{b}−1}ε_{\underline{b},j+p−\underline{b}}]

yields the commutativity

( Ξ ^{\overline{}^{α}} Ξα ) = ( Ξ ^{\overline{}^{χ_{0}}} T ^{\overline{}^{∼}} Ξα ) .

We claim exactness of Ξ ^{\overline{}^{β}} Ξ ^{\overline{}^{α}} Ξ. To prove that βα = 0, we calculate

μ̃_{p−1,p−\overline{b}} μ_{1,\overline{b}} = s^{b−1} ∑_{i∈[0,p−1]} iε_{\overline{b},p−\overline{b}} ε_{i,\overline{b}}
=^{(1.6)} s^{b−1} ∑_{i∈[0,p−1]} i∂_{i,0} ε_{\overline{b},p}
= 0 .

Now, let A ⊆ Λ be the kernel of the Λ-linear map Λ \overline{}Λ, 1 \overline{}μ̃_{p−1,p−\overline{b}}. To prove that Im β = Kern α, we shall show that the S-linear colengths of both submodules in A coincide. For j, k ∈ [0, p − 1], we have

ε_{j,k}α = s^{b−\underline{b}−1}ε_{\overline{b},p−\overline{b}} ε_{j,k} =^{(1.6)} s^{b−\underline{b}−1}∂_{j,0} ε_{\overline{b},p−\overline{b}+k} ,

and therefore A has the S-linear basis (ε_{j,k})_{j∈[1,p−1], k∈[0,p−1]}. By (1.17), Kern α = Ξ ∩ A has the S-linear basis (μ_{l,j})_{l∈[1,p−1], j∈[0,p−1]}, whence the colength of Kern α in A equals bp(p − 1)/2 (cf. pf. of 1.17). On the other hand, using the S-linear basis

(s^{1+\underline{bl−j−1}} ∑_{k∈[0,p−1]} (\begin{matrix}k \\ l \end{matrix}) ε_{\overline{k+b+1},j})_{l,j∈[0,p−1]}

of Ξ (1.18), we obtain

(s^{1+\underline{bl−j−1}} ∑_{k∈[0,p−1]} (\begin{matrix}k \\ l \end{matrix}) ε_{\overline{k+b+1},j}) β

= s^{\underline{b}+1+\underline{bl−j−1}} ∑_{i∈[0,p−1]} ∑_{k∈[0,p−1]} i (\begin{matrix}k \\ l \end{matrix}) ε_{i,\overline{b}} ε_{\overline{k+b+1},j}
=^{(1.6)} s^{\underline{b}+1+\underline{bl−j−1}} ∑_{i∈[0,p−1]} ∑_{k∈[0,p−1]} i (\begin{matrix}k \\ l \end{matrix}) ∂_{\overline{k+b+1},\overline{i+b}} ε_{i,\overline{b}+j}
= s^{\underline{b}+1+\underline{bl−j−1}} ∑_{i∈[l+1,p−1]} i (\begin{matrix}i−1 \\ l \end{matrix}) ε_{i,\overline{b}+j}
= s^{\underline{b}+1+\underline{bl−j−1}+\underline{\overline{b}+j}} ∑_{i∈[l+1,p−1]} i (\begin{matrix}i−1 \\ l \end{matrix}) ε_{i,\overline{b+j}} .

This yields the colength of Im β in A to be equal to

∑_{j∈[0,p−1]} ∑_{l∈[0,p−2]}(\underline{\overline{b} + j} + \underline{b} + 1 + \underline{bl − j − 1})
=^{(1.12)} \overline{b}(p − 1) + (\underline{b} + 1)p(p − 1) + ∑_{l∈[0,p−2]}(bl − p)
= bp(p − 1)/2 ,

too. Thus Im β = Kern α.

Example 4.3 (cont.) For Ξ ≃ Z_{(3)}[π_{2}] ≀ C_{3}, we have b = 1 and \underline{b} = 0, thus obtaining the projective resolution

P = (⋯ \overline{}Ξ ^{\overline{}^{[ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 3·2 & 0 & 0 \end{matrix} ]·(−)}} Ξ ^{\overline{}^{[ \begin{matrix}0 & 0 & 0 \\ 3·1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ]·(−)}} Ξ ^{\overline{}^{[ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 3·2 & 0 & 0 \end{matrix} ]·(−)}} \underbrace{Ξ}_{degree 0} \overline{}0 \overline{}⋯)

of T, with quasiisomorphism given by Ξ ^{\overline{}^{χ_{0}}} T in degree 0, sending ξ = [\begin{matrix}a_{0,0} & a_{0,1} & a_{0,2} \\ 3a_{1,2} & a_{1,0} & a_{1,1} \\ 3a_{2,1} & 3a_{2,2} & a_{2,0} \end{matrix} ] ∈ Ξ to t^{0}ξ = a_{0,0}t^{0} + a_{0,1}t^{1} + a_{0,2}t^{2}. Representing elements of T as row vectors with entries in S with respect to the basis (t^{0}, t^{1}, t^{2}), the map χ_{0} is given by [^{1 0 0}] · (−).

4.2 The Yoneda ring

Remark 4.4 An S-linear basis of _{Ξ}(Ξ, T) is given by (χ_{i})_{i∈[0,p−1]}. The restriction map

Λ(Λ, T) \overline{}_{Ξ}(Ξ, T)

is an isomorphism.

Proposition 4.5 We have

Ext_{T ≀G}^{0}(T, T) = S⟨χ_{0}⟩
≃ S ,
Ext_{T ≀G}^{2i}(T, T) = S⟨χ_{0}⟩/S⟨s^{b−\underline{b}} · χ_{0}⟩
≃ S/s^{b−\underline{b}}  for i ≥ 1 , and
Ext_{T ≀G}^{2i+1}(T, T) = \frac{S⟨ χ_{0}, …, χ_{\overline{b}−1}, χ_{\overline{b}+1}, …, χ_{p−1}⟩}{S⟨s^{\underline{b}+1}χ_{0}, …, s^{\underline{b}+1}χ_{\overline{b}−1}, s\underline{b}χ_{\overline{b}+1}, …, s\underline{b}χ_{p−1}⟩}
≃ ( ⨁_{k∈[0,\overline{b}−1]}S/s^{\underline{b}+1}) ⊕ ( ⨁_{k∈[\overline{b}+1,p−1]}S/s\underline{b})  for i ≥ 0 .

The element represented by χ_{0} in Ext_{T ≀G}^{2i}(T, T) shall be written χ_{0}^{(2i)}, the element represented by χ_{j} in Ext_{T ≀G}^{2i+1}(T, T) shall be written χ_{j}^{(2i+1)}, where i ≥ 0, j ∈ [0, p − 1] ∖ {1}. We calculate _{Ξ}(Ξ, T) ^{\overline{}^{α^{∗}}} _{Ξ}(Ξ, T), χ \overline{}αχ. Given k ∈ [0, p − 1], we get

1_{Ξ}(χ_{k}α^{∗}) = (1_{Ξ}α)χ_{k}
= (s^{b−\underline{b}−1}ε_{\overline{b},p−\overline{b}})χ_{k}
= s^{b−\underline{b}−1}(1_{Ξ} χ_{k})ε_{\overline{b},p−\overline{b}}
= ∂_{\overline{b},k}s^{b−\underline{b}}
= ∂_{\overline{b},k}s^{b−\underline{b}}(1_{Ξ} χ_{0}) ,

i.e.

χ_{k}α^{∗} = ∂_{\overline{b},k}s^{b−\underline{b}} · χ_{0} .

We calculate _{Ξ}(Ξ, T) ^{\overline{}^{β^{∗}}} _{Ξ}(Ξ, T), χ \overline{}βχ. Given k ∈ [0, p − 1], we get

1_{Ξ}(χ_{k}β^{∗}) = (1_{Ξ}β)χ_{k}
= (s\underline{b} ∑_{i∈[0,p−1]} iε_{i,\overline{b}})χ_{k}
= s\underline{b} ∑_{i∈[0,p−1]} i∂_{i,k}t\overline{k+b}s\underline{k+\overline{b}}
= ks^{\underline{k+\overline{b}}+\underline{b}}(1_{Ξ} χ\overline{k+b}) ,

i.e.

χ_{k}β^{∗} = ks^{\underline{k+\overline{b}}+\underline{b}} · χ\overline{k+b} .

The shape of the Ext-groups now follows by (4.2). Lemma 4.6 We have

s^{b+\underline{j−2b}} ((\overline{b − j})^{−1}ε_{\overline{2b−j},\overline{j−2b}} + (\overline{j − b})^{−1}ε_{\overline{b},\overline{j−2b}}) ∈ Ξ

for j ∈ [0, p − 1] ∖ \{\overline{b}\}.

First of all, we have b + \underline{j − 2b} ≥ b + \underline{−2b} = b − 1 − \underline{2b − 1} and p(b − 1 − \underline{2b − 1}) =

(p − 2)(b − 1) + \overline{2b − 1} − 1 ≥ 0, both if b = 1 or if b > 1. So the element in question is in Λ.

By (1.17), we have to prove that for l ∈ [0, p − 1],

v_{s}((−1)\overline{2b−j} ( l ; \overline{2b−j}) (\overline{b − j})^{−1} + (−1)\overline{b} ( l ; \overline{b}) (\overline{j − b})^{−1}) ≥ 1+\underline{bl − \overline{j − 2b} − 1}−(b+\underline{j − 2b}) .

If l ∈ [0, p − 2], then

1 + \underline{bl − \overline{j − 2b} − 1} ≤ 1 + \underline{b(p − 2) − \overline{j − 2b} − 1} = b + \underline{j − 2b} .

If l = p − 1 then

1 + \underline{b(p − 1) − \overline{j − 2b} − 1} − (b + \underline{j − 2b}) = −\underline{j − b} .

Now (−1)^{h} (\begin{matrix}p−1 \\ h \end{matrix}) ≡_{p} 1 for h ∈ [0, p − 1] and (\overline{b − j})^{−1} + (\overline{j − b})^{−1} ≡_{p} 0 together with v_{s}(p) ≥ b − \underline{b} ≥ −\underline{j − b}, both if b = 1 or if b > 1, yield the result.

Let

[figure: Ξ; μ_{j}; Ξ; :; 1; μ_{0,j} = ∑_{i∈[0,p−1]} ε_{i,j}; Ξ; ν_{j}; Ξ; :; 1; s^{b+\underline{j−2b}} ((\overline{b − j})^{−1}ε_{\overline{2b−j},\overline{j−2b}} + (\overline{j − b})^{−1}ε_{\overline{b},\overline{j−2b}})]

be Ξ-linear maps for j ∈ [0, p − 1] ∖ \{\overline{b}\}. By (4.6), the map ν_{j} is well defined.

Lemma 4.7 For j ∈ [0, p − 1] ∖ \{\overline{b}\}, we obtain

\begin{aligned}ν_{j}β & = & αμ_{j} \\ μ_{j}α & = & βν_{j} \\ μ_{j}χ_{0} & = & χ_{j}. \end{aligned}

That is, for i ≥ 0 we obtain a representative

[figure: ⋯; Ξ; β; Ξ; α; Ξ; β; Ξ; α; \overbrace{Ξ}^{degree 2i + 1}; β; Ξ; α; ⋯; μ_{j}; ν_{j}; μ_{j}; ν_{j}; μ_{j}; ⋯; Ξ; α; Ξ; β; Ξ; α; Ξ; β; Ξ; 0; ⋯; χ_{j}; χ_{0}; T .]

in _{K^{−}(Ξ)}(P, P [2i + 1]) of χ_{j}^{(2i+1)} ∈ Ext_{T ≀G}^{2i+1}(T, T).

Moreover, for i ≥ 0 we obtain a representative

[figure: ⋯; Ξ; α; Ξ; β; Ξ; α; Ξ; β; \overbrace{Ξ}^{degree 2i}; α; Ξ; β; ⋯; 1_{Ξ}; 1_{Ξ}; 1_{Ξ}; 1_{Ξ}; 1_{Ξ}; ⋯; Ξ; α; Ξ; β; Ξ; α; Ξ; β; Ξ; 0; ⋯; χ_{0}; χ_{0}; T .]

in _{K^{−}(Ξ)}(P, P [2i]) of χ_{0}^{(2i)} ∈ Ext_{T ≀G}^{2i}(T, T). We claim that ν_{j}β = αμ_{j}. On the one hand, we obtain

1_{Ξ}(ν_{j}β) = s^{\underline{b}+b+\underline{j−2b}} ∑_{i∈[0,p−1]} (i(\overline{b − j})^{−1}ε_{i,\overline{b}} ε_{\overline{2b−j},\overline{j−2b}} + i(\overline{j − b})^{−1}ε_{i,\overline{b}} ε_{\overline{b},\overline{j−2b}})
=^{(1.6)} s^{\underline{b}+b+\underline{j−2b}} ∑_{i∈[0,p−1]} i(\overline{b − j})^{−1}∂_{\overline{2b−j},\overline{i+b}} ε_{i,\overline{b}+\overline{j−2b}}
= s^{\underline{b}+b+\underline{j−2b}} ε_{\overline{b−j},\overline{b}+\overline{j−2b}}
= s^{b−\underline{b}−1}ε_{\overline{b−j},j+p−\overline{b}} .

On the other hand, we obtain

1_{Ξ}(αμ_{j}) = s^{b−\underline{b}−1} ∑_{i∈[0,p−1]} ε_{i,j} ε_{\overline{b},p−\overline{b}}
=^{(1.6)} s^{b−\underline{b}−1} ∑_{i∈[0,p−1]} ∂_{\overline{b},\overline{i+j}} ε_{i,j+p−\overline{b}}
= s^{b−\underline{b}−1}ε_{\overline{b−j},j+p−\overline{b}} .

We claim that μ_{j}α = βν_{j}. On the one hand, we obtain

1_{Ξ}(βν_{j}) = s^{\underline{b}+b+\underline{j−2b}} ∑_{i∈[0,p−1]} (i(\overline{b − j})^{−1}ε_{\overline{2b−j},\overline{j−2b}} ε_{i,\overline{b}} + i(\overline{j − b})^{−1}ε_{\overline{b},\overline{j−2b}} ε_{i,\overline{b}})
=^{(1.6)} s^{\underline{b}+b+\underline{j−2b}} ∑_{i∈[0,p−1]} i(\overline{j − b})^{−1}∂_{i,\overline{j−b}} ε_{\overline{b},\overline{b}+\overline{j−2b}}
= s^{\underline{b}+b+\underline{j−2b}} ε_{\overline{b},\overline{b}+\overline{j−2b}}
= s^{b−\underline{b}−1}ε_{\overline{b},j−\overline{b}+p} .

On the other hand, we obtain

1_{Ξ}(μ_{j}α) = s^{b−\underline{b}−1} ∑_{i∈[0,p−1]} ε_{\overline{b},p−\overline{b}}ε_{i,j}
=^{(1.6)} s^{b−\underline{b}−1} ∑_{i∈[0,p−1]} ∂_{i,0}ε_{\overline{b},j−\overline{b}+p}
= s^{b−\underline{b}−1}ε_{\overline{b},j−\overline{b}+p} .

We claim that μ_{j}χ_{0} = χ_{j}. In fact,

1_{Ξ}(μ_{j}χ_{0}) = ∑_{i∈[0,p−1]} t^{0}ε_{i,j}
= ∑_{i∈[0,p−1]} ∂_{i,0}t^{j}
= t^{j}
= 1_{Ξ} χ_{j} .

Example 4.8 (cont.) If Ξ ≃ Z_{(3)}[π_{2}] ≀ C_{3}, then b = \overline{b} = 1; ν_{0} = [\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1/2 \\ 3 & 0 & 0 \end{matrix}] · (−),

μ_{0} = [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] · (−); ν_{2} = [\begin{matrix}3/2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 0 \end{matrix} ] · (−), μ_{2} = [\begin{matrix}0 & 0 & 1 \\ 3 & 0 & 0 \\ 0 & 3 & 0 \end{matrix}] · (−).

Theorem 4.9 We have isomorphisms of graded S-algebras

\begin{matrix}\frac{S[h_{0}^{(1)}, …, h_{\overline{b}−1}^{(1)}; h_{\overline{b}+1}^{(1)}, …, h_{p−1}^{(1)}; h_{0}^{(2)}]}{( \begin{matrix}s^{\underline{b}+1}h_{0}^{(1)}, …, s^{\underline{b}+1}h_{\overline{b}−1}^{(1)}; s\underline{b}h_{\overline{b}+1}^{(1)}, …, s\underline{b}h_{p−1}^{(1)}; s^{b−\underline{b}}h_{0}^{(2)}; \\ h_{j}^{(1)} · h_{k}^{(1)} − ∂_{\overline{j+k},\overline{2b}} s^{b+\underline{j+k−2b}} (\overline{b − j})^{−1}h_{0}^{(2)} \\ for j, k ∈ [0, p − 1] ∖ \{\overline{b}\} \end{matrix} )} & ^{\overline{}^{∼}} & Ext_{T ≀G}^{∗}(T, T) ^{\overline{}^{∼}} H^{∗}(G, T; S) \\ h_{j}^{(1)} & \overline{} & χ_{j}^{(1)} , j ∈ [0, p − 1] ∖ \{\overline{b}\} \\ h_{0}^{(2)} & \overline{} & χ_{0}^{(2)} , \end{matrix}

as quotient of the graded commutative polynomial ring S[h_{0}^{(1)}, …, h_{\overline{b}−1}^{(1)}; h_{\overline{b}+1}^{(1)}, …, h_{p−1}^{(1)}; h_{0}^{(2)}]

with grading determined by deg h_{j}^{(1)} = 1 for j ∈ [0, p − 1] ∖ \{\overline{b}\} and deg h_{0}^{(2)} = 2.

The isomorphism Ext_{T ≀G}^{∗}(T, T) ^{\overline{}^{∼}} H^{∗}(G, T; S) of graded S-algebras with respect to the Yoneda resp. with respect to the cup product is a consequence of (3.1).

We shall exhibit the ring structure on the graded S-module Ext_{T ≀G}^{∗}(T, T) (cf. 4.5). By (4.7), we obtain

\begin{aligned}χ_{0}^{(2i)} & · & χ_{0}^{(2j)} & = & χ_{0}^{(2i+2j)} & ∈ & Ext_{T ≀G}^{2i+2j}(T, T) \\ χ_{0}^{(2i)} & · & χ_{k}^{(2j+1)} & = & χ_{k}^{(2i+2j+1)} & ∈ & Ext_{T ≀G}^{2i+2j+1}(T, T) \end{aligned}

for i, j ≥ 0 and k ∈ [0, p − 1] ∖ \{\overline{b}\}. It remains to calculate χ_{j}^{(1)} · χ_{k}^{(1)} ∈ Ext_{T ≀G}^{2}(T, T) for j, k ∈ [0, p − 1] ∖ \{\overline{b}\}. Using (4.7) to represent χ_{j} in _{K^{−}(Ξ)}(P, P [1]), this product is represented by the 2-cocycle ν_{j}χ_{k} ∈ _{Ξ}(Ξ, T). We obtain

1_{Ξ}(ν_{j}χ_{k}) = t^{k}s^{b+\underline{j−2b}} ((\overline{b − j})^{−1}ε_{\overline{2b−j},\overline{j−2b}} + (\overline{j − b})^{−1}ε_{\overline{b},\overline{j−2b}})
= t^{k}s^{b+\underline{j−2b}} (\overline{b − j})^{−1}ε_{\overline{2b−j},\overline{j−2b}}
= s^{b+\underline{j−2b}} (\overline{b − j})^{−1}∂_{k,\overline{2b−j}} t\overline{k+j−2b}s\underline{k+\overline{j−2b}}
= s^{b+\underline{j+k−2b}} (\overline{b − j})^{−1}∂_{\overline{j+k},\overline{2b}} t^{0} ,

i.e. ν_{j}χ_{k} = ∂_{\overline{j+k},\overline{2b}} s^{b+\underline{j+k−2b}} (\overline{b − j})^{−1}χ_{0}. Hence

χ_{j}^{(1)} · χ_{k}^{(1)} = ∂_{\overline{j+k},\overline{2b}} s^{b+\underline{j+k−2b}} (\overline{b − j})^{−1}χ_{0}^{(2)} .

Remark 4.10 Since \overline{j + k} = \overline{2b} implies (\overline{b − j})^{−1} ≡_{p} −(\overline{b − k})^{−1}, and since v_{s}(p) ≥ b − \underline{b}, we obtain the graded commutativity of Ext_{T ≀G}^{∗}(T, T) without reverting to the graded commutativity of the cup product. For instance, if b = p + 1, then χ_{0}^{(1)} · χ_{2}^{(1)} = s^{p−1}χ_{0}^{(2)} ≠ 0. In particular, Ext_{T ≀G}^{∗}(T, T) is not commutative in this case.

Corollary 4.11 If b = 1, then

S[h^{(1)}, h^{(2)}]/(sh^{(1)}, sh^{(2)}, (h^{(1)})^{2}) ^{\overline{}^{∼}} Ext_{T ≀G}^{∗}(T, T) ^{\overline{}^{∼}} H^{∗}(G, T; S) ,

as quotient of the commutative polynomial ring S[h^{(1)}, h^{(2)}] with grading determined by deg h^{(1)} = 1, deg h^{(2)} = 2.

In fact, we have \underline{b} = 0. There are no nonzero products of homogeneous elements of odd degree, so we may use the commutative polynomial ring.

Example 4.12 (cont.) We have H^{∗}(C_{3}, Z_{(3)}[π_{2}]; Z_{(3)}) ≃ Z_{(3)}[h^{(1)}, h^{(2)}]/(3h^{(1)}, 3h^{(2)}, (h^{(1)})^{2}).

Note that H^{0}(C_{3}, Z_{(3)}[π_{2}]; Z_{(3)}) ≃ Z_{(3)}, and that H^{i}(C_{3}, Z_{(3)}[π_{2}]; Z_{(3)}) ≃ F_{3} for i ≥ 1.

5 Applications

We give some applications, refraining, however, from a repetition of (4.9) in different disguises.

5.1 Lubin-Tate extensions

The results apply to certain of the extensions of local fields described by Lubin and Tate [18]. An introduction to their theory is also given in [28, p. 146 ff.].

Let p ≥ 3 be a prime. Let B be a local field with discrete valuation ring R, whose maximal ideal is generated by π. Assume that R/π ≃ F_{p}. We choose the Lubin-Tate series f(X) = X^{p} + πX ∈ R[[X]], and obtain the unique commutative formal group

F (X, Y) = X + Y
− (π − π^{p})^{−1}((X + Y)^{p} − (X^{p} + Y^{p}))
− p(π − π^{p})^{−1}(π − π^{2p−1})^{−1} ( \begin{matrix}& π^{p−1}(X + Y)^{p−1}(X^{p} + Y^{p}) \\ − & (X + Y)^{p−1}((X + Y)^{p} − (X^{p} + Y^{p})) \\ − & π^{p−1}(X^{2p−1} + Y^{2p−1}) \end{matrix} )
+ O(degree 3p − 2) ∈ R[[X, Y]]

such that F (f(X), f(Y)) = f(F (X, Y)). There is an injective ring morphism

[figure: R; End F; a; [a](X) ,]

where [a](X) ∈ R[[X]] is uniquely determined by [a](X) ≡_{X^{2}} aX and the endomorphism property F ([a](X), [a](Y)) = [a](F (X, Y)). So for instance, [π](X) = f(X) = X^{p} + πX. We write P_{n}(X) := [π^{n}](X) ∈ R[X] for n ≥ 0, so that P_{0}(X) = X, P_{1}(X) = f(X) = X^{p} + πX and P_{n}(X) = P_{n−1}(X)^{p} + πP_{n−1}(X). Moreover, P_{n}(0) = 0, deg P_{n}(X) = p^{n}, P_{n}(X) ≡_{π} X^{p^{n}} and P′_{n}(X) ≡_{p} π^{n}.

Let B̄ be an algebraic closure of B, and let 𝔪̄ = {x ∈ B̄ : N_{B(x)|B}(x) ∈ Rπ} ⊆ B̄ be the maximal ideal of its valuation ring, which becomes an abelian group (𝔪̄, ∗) via x ∗ y := F (x, y). Moreover, 𝔪̄ becomes an R-module via

\begin{matrix}R & \overline{} & End(𝔪̄, ∗) \\ a & \overline{} & ( [a] : x \overline{}[a] · x := [a](x) ) . \end{matrix}

For n ≥ 1, we let

μ_{n} := ann_{[π^{n}]}𝔪̄ = {x ∈ K̄ : P_{n}(x) = 0} .

By separability of P_{n}(X), we have #μ_{n} = p^{n} for each n ≥ 1, whence μ_{n} ≃ R/π^{n} as Rmodules. Let ϑ_{n} be an R-linear generator of μ_{n}, chosen in such a way that [π](ϑ_{n}) = ϑ_{n−1}. We have μ_{ϑ_{n},B}(X) = P_{n}(X)/P_{n−1}(X) = P_{n−1}(X)^{p−1} +π, whence B(μ_{n}) = B(ϑ_{n}) is galois over B with

\begin{matrix}(R/π^{n})^{∗} & ^{\overline{}^{∼}} & Gal(B(μ_{n})|B) \\ u & \overline{} & ( ⟨u⟩ : ϑ_{n} \overline{}[u](ϑ_{n}) ) . \end{matrix}

Now R[ϑ_{n}]|R is purely ramified, and as different we obtain

𝔇_{R[ϑ_{n}]|R} = (μ′_{ϑ_{n},B}(ϑ_{n}))
= (P′_{n}(ϑ_{n})/P_{n−1}(ϑ_{n}))
= (P′_{n−1}(ϑ_{n})(pϑ_{1}^{p−1} + π)/ϑ_{1})
= (P′_{n−2}(ϑ_{n})(pϑ_{2}^{p−1} + π)(pϑ_{1}^{p−1} + π)/ϑ_{1})
= ⋯
= (ϑ_{1}^{−1} ∏_{i∈[1,n]}(pϑ_{i}^{p−1} + π))
= (ϑ_{1}^{−1}π^{n}) ,

whence 𝔇_{R[ϑ_{n}]|R[ϑ_{n−1}]} = (π) = (ϑ_{n}^{p^{n−1}(p−1)}).

Example 5.1 Let n ≥ 2. We may apply (1.19, 4.9) to

S = R[ϑ_{n−1}]  s = ϑ_{n−1}
T = R[ϑ_{n}]  t = ϑ_{n}
b = p^{n−1} − 1
\underline{b} = p^{n−2} − 1  \overline{b} = p − 1 .

The value of b results from the different by the formula v_{t}(𝔇_{T |S}) = (p − 1)(b + 1) (1.14). Moreover, v_{s}(p) ≥ v_{s}(π) = p^{n−2}(p − 1) = b − \underline{b}.

We have F_{p}^{∗} [figure] (R/π^{n})^{∗} by sending j \overline{}j^{p^{n−1}}. For n ≥ 1, we let

π_{n} := ∏_{j∈F_{p}^{∗}} ϑ_{n}^{⟨j^{p^{n−1}}⟩} = ∏_{j∈[1,p−1]}[j^{p^{n−1}}](ϑ_{n}) .

Then R[π_{n}] is purely ramified over R, of degree p^{n−1} and with maximal ideal generated by π_{n}. In particular, π_{1} = π. Moreover, π_{n} = N_{B(ϑ_{n})|B(π_{n})}(ϑ_{n}).

As different, we obtain

𝔇_{R[π_{n}]|R[π_{n−1}]} = 𝔇_{R[ϑ_{n}]|R[π_{n}]}^{−1}𝔇_{R[ϑ_{n}]|R[ϑ_{n−1}]}𝔇_{R[ϑ_{n−1}]|R[π_{n−1}]}
= (ϑ_{n}^{p−2})^{−1}(ϑ_{n}^{p^{n−1}(p−1)})(ϑ_{n}^{p(p−2)})
= (π_{n}^{p^{n−1}+p−2})

[28, III.§3, prop. 13].

Example 5.2 Let n ≥ 2, p ≥ 3. We may apply (1.19, 4.9, 2.10) to

S = R[π_{n−1}]  s = π_{n−1}
T = R[π_{n}]  t = π_{n}
U = R[ϑ_{n}]  u = ϑ_{n}
b = (p^{n−1} − 1)/(p − 1)
\underline{b} = (p^{n−2} − 1)/(p − 1)  \overline{b} = 1 ,

The value of b results from the different by the formula v_{t}(𝔇_{T |S}) = (p − 1)(b + 1) (1.14). Moreover, v_{s}(p) ≥ v_{s}(π) = p^{n−2} = b − \underline{b}.

In particular, (3.1, 2.10) yield isomorphisms of graded S-algebras

H^{∗}(C_{p} × C_{p−1}, U; S) ≃ Ext_{U≀(C_{p}×C_{p−1})}^{∗}(U, U) ≃ Ext_{T ≀C_{p}}^{∗}(T, T) ≃ H^{∗}(C_{p}, T; S) .

5.2 Cyclotomic number field extensions

Passing to completions without changing cohomology, we may consider cyclotomic number field extensions as particular Lubin-Tate extensions. For sake of illustration, we recall the cyclotomic framework; in it, there is no need for completion, since the formal group law is given by the polynomial F (X, Y) = X + Y + XY. Strictly speaking, since in section 5.2 we choose a different Lubin-Tate series as in section 5.1, viz. (X + 1)^{p} − 1 instead of X^{p} + pX, we are not directly specializing to this cyclotomic case. So keeping the notation of section 5.1 is a slight abuse.

Let p ≥ 3 be a prime. For n ≥ 1, we let ζ_{p^{n}} be a primitive p^{n}th root of unity over Q. We make choices in such a manner that ζ_{p^{n}}^{p} = ζ_{p^{n−1}} for n ≥ 2 and denote ϑ_{n} := ζ_{p^{n}} − 1. Let

π_{n} = ∏_{j∈[1, p−1]}(ζ_{p^{n}}^{j^{p^{n−1}}} − 1) .

Then Q(ϑ_{n}) = Q(ζ_{p^{n}}), Q(π_{n}) = Fix_{C_{p−1}} Q(ϑ_{n}) and π_{n} = N_{Q(ϑ_{n})|Q(π_{n})}(ϑ_{n}).

We have N_{Q(ϑ_{n})|Q(ϑ_{n−1})}(ϑ_{n}) = ϑ_{n−1} and N_{Q(π_{n})|Q(π_{n−1})}(π_{n}) = π_{n−1}. Note that π_{1} = p.

The integral closure of Z_{(p)} in Q(ϑ_{n}) is given by the discrete valuation ring Z_{(p)}[ϑ_{n}], with maximal ideal generated by ϑ_{n}, purely ramified over Z_{(p)}; the integral closure of Z_{(p)} in Q(π_{n}) is given by the discrete valuation ring Z_{(p)}[π_{n}], with maximal ideal generated by π_{n}, purely ramified over Z_{(p)}.

Example 5.3 Let n ≥ 2. We may apply (1.19, 4.9) to

S = Z_{(p)}[ϑ_{n−1}]  s = ϑ_{n−1}
T = Z_{(p)}[ϑ_{n}]  t = ϑ_{n}
b = p^{n−1} − 1
\underline{b} = p^{n−2} − 1  \overline{b} = p − 1 .

We remark that v_{s}(p) = p^{n−2}(p − 1) = b − \underline{b}.

Example 5.4 Let n ≥ 2, p ≥ 3. We may apply (1.19, 4.9, 2.10) to

S = Z_{(p)}[π_{n−1}]  s = π_{n−1}
T = Z_{(p)}[π_{n}]  t = π_{n}
U = Z_{(p)}[ϑ_{n}]  u = ϑ_{n}
b = (p^{n−1} − 1)/(p − 1)
\underline{b} = (p^{n−2} − 1)/(p − 1)  \overline{b} = 1 ,

where b is e.g. calculated using [28, VI.§1, prop. 3]. We remark that v_{s}(p) = p^{n−2} = b−\underline{b}. In particular, (3.1, 2.10) yield

H^{∗}(C_{p} × C_{p−1}, U; S) ≃ H^{∗}(C_{p}, T; S) .

Hence, for instance,

H^{∗}((Z/p^{2})^{∗}, Z_{(p)}[ζ_{p^{2}}]; Z_{(p)}) ≃ Z_{(p)}[h^{(1)}, h^{(2)}]/(ph^{(1)}, ph^{(2)}, (h^{(1)})^{2})

(cf. 4.11).

Remark 5.5 In the same manner, (1.19, 4.9, 2.10) may be applied to certain cyclotomic function field extensions as defined by Carlitz and Hayes (cf. [9], [14]; see also [21, sec. 6.1]). Up to completion, these also form a particular case of Lubin-Tate-extensions; again, there is no need for completion, the formal group law being given by F (X, Y) = X + Y.

A The case C_{p^{2}}: a conjecture and an experiment

So far, we have essentially only treated the case of an extension with galois group C_{p}. The galois group C_{p^{2}} seems to yield a somewhat more involved twisted group ring, which we would like to illustrate in the case of Z_{(p)}[π_{3}] ≀ C_{p^{2}}. The calculations were carried out using Magma [22].

A.1 The conjectural situation

Suppose given a prime p ≥ 3. We maintain the notation of section 5.2 concerning π_{n}. Let

\begin{aligned}S & = & Z_{(p)} , & s & = & π_{1} = p , \\ T & = & Z_{(p)}[π_{2}] , & t & = & π_{2} , \\ U & = & Z_{(p)}[π_{3}] , & u & = & π_{3} , \end{aligned}

and let G = C_{p^{2}} be generated by the restriction of ζ_{p^{3}} ^{\overline{}^{σ}} ζ_{p^{3}}^{p+1} from Q(ζ_{p^{3}}) to Q(π_{3}). (The role of U in this appendix differs from the role of U in section 2, where it has been a ‘cohomologically inessential’ extension of T.)

For some peculiar reason, t will not play a role at all. Instead, we consider a Sen element (cf. [27, Lem. 1])

v := ∏_{i∈[0,p−1]} u^{σ^{i}} .

The S-linear colength of the Wedderburn embedding

U ≀ C_{p^{2}} [figure: ω] Γ := End_{S} U

is p^{2}(p^{2} + (p^{2} − p − 2)/2) [20, (2.17)]. We fix the S-linear basis

(u^{i}v^{j})_{i∈[0,p−1], j∈[0,p−1]} = (u^{0}v^{0}, u^{0}v^{1}, …, u^{0}v^{p−1}, u^{1}v^{0}, …, u^{1}v^{p−1}, …, u^{p−1}v^{0}, …, u^{p−1}v^{p−1})

of U with respect to which we represent elements of Γ as matrices, i.e. by means of which we identify Γ = S^{p^{2}×p^{2}}.

Remark A.1 We have

\begin{aligned}v_{u}(u^{σ} − u) & = & 1 + 1 \\ v_{u}(v^{σ} − v) & = & 1 + 2p . \end{aligned}

The second congruence is equivalent to v_{u}(u^{σ^{p}} − u) = 1 + (p + 1), so that both assertions follow from [28, VI.§1, prop. 3].

As usual, let Ξ denote the image of the Wedderburn embedding

\begin{matrix}U ≀ C_{p^{2}} & ^{\overline{}^{ω}} & Γ \\ u & \overline{} & (u̇ : x \overline{}xu) \\ σ & \overline{} & (σ̇ : x \overline{}x^{σ}) , \end{matrix}

and let

Λ := {f ∈ Γ : (Uu^{i})f ⊆ Uu^{i} for all i ∈ [0, p − 1]} ⊆ Γ .

By (A.1) we obtain the intermediate ring

Ξ ⊆ Λ^{D} := Λ((u̇, v̇), (p − 1, p − 1), (2, 2p + 1))_{u̇Λ}
= {f ∈ Λ : Di ; u̇ ∘ Dj ; v̇(f) ≡_{u̇^{2i+(2p+1)j}Λ} 0 for all i, j ∈ [0, p − 1]}
⊆ Λ ,

cf. (1.1). Presumably, this is the smallest intermediate ring between Ξ and Λ that can be defined by derivations.

Conjecture A.2

(i) Given τ ∈ C_{p^{2}}, we conjecture that

2(v − v^{τ}) + (u^{τ})^{p}(u − u^{τ})u^{p−1} + ((u^{τ})^{2p−1} − (u^{τ})^{2p})(u − u^{τ}) ≡_{u^{2p+3}} 0 .

(ii) Moreover, we conjecture that

Λ^{D, E} := {f ∈ Λ^{D} : Di ; u̇ ∘ E_{u̇,v̇}^{j}(f) ≡_{u̇^{2i+(2p+3)j}Λ} 0 for all i, j ∈ [0, p − 1]}

contains Ξ, where

E_{u̇,v̇}(f) := 2Dv̇(f) + u̇^{p}Du̇(f)u̇^{p−1} + (u̇^{2p−1} − u̇^{2p})Du̇(f) .

Remark A.3

(i) Conjecture (A.2.i) holds for p ∈ {3, 5, 7}.

(ii) If p = 3, we obtain the colengths

Ξ = Λ^{D, E} ⊆^{9·(3−1)} Λ^{D} ⊆^{9·(3·(3+1)/2−1)} Λ ⊆^{9·(9−1)/2} Γ .

(iii) If p = 5, we obtain the colengths

Ξ ⊆^{25·(5−1)(5−3)/2} Λ^{D, E} ⊆^{25·(5−1)} Λ^{D} ⊆^{25·(5·(5+1)/2−1)} Λ ⊆^{25·(25−1)/2} Γ .

(iv) According to our wishful thinking, (A.2.i) should be part of a series of congruences in U that completely describes Ξ by adding the according congruences in Λ to the provisional definition of Λ^{D, E} given in (A.2.ii).

(v) Since E_{u̇,v̇}^{j} is not a derivation, we do not know whether (A.2.i) implies (A.2.ii).

A.2 Simplifying u̇, v̇

A.2.1 The case p = 3

Suppose p = 3, i.e. S = Z_{(3)}, u = π_{3} = (ζ_{27} − 1)(ζ_{27}^{−1} − 1), v = u u^{σ}u^{σ^{2}} = (ζ_{27} − 1)(ζ_{27}^{−1} − 1)(ζ_{27}^{4} − 1)· ·(ζ_{27}^{−4} − 1)(ζ_{27}^{16} − 1)(ζ_{27}^{−16} − 1) and U = Z_{(3)}[π_{3}].

With respect to the basis (u^{0}v^{0}, u^{0}v^{1}, u^{0}v^{2}, u^{1}v^{0}, u^{1}v^{1}, u^{1}v^{2}, u^{2}v^{0}, u^{2}v^{1}, u^{2}v^{2}), the multiplication by u on U is given by

\begin{matrix}u̇ = [\begin{matrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ − \frac{2433}{7217} & \frac{3119}{1031} & − \frac{2151}{7217} & − \frac{6249}{7217} & − \frac{12225}{1031} & \frac{1168}{1031} & \frac{19680}{7217} & \frac{25540}{7217} & − \frac{306}{1031} \\ − \frac{4050}{7217} & \frac{3777}{1031} & \frac{593}{7217} & \frac{18804}{7217} & − \frac{20163}{1031} & \frac{987}{1031} & − \frac{6885}{7217} & \frac{64005}{7217} & − \frac{278}{1031} \\ − \frac{111}{7217} & − \frac{1026}{1031} & \frac{50145}{7217} & \frac{18189}{7217} & − \frac{14604}{1031} & − \frac{12306}{1031} & − \frac{6684}{7217} & \frac{29259}{7217} & \frac{6603}{1031} \end{matrix} ] \\ ≡_{u̇^{6}Λ} [\begin{matrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & −1 & 0 & −1 & 0 \\ 0 & 0 & 1 & −3 & 0 & 0 & 0 & 0 & −1 \\ 3 & 0 & 0 & 0 & −3 & 0 & −3 & 0 & 0 \end{matrix} ] =: \begin{matrix}... \\ u \end{matrix} ≡_{u̇^{3}Λ} [\begin{matrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ] =: ü , \end{matrix}

the multiplication by the Sen-element v by

\begin{matrix}v̇ = [ \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{2256}{1031} & − \frac{19548}{1031} & \frac{31788}{1031} & \frac{2484}{1031} & − \frac{3456}{1031} & − \frac{31626}{1031} & − \frac{495}{1031} & − \frac{7083}{1031} & \frac{10224}{1031} \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \frac{27891}{7217} & − \frac{37620}{1031} & \frac{494226}{7217} & \frac{69981}{7217} & − \frac{19980}{1031} & − \frac{97587}{1031} & − \frac{11043}{7217} & − \frac{186021}{7217} & \frac{35910}{1031} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \frac{104247}{7217} & − \frac{137862}{1031} & \frac{1734570}{7217} & \frac{186300}{7217} & − \frac{7866}{1031} & − \frac{385245}{1031} & − \frac{15474}{7217} & − \frac{809595}{7217} & \frac{140031}{1031} \end{matrix} ] \\ ≡_{u̇^{9+3·(3+1)/2}Λ} [\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 9 & 0 & 3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 9 & 0 & 3 & 0 & 0 \end{matrix} ] =: \begin{matrix}... \\ v \end{matrix} ≡_{u̇^{9+3}Λ} [\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \end{matrix} ] =: v̈ , \end{matrix}

and the operation of σ by

σ̇ = [\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ − \frac{24876}{7217} & \frac{25976}{1031} & − \frac{31737}{7217} & \frac{55068}{7217} & − \frac{25392}{1031} & \frac{5540}{1031} & − \frac{16308}{7217} & \frac{36132}{7217} & − \frac{1176}{1031} \\ − \frac{62766}{1031} & \frac{413856}{1031} & − \frac{149176}{1031} & \frac{117030}{1031} & − \frac{647484}{1031} & \frac{228306}{1031} & − \frac{34326}{1031} & \frac{147615}{1031} & − \frac{51159}{1031} \\ 0 & 0 & 0 & 4 & 0 & 0 & −1 & 0 & 0 \\ − \frac{35412}{7217} & \frac{29460}{1031} & − \frac{75624}{7217} & \frac{164160}{7217} & − \frac{57244}{1031} & \frac{17712}{1031} & − \frac{51540}{7217} & \frac{97432}{7217} & − \frac{4033}{1031} \\ − \frac{845298}{7217} & \frac{815757}{1031} & − \frac{3392322}{7217} & \frac{2508327}{7217} & − \frac{2001978}{1031} & \frac{810182}{1031} & − \frac{796137}{7217} & \frac{3442200}{7217} & − \frac{184868}{1031} \\ − \frac{210}{1031} & − \frac{2776}{1031} & − \frac{206}{1031} & \frac{13095}{1031} & \frac{563}{1031} & \frac{679}{1031} & − \frac{2419}{1031} & − \frac{347}{1031} & − \frac{162}{1031} \\ − \frac{9945}{1031} & \frac{52791}{1031} & − \frac{35236}{1031} & \frac{72018}{1031} & − \frac{171069}{1031} & \frac{62570}{1031} & − \frac{21546}{1031} & \frac{43586}{1031} & − \frac{14447}{1031} \\ − \frac{2471781}{7217} & \frac{2428305}{1031} & − \frac{12112977}{7217} & \frac{8656818}{7217} & − \frac{7025124}{1031} & \frac{2951589}{1031} & − \frac{2779557}{7217} & \frac{12296484}{7217} & − \frac{676060}{1031} \end{matrix} ] .

We observe that we may replace u̇, v̇ by ü, v̈ resp. by \begin{matrix}... \\ u \end{matrix}, \begin{matrix}... \\ v \end{matrix} to obtain

Λ^{D} = Λ((ü, v̈), (2, 2), (2, 7))_{u̇Λ} = {f ∈ Λ : Di ; ü ∘ Dj ; v̈(f) ≡_{u̇^{2i+7j}Λ} 0 for all i, j ∈ [0, 2]}

and

Ξ = Λ^{D, E} = {f ∈ Λ^{D} : D ..._{u}^{i} ∘ E_{..._{u} , ..._{v}}^{j}(f) ≡_{u̇^{2i+9j}Λ} 0 for all i, j ∈ [0, 2]} .

A.2.2 The case p = 5

Suppose p = 5, i.e. S = Z_{(5)}, u = π_{3} = (ζ_{125} − 1)(ζ_{125}^{−1} − 1)(ζ_{125}^{57} − 1)(ζ_{125}^{−57} − 1), ζ_{125}^{σ} = ζ_{125}^{6}, v = u u^{σ}u^{σ^{2}}u^{σ^{3}}u^{σ^{4}} and U = Z_{(5)}[π_{3}]. With respect to the basis

( u^{0}v^{0}, u^{0}v^{1}, u^{0}v^{2}, u^{0}v^{3}, u^{0}v^{4},
u^{1}v^{0}, u^{1}v^{1}, u^{1}v^{2}, u^{1}v^{3}, u^{1}v^{4},
u^{2}v^{0}, u^{2}v^{1}, u^{2}v^{2}, u^{2}v^{3}, u^{2}v^{4},
u^{3}v^{0}, u^{3}v^{1}, u^{3}v^{2}, u^{3}v^{3}, u^{3}v^{4},
u^{4}v^{0}, u^{4}v^{1}, u^{4}v^{2}, u^{4}v^{3}, u^{4}v^{4}, ) ,

the matrix describing the multiplication by u on U reduces to

\begin{aligned}u̇ & ≡_{u̇^{31}Λ} & [\begin{matrix}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ −5 & −9 & 0 & 3 & 3 & 10 & 0 & 0 & 3 & 1 & −10 & 10 & 0 & 1 & 2 & 0 & −10 & −1 & 1 & 1 & −5 & −6 & 1 & 0 & 2 \\ 15 & −5 & −9 & 0 & 3 & 5 & 10 & 0 & 0 & 3 & 10 & −10 & 10 & 0 & 1 & 5 & 0 & 10 & −1 & 1 & 10 & −5 & −6 & 1 & 0 \\ 15 & 15 & −5 & −9 & 0 & 15 & 5 & 10 & 0 & 0 & 5 & 10 & −10 & 10 & 0 & 5 & 5 & 0 & 10 & −1 & 0 & 10 & −5 & −6 & 1 \\ 0 & 15 & 15 & −5 & −9 & 0 & 15 & 5 & 10 & 0 & 0 & 5 & 10 & −10 & 10 & −5 & 5 & 5 & 0 & 10 & 5 & 0 & 10 & −5 & −6 \\ −45 & 0 & 15 & 15 & −5 & 0 & 0 & 15 & 5 & 10 & 50 & 0 & 5 & 10 & −10 & 50 & −5 & 5 & 5 & 0 & −5 & 5 & 0 & 10 & −5 \end{matrix} ] =: \begin{matrix}... \\ u \end{matrix} \\ & ≡_{u̇^{5}Λ} & [\begin{matrix}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ] =: ü , \end{aligned}

and the matrix describing the multiplication by the Sen-element v reduces to

v̇ ≡_{u̇^{25+5·(5+1)/2}Λ} [\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 5 & 75 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −25 & 0 & 0 & 0 & 25 & 25 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 25 & 25 & 0 & 0 & 5 & 75 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −50 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & −50 & 0 & 0 & 0 & 25 & 25 & 0 & 0 & 5 & 75 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −50 & 0 & 0 & 0 & 25 & 25 & 0 & 0 & 5 & 75 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −50 & 0 & 0 & 0 & 25 & 25 & 0 & 0 & 5 & 75 & 0 & 0 & 0 \end{matrix} ] =: \begin{matrix}... \\ v \end{matrix}

≡_{u̇^{25+5}Λ} [\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 \end{matrix} ] =: v̈ .

We observe that we may replace u̇, v̇ by ü, v̈ resp. by \begin{matrix}... \\ u \end{matrix}, \begin{matrix}... \\ v \end{matrix} to obtain

Λ^{D} = Λ((ü, v̈), (4, 4), (2, 11))_{u̇Λ} = {f ∈ Λ : Di ; ü ∘ Dj ; v̈(f) ≡_{u̇^{2i+11j}Λ} 0 for all i, j ∈ [0, 4]}

and

Λ^{D, E} = {f ∈ Λ^{D} : D ..._{u}^{i} ∘ E_{..._{u} , ..._{v}}^{j}(f) ≡_{u̇^{2i+13j}Λ} 0 for all i, j ∈ [0, 4]} .

A.3 A spectral sequence

Alternatively, there is a Lyndon-Hochschild-Serre-Grothendieck spectral sequence that might perhaps help in calculating cohomology in the case C_{p^{2}} using the result in the case C_{p} instead of using the Wedderburn embedding of U ≀ C_{p^{2}} (cf. preceding sections). Due to varying ground rings, we have to apply a (hardly visible) modification to the usual Lyndon-Hochschild-Serre spectral sequence.

Let S ⊆ T ⊆ U be an iterated finite extension of discrete valuation rings, U|S galois with H = Gal(U|S), T |S galois with G = Gal(T |S). Let N be the kernel of the restriction map H \overline{}G, so N = Gal(U|T) and G ≃ H/N. In this section, modules are not necessarily finitely generated.

The Grothendieck spectral sequence of the composition

Mod- U ≀ H ^{\overline{}^{^{U≀N}(U,−)}} Mod- T ≀ G ^{\overline{}^{^{T ≀G}(T,−)}} Mod- S

is given by

(∗)  E_{2}^{m,n} := Ext_{T ≀G}^{m}(T, Ext_{U≀N}^{n}(U, X)) ,

where X ∈ mod- U ≀ H, m, n ≥ 0. For X ∈ Mod- U ≀ H, the T ≀ G-module structure on the image U≀N(U, X) is induced by the _{U≀N}(U, X) ≃ _{T N}(T, X) and the left T ≀ G-module structure on T.

To prove that it converges to Ext_{U≀H}^{m+n}(U, X), it suffices to show that an injective U ≀H-module I is mapped to an injective module. In fact, for Y ∈ Mod- T ≀ G we calculate

T ≀G(Y, _{U≀N}(U, I)) ≃ _{T ≀G}(Y, _{T N}(T, I))
≃ _{T N}(Y ⊗_{T ≀G} T, I)
≃ _{T N}(Y ⊗_{T ≀G} T, _{U≀H}(U ≀ H, I))
≃ _{U≀H}(Y ⊗_{T ≀G} T ⊗_{T N} U ≀ H, I) ,

so that the assertion follows by injectivity of I and by projectivity as a left T ≀ G-module of

\begin{aligned}T ⊗_{T N} U ≀ H & _{T ≀G}≃ & T ⊗_{T N} (T ≀ H)^{(#N)} \\ & _{T ≀G}≃ & (T ≀ G)^{(#N)} . \end{aligned}

Using adjunction, we may rewrite (∗) in the familiar shape

(∗∗)  E_{2}^{m,n} = H^{m}(G, H^{n}(N, X; T); S) ⟹ H^{m+n}(H, X; S) ,

applicable to X ∈ Mod- U ≀ H — so e.g. to X = U. Concerning the cup product, cf. [4, sec. 3.9]. Now, if H = C_{p^{2}}, N = C_{p}, G = C_{p} and X = U, and our remaining conditions are satisfied (pure ramification, v(p) big enough), then (4.5) already calculates E_{2}^{m,0} for m ≥ 0. The first step to take when pursuing this spectral sequence approach, using (∗) rather than (∗∗), would be to calculate Ext_{U≀N}^{n}(U, U) as a T ≀ G-module for n ≥ 1. We do not know whether this approach is actually viable.

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Matthias Künzer

German University of Cairo

New Cairo City

Main Entrance El Tagamoa El Khames

Egypt

Matthias.Kuenzer@guc.edu.eg

Harald Weber Universität Stuttgart

Institut für Algebra und Zahlentheorie Pfaffenwaldring 57 D-70569 Stuttgart

Harald.Weber@iaz.uni-stuttgart.de