On the cyclotomic Dedekind embedding and the cyclic Wedderburn embedding

M. Künzer, H. Weber

March 20, 2001

Abstract

Let n ⩾ 1 and let p be a prime. Expand j ∈ [0, p^{n} − 1]\(p) p-adically as j = ∑_{s⩾0} a_{s}p^{s} with a_{s} ∈ [0, p−1]. The #([0, j]\(p))th Z_{(p)}[ζ_{p^{n}}]-linear elementary divisor of the cyclotomic Dedekind embedding

Z_{(p)}[ζ_{p^{n}}] ⊗_{Z_{(p)}} Z_{(p)}[ζ_{p^{n}}] [figure] ∏_{i∈(Z/p^{n})^{∗}} Z_{(p)}[ζ_{p^{n}}]

has valuation

−1 + ∑_{s⩾0}(a_{s}(s + 1) − a_{s+1}(s + 2)) p^{s}

at 1 − ζ_{p^{n}}. There is a similar result for the related cyclic Wedderburn embedding.

0 Introduction

0.0 Contents

0 Introduction  1
0.0 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2.1 Hochschild (co)homology with twisted coefficients . . . . . . . . . . . . . . . . . . 3
0.2.2 Elementary divisors of the cyclotomic Dedekind embedding . . . . . . . . . . . . . 4
0.2.3 The cyclic Wedderburn embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2.4 The absolute cyclic Wedderburn embedding . . . . . . . . . . . . . . . . . . . . . . 5
0.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1 Vandermonde  6
1.1 Diagonalization of Vandermonde matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Elementary divisors of Vandermonde matrices over discrete valuation rings . . . . . . . . 9
1.3 A projective resolution of T over T ⊗_{S} T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 The local ring T ⊗_{S} T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 The cyclotomic Dedekind embedding  13
3 The cyclic Wedderburn embedding  16
3.1 The q-Pascal method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Pedestrian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Invoking q-Saalschütz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The general Vandermonde method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 The Pascal method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 First order Pascal ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.2 Second order Pascal ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Comparison of methods in an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 The radical series of (W_{p^{n}}^{(1)})_{(p)}  27
5 The absolute cyclic Wedderburn embedding  30
5.1 An inversion formula for QC_{p^{n}} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 An inversion formula for QC_{m} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3 The Kervaire-Murthy pullback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.4 Kervaire-Murthy ties for ZC_{p^{n}} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5 Kervaire-Murthy ties for ZC_{m} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 References  43

0.1 Objectives

In this note, we consider an extension of discrete valuation rings S ⊆ T, with s ∈ S and t ∈ T generating the respective maximal ideals. Let K be the field of fractions of S, let L be the field of fractions of T and assume L over K to be finite galois of Galois group G of order m = |G|. Moreover, assume T = S[ϑ] for some ϑ ∈ T. We restrict the Dedekind isomorphism

[figure: L ⊗_{K} L; ∼; ∏_{σ∈G}L; x ⊗ y; (xy^{σ})_{σ∈G},]

the injectivity of which ensues from Dedekind’s Lemma, to the locally integral situation

T ⊗_{S} T [figure] ∏_{σ∈G}T.

This embedding is no longer an isomorphism in general; in fact, the square of its T -linear determinant is just the discriminant of T over S. Being a T -linear map, we can ask for its elementary divisors, or, more precisely, for an elementary divisor diagonalization.

Once such a diagonalization obtained, we may use the image of T ⊗_{S} T inside ∏_{σ∈G} T, instead of T ⊗_{S} T itself, to calculate Hochschild (co)homology of T. For instance, we may ask for these (co)homology groups with coefficients in T, equipped with a galois twisted T -T -bimodule structure (the untwisted case being due to M. Larsen and A. Lindenstrauss [LL 92, 1.6.2] and to the Buenos Aires Cyclic Homology group [BACH 91, prop. 1.3]).

The cyclotomic Dedekind embedding shall be studied in more detail, where T = Z_{(p)}[ζ_{p^{n}}] and S = Z_{(p)}, for n ⩾ 1, p a prime, and where, in general for m ⩾ 1, ζ_{m} denotes a primitive mth root of unity over Q.

This cyclotomic case leads to a related question, which is to give an elementary divisor diagonalization for the cyclic Wedderburn embedding. Given m ⩾ 1, and letting c_{m} denote a generator of the cyclic group C_{m} of m elements, this embedding results from restricting the rational Wedderburn isomorphism

[figure: Q(ζ_{m})C_{m}; ∼; ∏_{j∈[0,m−1]}Q(ζ_{m}); c_{m}; (ζ_{m}^{j})_{j∈[0,m−1]},]

to the integral situation

Z[ζ_{m}]C_{m} [figure] ∏_{j∈[0,m−1]}Z[ζ_{m}],

which is likewise no longer an isomorphism for m > 1, the square of its Z[ζ_{m}]-linear determinant being ±m^{m}. Now the cyclotomic Dedekind embedding may be viewed, considered as a diagram, as a quotient of the corresponding cyclic Wedderburn embedding.

This consideration in turn gives rise to the question for a closed description of the image of the cyclic Wedderburn embedding in the absolute case, i.e. of

[figure: ZC_{m}; ∏_{d | m}Z[ζ_{d}]; c_{m}; (ζ_{d})_{d | m}.]

0.2 Results

0.2.1 Hochschild (co)homology with twisted coefficients

We choose a total ordering on G, (σ_{0}, …, σ_{m−1}), in such a way that σ_{0} = 1_{T}, and such that

∑_{i∈[0,j−1]}v_{t}(ϑ^{σ_{j}} − ϑ^{σ_{i}}) ⩽ ∑_{i∈[0,j−1]}v_{t}(ϑ^{σ_{k}} − ϑ^{σ_{i}})

for each j ∈ [0, m − 1] and each k ∈ [j + 1, m − 1], where v_{t} denotes the t-adic valuation. Let T_{i} denote T, considered as a T -T -bimodule with untwisted left multiplication and σ_{i}-twisted right multiplication by T. Let φ := ∑_{k∈[1,m−1]} v_{t}(ϑ − ϑ^{σ_{k}}).

The Hochschild homology of T, over the ground ring S and with coefficients in T_{i}, i ∈ [0, m − 1], is given in dimension j ⩾ 0 by

H_{j}(T, T_{0}; S) ≃ \{ \begin{matrix}T & for j = 0 \\ T/t^{φ}T & for j odd \\ 0 & for j even, j ⩾ 2 \end{matrix}

and by

H_{j}(T, T_{i}; S) ≃ \{ \begin{matrix}0 & for j odd \\ T/(ϑ^{σ_{i}} − ϑ)T & for j even \end{matrix}

for i ∈ [1, m − 1]. The Hochschild cohomology is given by

H^{j}(T, T_{0}; S) ≃ \{ \begin{matrix}T & for j = 0 \\ 0 & for j odd \\ T/t^{φ}T & for j even, j ⩾ 2 \end{matrix}

and by

H^{j}(T, T_{i}; S) ≃ \{ \begin{matrix}T/(ϑ^{σ_{i}} − ϑ)T & for j odd \\ 0 & for j even \end{matrix}

for i ∈ [1, m − 1] (1.17).

0.2.2 Elementary divisors of the cyclotomic Dedekind embedding

Let n ⩾ 1 and let p be a prime. Expand j ∈ [0, p^{n}−1]\(p) p-adically as j = ∑_{s⩾0} a_{s}p^{s} with a_{s} ∈ [0, p − 1]. The #([0, j]\(p))th Z_{(p)}[ζ_{p^{n}}]-linear elementary divisor of the cyclotomic Dedekind embedding

[figure: Z_{(p)}[ζ_{p^{n}}]; ⊗_{Z_{(p)}}; Z_{(p)}[ζ_{p^{n}}]; δ_{p^{n}}; ∏_{i∈(Z/p^{n})^{∗}} Z_{(p)}[ζ_{p^{n}}]; ζ_{p^{n}}^{k}; ⊗; ζ_{p^{n}}^{l}; (ζ_{p^{n}}^{k+il}),]

where k, l ∈ [0, p^{n−1}(p − 1) − 1], has valuation

−1 + ∑_{s⩾0}(a_{s}(s + 1) − a_{s+1}(s + 2)) p^{s}

at 1 − ζ_{p^{n}} (2.7).

0.2.3 The cyclic Wedderburn embedding

Expand j ∈ [0, p^{n} − 1] p-adically as j = ∑_{s⩾0} a_{s}p^{s} with a_{s} ∈ [0, p − 1]. The (j + 1)st Z_{(p)}[ζ_{p^{n}}]-linear elementary divisor of the localized cyclic Wedderburn embedding

[figure: Z_{(p)}[ζ_{p^{n}}]C_{p^{n}}; ω_{p^{n}}; ∏_{j∈Z/p^{n}} Z_{(p)}[ζ_{p^{n}}]; c_{p^{n}}; (ζ_{p^{n}}^{j})_{j∈Z/p^{n}}]

has valuation

∑_{s⩾0}(a_{s} − a_{s+1})(s + 1)p^{s}

at 1 − ζ_{p^{n}} (3.15). We can render this result a bit more precise and a bit more general, in that we can diagonalize the cyclic Wedderburn embedding

[figure: Z[ζ_{m}]C_{m}; ω_{m}; ∏_{j∈Z/m} Z[ζ_{m}]; c_{m}; (ζ_{m}^{j})_{j∈Z/m}]

for m ⩾ 1, regardless whether Z[ζ_{m}] is a principal ideal domain (3.8). For l ∈ [0, m − 1], the (l + 1)th diagonal entry becomes

\frac{mζ_{m}^{(l^{2})}}{∏_{j∈[1,l]}(1 − ζ_{m}^{j})}.

0.2.4 The absolute cyclic Wedderburn embedding

It suffices to consider the primary parts of m separately, i.e. we may assume m = p^{n} (cf. 5.18). In this case, the absolute cyclic Wedderburn embedding is given by

[figure: ZC_{p^{n}}; ω_{Z,p^{n}}; ∏_{i∈[0,n]}Z[ζ_{p^{i}}]; c_{p^{n}}; (ζ_{p^{i}})_{i∈[0,n]}.]

We derive from the pullback of Kervaire and Murthy [KM 77] the following triangular system of ties, i.e. of congruences of tuple entries, that describe its image.

Let φ denote Euler’s function. For m ⩾ 1 and j ∈ Z, we let [j]_{m} ∈ [0, m − 1] be such that [j]_{m} ≡_{m} j. Let ∂ denote Kronecker’s delta. The image of the absolute Wedderburn embedding is given by

(ZC_{p^{n}})ω_{Z,p^{n}} = { ( ∑_{j∈[0,φ(p^{i})−1]}x_{i,j}ζ_{p^{i}}^{j})_{i∈[0,n]} , x_{i,j} ∈ Z |

for l ∈ [1, n] and j ∈ [0, φ(p^{n−l}) − 1] we have x_{n−l,j} ≡_{p^{l}} ∑_{i∈[0,l−1]}p^{l−1−i}·

· ∑_{k∈[1,p−1]} (x_{n−i,j−p^{n−l}+kp^{n−1−i}} − (1 − ∂_{l,n})x_{n−i,[j]_{p^{n−l−1}}−p^{n−l−1}+kp^{n−1−i}}) }

⊆ ∏_{i∈[0,n]}Z[ζ_{p^{i}}].

The elementary divisors of ω_{Z,p^{n}} over Z are given by p^{i} with multiplicity φ(p^{n−i}) for i ∈ [0, n] (5.14).

Kleinert gives a system of ties that describes the image of the absolute Wedderburn embedding ZC_{m}^{\overline{}^{ω_{Z,m}}} ∏_{d|m} Z[ζ_{d}] in terms of certain prime ideals of the rings Z[ζ_{d}], d|m, in case m is a positive squarefree integer [Kl 81, p. 550]. In loc. cit., this description is used as a tool to study units in dihedral group rings.

0.3 Acknowledgements

We thank T. Zink for a hint how to simplify our elementary divisor calculations, see §§1.1, 1.2. We thank J. Stokman for the deduction of the diagonalization of the Wedderburn embedding from the q-Saalschütz Theorem, see §3.1.2. We thank several colleagues for comments on an earlier version. The second author thanks P. Littelmann for kind hospitality in Strasbourg, where this earlier version has been written, and where he has been supported by the EU TMR-network ‘Algebraic Lie Representations’, grant no. ERB FMRX-CT97-0100.

Notation 0.1

(o) Composition of maps is written on the right, ^{\overline{}^{a}} ^{\overline{}^{b}} = ^{\overline{}^{ab}} .

(i) If φ is a map of A-modules, A a commutative ring, and 𝔭 a prime ideal of A, we sometimes denote the localization φ_{𝔭} merely by φ. If m ⩾ 1, we denote the ring of m × m-matrices with entries in A by (A)_{m}.

(ii) If x, y are elements of some set, we let ∂_{x,y} := 1 if x = y and ∂_{x,y} := 0 if x ≠ y.

(iii) For integers a, b, we denote [a, b] := {x ∈ Z | a ⩽ x ⩽ b} and [a, b[ := {x ∈ Z | a ⩽ x < b}.

(iv) For a, b ⩾ 0, we let the binomial coefficient (\begin{matrix}a \\ b \end{matrix}) be equal to zero if b > a.

(v) Let ⊗ = ⊗_{Z} (i.e. ⊗ = ⊗_{Q} over Q).

Let m ⩾ 1.

(vi) For a prime p, we let m[p] := p^{v_{p}(m)} denote the p-part of m.

(vii) Let φ denote Euler’s function, φ(m) := m ∏_{p|m}(1 − p^{−1}). Let Φ_{m}(X) denote the m-th cyclotomic polynomial, i.e. the irreducible factor of X^{m} − 1 ∈ Q[X] that does not divide X^{m′} − 1 for any m′|m, m′ ≠ m. Hence deg Φ_{m}(X) = φ(m).

(viii) Let ζ_{m} denote a primitive mth root of unity over Q, with minimal polynomial μ_{ζ_{m},Q}(T) = Φ_{m}(T). Let c_{m} denote a generator of the cyclic group C_{m} of m elements.

1 Vandermonde

We diagonalize the polynomial Vandermonde matrix rationally. Specializing under the assumption that the diagonalizing matrices become integral, we obtain an elementary divisor form, i.e. a diagonal matrix in which each subsequent diagonal entry is divisible by its predecessor.

1.1 Diagonalization of Vandermonde matrices

A hint of T. Zink how to simplify our previous efforts led to the approach in this and in the next subsection.

Notation 1.1 Let m ⩾ 1. Let x = (x_{0}, …, x_{m−1}) be a tuple of indeterminates, let a, b ∈ [0, m − 1] and let d ∈ Z. We consider in Z[x_{0}, …, x_{m−1}] the symmetric polynomials

P_{d,[a,b]} := \{ \begin{matrix}∑ ; s_{i}⩾0, ∑_{i∈[a,b]} s_{i} = dx_{a}^{s_{a}}x_{a+1}^{s_{a+1}} ⋯ x_{b}^{s_{b}} & if d ⩾ 0 \\ 0 & if d < 0 \end{matrix}

E_{d,[a,b[} := \{ \begin{matrix}∑ ; s_{i}∈\{0,1\}, ∑_{i∈[a,b[} s_{i} = dx_{a}^{s_{a}}x_{a+1}^{s_{a+1}} ⋯ x_{b−1}^{s_{b−1}} & if d ⩾ 0 \\ 0 & if d < 0. \end{matrix}

In particular, we have

\begin{aligned}P_{d,∅} & = & ∂_{d,0} \\ E_{d,∅} & = & ∂_{d,0}. \end{aligned}

Moreover, for i ∈ [0, m − 1] we denote

y_{i} := ∏_{k∈[0,i[}(x_{i} − x_{k}) = ∑_{j∈[0,i]}(−1)^{i−j}E_{i−j,[0,i[}x_{i}^{j},

considered as element of Z[x_{0}, …, x_{m−1}]. For i, j ∈ [0, m − 1], we denote the Lagrange interpolation function by

L_{i,j} := \{ \begin{matrix}\frac{∏_{k∈[0,j[\\{i\}}(x_{j} − x_{k})}{∏_{k∈[0,j[\\{i\}}(x_{i} − x_{k})} & if i < j \\ 0 & if i ⩾ j \end{matrix} ,

and as a variant

M_{i,j} := \frac{∏_{k∈[0,i[}(x_{j} − x_{k})}{∏_{k∈[0,i[}(x_{i} − x_{k})},

considered as elements of Q(x_{0}, …, x_{m−1}). These polynomials furnish the matrices

\begin{aligned}I & := & (∂_{i,j})_{i×j∈[0,m−1]×[0,m−1]} \\ V_{x} & := & (x_{j}^{i})_{i×j∈[0,m−1]×[0,m−1]} \\ L_{x} & := & (L_{i,j})_{i×j∈[0,m−1]×[0,m−1]} \\ M_{x} & := & (M_{i,j})_{i×j∈[0,m−1]×[0,m−1]} \\ P_{x} & := & (P_{i−j,[0,j]})_{i×j∈[0,m−1]×[0,m−1]} \\ E_{x} & := & ((−1)^{i−j}E_{i−j,[0,i[})_{i×j∈[0,m−1]×[0,m−1]} \\ Y_{x} & := & (∂_{i,j}y_{i})_{i×j∈[0,m−1]×[0,m−1]} \end{aligned}

in (Q(x_{0}, …, x_{m−1}))_{m}.

Lemma 1.2 For 0 ⩽ a ⩽ b ⩽ m and d ∈ Z, we have

\begin{aligned}P_{d,[a,b]} & = & P_{d,[a+1,b]} + x_{a}P_{d−1,[a,b]} & = & P_{d,[a,b−1]} + x_{b}P_{d−1,[a,b]} \\ E_{d,[a,b[} & = & E_{d,[a+1,b[} + x_{a}E_{d−1,[a+1,b[} & = & E_{d,[a,b−1[} + x_{b}E_{d−1,[a,b−1[}, \end{aligned}

where for the second equation, we stipulate in addition that a < b.

Lemma 1.3 For i, k ∈ Z and 0 ⩽ a ⩽ b < c ⩽ m, we obtain

∑_{j∈Z}(−1)^{j}E_{i−j,[a,c[}P_{j−k,[a,b]} = (−1)^{k}E_{i−k,[b+1,c[}.

In fact,

∑_{j∈Z}(−1)^{j}E_{k−j,[a,c[}P_{j−i,[a,b]} =^{(1.2)} ∑_{j∈Z}(−1)^{j}E_{k−j,[a+1,c[}P_{j−i,[a,b]}
+ ∑_{j∈Z}(−1)^{j}x_{a}E_{k−j−1,[a+1,c[}P_{j−i,[a,b]}
= ∑_{j∈Z}(−1)^{j}E_{k−j,[a+1,c[}P_{j−i,[a,b]}
+ ∑_{j∈Z}(−1)^{j−1}E_{k−j,[a+1,c[}x_{a}P_{j−1−i,[a,b]}
=^{(1.2)} ∑_{j∈Z}(−1)^{j}E_{k−j,[a+1,c[}P_{j−i,[a,b]}
+ ∑_{j∈Z}(−1)^{j−1}E_{k−j,[a+1,c[}P_{j−i,[a,b]}
− ∑_{j∈Z}(−1)^{j−1}E_{k−j,[a+1,c[}P_{j−i,[a+1,b]}
= ∑_{j∈Z}(−1)^{j}E_{k−j,[a+1,c[}P_{j−i,[a+1,b]}.

Lemma 1.4 We have E_{x}P_{x} = I. Given i, k ∈ [0, m − 1], we may assume i > k and obtain

(E_{x}P_{x})_{i,k} = ∑_{j∈[0,m−1]}(−1)^{i−j}E_{i−j,[0,i[}P_{j−k,[0,k]}
=^{(1.3)} (−1)^{i−k}E_{i−k,[k+1,i[}
= 0.

Lemma 1.5 We have M_{x}(I − L_{x}) = I. Given i, k ∈ [0, m − 1], we may assume i < k and need to show that

M_{i,k} − ∑_{j∈[i,k[} M_{i,j}L_{j,k} = 0.

This expression being an element of Q(x_{0}, …, x_{k−1}, x_{k+1}, x_{m−1})[x_{k}] of degree in x_{k} less than k, we are reduced to plug in x_{k} = x_{l} for l ∈ [0, k − 1].

Lemma 1.6 We have E_{x}V_{x} = Y_{x}M_{x}. Given i, k ∈ [0, m − 1], we have

∑_{j∈[0,m−1]}(−1)^{i−j}E_{i−j,[0,i[}x_{k}^{j} = ∏_{j∈[0,i[}(x_{k} − x_{j}) = y_{i}M_{i,k}.

Proposition 1.7 We have E_{x}V_{x}(I − L_{x}) = Y_{x}. This follows from (1.5, 1.6).

Remark 1.8 We have V_{x} = P_{x}Y_{x}M_{x}.

This follows from (1.4, 1.6).

Example 1.9 Letting m = 4, we obtain

E_{x}V_{x}(I − L_{x})

= [ \begin{matrix}1 & 0 & 0 & 0 \\ −x_{0} & 1 & 0 & 0 \\ x_{0}x_{1} & −(x_{0}+x_{1}) & 1 & 0 \\ −x_{0}x_{1}x_{2} & x_{0}x_{1}+x_{0}x_{2}+x_{1}x_{2} & −(x_{0}+x_{1}+x_{2}) & 1 \end{matrix} ] · [ \begin{matrix}1 & 1 & 1 & 1 \\ x_{0} & x_{1} & x_{2} & x_{3} \\ x_{0}^{2} & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} \\ x_{0}^{3} & x_{1}^{3} & x_{2}^{3} & x_{3}^{3} \end{matrix} ] · [ \begin{matrix}1 & −1 & − \frac{x_{2}−x_{1}}{x_{0}−x_{1}} & − \frac{(x_{3}−x_{1})(x_{3}−x_{2})}{(x_{0}−x_{1})(x_{0}−x_{2})} \\ 0 & 1 & − \frac{x_{2}−x_{0}}{x_{1}−x_{0}} & − \frac{(x_{3}−x_{0})(x_{3}−x_{2})}{(x_{1}−x_{0})(x_{1}−x_{2})} \\ 0 & 0 & 1 & − \frac{(x_{3}−x_{0})(x_{3}−x_{1})}{(x_{2}−x_{0})(x_{2}−x_{1})} \\ 0 & 0 & 0 & 1 \end{matrix} ]

= [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & (x_{1}−x_{0}) & 0 & 0 \\ 0 & 0 & (x_{2}−x_{0})(x_{2}−x_{1}) & 0 \\ 0 & 0 & 0 & (x_{3}−x_{0})(x_{3}−x_{1})(x_{3}−x_{2}) \end{matrix} ] = Y_{x}.

1.2 Elementary divisors of Vandermonde matrices over discrete valuation rings

Setup 1.10 Let T be a discrete valuation ring with fraction field L = frac T. Let t ∈ T be a generator of the maximal ideal, and let v_{t} be the according valuation.

Definition 1.11 Let m ⩾ 1. Given a tuple ξ = (ξ_{0}, …, ξ_{m−1}) of pairwise distinct elements of T, we say that ξ is minimally ordered, if

∑_{i∈[0,j−1]}v_{t}(ξ_{j} − ξ_{i}) ⩽ ∑_{i∈[0,j−1]}v_{t}(ξ_{k} − ξ_{i})

for each j ∈ [0, m−1] and each k ∈ [j +1, m−1]. Note that any tuple of pairwise distinct elements of T can be reordered (non-uniquely, in general) to a minimally ordered tuple.

Lemma 1.12 If ξ is minimally ordered, then L_{ξ} ∈ (T)_{m} (cf. 1.1). In particular, the (i + 1)st T -linear elementary divisor of V_{ξ}, where i ∈ [0, m − 1], has valuation

∑_{j∈[0,i−1]} v_{t}(ξ_{i} − ξ_{j}).

By (1.5), L_{ξ} is contained in (T)_{m} if and only if M_{ξ} is contained in (T)_{m}. The assertion on the elementary divisors now ensues from (1.7).

Setup 1.13 Let S ⊆ T be a finite extension of discrete valuation rings, with fraction fields K := frac S and L := frac T. Let s ∈ S and t ∈ T be generators of the respective maximal ideals, and v_{s} and v_{t} the respective valuations. Assume L over K to be galois of degree m = [L : K] with Galois group G. Assume that

T = S[ϑ]

for some ϑ ∈ T. Choose an ordering

[figure: [0, m − 1]; ∼; G; i; σ_{i}]

such that the tuple τ := (ϑ^{σ_{1}}, …, ϑ^{σ_{m−1}}) is minimally ordered, starting with σ_{0} = 1_{T}.

Proposition 1.14 For i, j ∈ [0, m − 1], we let L_{i,j}(τ) denote the specialization of L_{i,j} along x \overline{}τ (cf. 1.1). The following assertions hold.

(i) The (i + 1)st T -linear elementary divisor, i ∈ [0, m − 1], of the Dedekind embedding

[figure: T; ⊗_{S}; T; δ_{T/S}; ∏_{σ∈G} T; x; ⊗; y; (xy^{σ})_{σ∈G}]

has valuation

φ_{i} := ∑_{j∈[0,i−1]} v_{t}(ϑ^{σ_{i}} − ϑ^{σ_{j}}).

In particular, (φ_{i})_{i∈[0,m−1]} does neither depend on the choice of ϑ nor on the minimal ordering chosen on τ.

(ii) The image of T ⊗_{S} T under δ_{T/S}, which is an isomorphic copy of T ⊗_{S} T, allows a description via ties, i.e. via congruences of tuple entries, as

\begin{matrix}(T ⊗_{S} T)δ_{T/S} \\ = & \{(η_{j})_{j∈[0,m−1]} | η_{i} − ∑_{j∈[0,i−1]}η_{j}L_{j,i}(τ) ∈ T t^{φ_{i}} for i ∈ [0, m − 1]\} \\ ⊆ & ∏_{j∈[0,m−1]} T. \end{matrix}

(iii) A T -linear basis of (T ⊗_{S} T)δ_{T/S} of triangular shape is given by the following tuple of elements of ∏_{j∈[0,m−1]} T.

(( ∏_{k∈[0,i−1]}(ϑ^{σ_{j}} − ϑ^{σ_{k}}))_{j∈[0,m−1]})_{i∈[0,m−1]} .

Ad (i). This follows from (1.12).

Ad (ii). An element η of ∏_{i∈[0,m−1]} T, considered as a row vector, is contained in the image of δ_{T/S} if and only if its rational inverse image, written as a row vector in the L-linear basis (1 ⊗ ϑ^{0}, …, 1 ⊗ ϑ^{m−1}) of L ⊗_{K} L, is in T ⊗_{S} T, i.e. if and only if ηV_{τ}^{−1} has entries in T. Which, in turn, is equivalent to η(I − L_{τ})Y_{τ}^{−1} having entries in T (1.7).

Ad (iii). We use E_{τ}V_{τ} = Y_{τ}M_{τ} (1.6).

1.3 A projective resolution of T over T ⊗_{S} T

In the introduction to [LL 92], several sources for a projective resolution of T over T ⊗_{S} T are indicated. We give still another alternative way to view such a projective resolution, using our isomorphic copy (T ⊗_{S} T)δ_{T/S}. We include the case of Hochschild (co)homology with galois twisted coefficients, for there a shift in the 2-periodic vanishing of these groups occurs when compared to the untwisted case.

Notation 1.15 We denote Λ := (T ⊗_{S} T)δ_{T/S}, i.e.

T ⊗_{S} T \overline{}^{δ_{T/S}}_{∼} Λ [figure] ∏_{σ∈G} T.

For i ∈ [0, m − 1], we dispose of the projection map

[figure: Λ; π_{i}; T; (η_{j})_{j∈[0,m−1]}; η_{i},]

which is a ring morphism, and by means of which T becomes a module over Λ, denoted by T_{i}.

If we identify along δ_{T/S}, the module operation of x ⊗ y ∈ T ⊗_{S} T on z ∈ T_{i} is given by z·(x⊗y) := xzy^{σ_{i}}. I.e. T_{i} may be viewed as T equipped with a structure as a galois twisted T -T -bimodule. In particular, T_{0} may be viewed as T equipped with the structure as an untwisted T -T -bimodule, that is z · (x ⊗ y) := xzy. Note that π_{0} is just the multiplication map.

In Λ, we have the elements

\begin{aligned}a & := & (∏_{k∈[1,m−1]}(ϑ − ϑ^{σ_{k}}), 0, …, 0) \\ b & := & (0, ϑ^{σ_{1}} − ϑ, …, ϑ^{σ_{m−1}} − ϑ). \end{aligned}

The element a is in Λ by (1.14 ii), and b is in Λ by (1.14 iii). The multiplication map by a resp. by b shall be denoted by Λ ^{\overline{}^{α}} Λ resp. by Λ ^{\overline{}^{β}} Λ.

Lemma 1.16 We have a 2-periodic projective resolution

⋯ ^{\overline{}^{α}} Λ ^{\overline{}^{β}} Λ ^{\overline{}^{α}} Λ ^{\overline{}^{β}} Λ ^{\overline{}^{π_{0}}} T_{0}

of T_{0} over Λ.

From ab = 0 we take αβ = 0 and βα = 0. Moreover, βπ_{0} = 0. By (1.14 ii), we obtain that the image of α equals the kernel of β. To prove that the image of β equals the kernel of α (resp. of π_{0}), means, after identification along δ_{T/S}, to show that this kernel is T ⊗_{S} T - linearly generated by b = 1 ⊗ ϑ − ϑ ⊗ 1. In fact, suppose given ∑_{i∈[0,m−1]} u_{i} ⊗ ϑ^{i} such that ∑_{i∈[0,m−1]} u_{i}ϑ^{i} = 0, we may write ∑_{i∈[0,m−1]} u_{i} ⊗ ϑ^{i} = ∑_{i∈[0,m−1]} u_{i}(1 ⊗ ϑ^{i} − ϑ^{i} ⊗ 1), and 1 ⊗ ϑ^{i} − ϑ^{i} ⊗ 1 is a multiple of 1 ⊗ ϑ − ϑ ⊗ 1.

Proposition 1.17 (cf. [BACH 91, prop. 1.3], [LL 92, 1.6.2]) Let φ := ∑_{k∈[1,m−1]} v_{t}(ϑ − ϑ^{σ_{k}}). The Hochschild homology of T, over the ground ring S and with coefficients in T_{i}, i ∈ [0, m − 1], is given in dimension j ⩾ 0 by

H_{j}(T, T_{0}; S) ≃ \{ \begin{matrix}T & for j = 0 \\ T/t^{φ}T & for j odd \\ 0 & for j even, j ⩾ 2 \end{matrix}

and by

H_{j}(T, T_{i}; S) ≃ \{ \begin{matrix}0 & for j odd \\ T/(ϑ^{σ_{i}} − ϑ)T & for j even \end{matrix}

for i ∈ [1, m − 1]. The Hochschild cohomology is given by

H^{j}(T, T_{0}; S) ≃ \{ \begin{matrix}T & for j = 0 \\ 0 & for j odd \\ T/t^{φ}T & for j even, j ⩾ 2 \end{matrix}

and by

H^{j}(T, T_{i}; S) ≃ \{ \begin{matrix}T/(ϑ^{σ_{i}} − ϑ)T & for j odd \\ 0 & for j even \end{matrix}

for i ∈ [1, m − 1].

Tensoring the resolution in (1.16) with T_{i} over Λ yields the homology. Application of Hom_{Λ}(−, T_{i}) yields the cohomology.

1.4 The local ring T ⊗_{S} T

Assume ϑ = t.

Remark 1.18 The radical of Λ is given by 𝔯Λ = Λ ∩ ∏_{i∈[0,m−1]} tT. In fact, the (φ + 1)st power of this intersection is contained in tΛ (1.14 ii). And conversely, by (1.14 iii), this intersection has the T -linear basis

{(t, …, t)} ∪ {(_{∏k∈[0,i−1]}(t^{σ_{j}} − t^{σ_{k}}))_{j∈[0,m−1]} | i ∈ [1, m − 1]} ,

whence Λ/(Λ ∩ ∏_{i∈[0,m−1]} tT) is isomorphic to T_{0}/tT_{0}. In particular, T ⊗_{S} T is a local ring.

Identifying along δ_{T/S}, a T -linear basis of the radical 𝔯Λ is given by

(t ⊗ 1, 1 ⊗ t, 1 ⊗ t^{2}, …, 1 ⊗ t^{m−1}).

Given j ∈ Z, we write \underline{j} := max(j, 0). Suppose given i ⩾ 0. A basis of 𝔯^{i}Λ is given by

(t\underline{i−0} ⊗ 1, t\underline{i−1} ⊗ t, t\underline{i−2} ⊗ t^{2}, …, t\underline{i−(m−1)} ⊗ t^{m−1}).

In particular, we have

dim_{T/tT} 𝔯^{i}Λ/𝔯^{i+1}Λ = min(i + 1, m).

Cf. [Kü 99, E.2.3].

2 The cyclotomic Dedekind embedding

We shall apply (1.14) to the case of the extension Z_{(p)} ⊆ Z_{(p)}[ζ_{p^{n}}].

Setup 2.1 Let p be a prime, let n ⩾ 1. In the notation of (1.13), we place ourselves in the situation S = Z_{(p)}, s = p, T = Z_{(p)}[ζ_{p^{n}}] and t = ϑ = 1 − ζ_{p^{n}}.

We consider the Dedekind embedding

[figure: Z_{(p)}[ζ_{p^{n}}]; ⊗; Z_{(p)}[ζ_{p^{n}}]; δ_{p^{n}}; ∏_{j∈(Z/p^{n})^{∗}} Z_{(p)}[ζ_{p^{n}}]; ζ_{p^{n}}^{k}; ⊗; ζ_{p^{n}}^{l}; (ζ_{p^{n}}^{k}ζ_{p^{n}}^{jl})_{j∈(Z/p^{n})^{∗}}]

where k, l ∈ [0, p^{n} − 1]. With respect to the Z_{(p)}[ζ_{p^{n}}]-linear basis (1 ⊗ t^{i})_{i∈[0,(p−1)p^{n−1}−1]} of Z_{(p)}[ζ_{p^{n}}] ⊗ Z_{(p)}[ζ_{p^{n}}] and to the tuple basis on the right hand side, this embedding is given by the Vandermonde matrix V_{τ} for τ := (1 − ζ_{p^{n}}^{j})_{j∈[0,p^{n}−1]\(p)}.

Lemma 2.2 For i, j ∈ [0, p^{n} − 1] with i ≠ j, we have v_{t}(ζ_{p^{n}}^{i} − ζ_{p^{n}}^{j}) = (i − j)[p].

We may assume j = 0 and i = i[p]. Using v_{t}(p) = (p − 1)p^{n−1} (resp. using a direct calculation if p = 2, i = 2^{n−1}), the congruence ζ_{p^{n}}^{i} − 1 ≡_{p} (ζ_{p^{n}} − 1)^{i} yields v_{t}(ζ_{p^{n}}^{i} − 1) = v_{t}((ζ_{p^{n}} − 1)^{i}) = i.

Lemma 2.3 Suppose given j ⩾ 0, and write it p-adically as j = ∑_{l⩾0} a_{l}p^{l}, where a_{l} ∈ [0, p − 1]. Then

∑_{i∈[1,j]}i[p] = ∑_{k⩾0}(a_{k} − a_{k+1})(k + 1)p^{k}.

Denote left and right hand side of the claimed equation by l(j) and r(j), respectively. We have l(0) = 0 = r(0). Moreover, for j ⩾ 1 we have l(j) − l(j − 1) = j[p]. If v_{p}(j) = 0, then r(j) − r(j − 1) = 1, so that we may suppose v_{p}(j) ⩾ 1. Writing j − 1 = ∑_{k⩾0} a′_{k}p^{k}, a′_{k} ∈ [0, p − 1], we note that a′_{k} = p − 1 and a_{k} = 0 for k ⩽ v_{p}(j) − 1, a′_{k} = a_{k} − 1 for k = v_{p}(j) and a′_{k} = a_{k} for k ⩾ v_{p}(j) + 1. Therefore,

r(j) − r(j − 1) = ∑_{k⩾0}(a_{k} − a′_{k})(k + 1)p^{k} − ∑_{k⩾1}(a_{k} − a′_{k})kp^{k−1}
= (v_{p}(j) + 1)j[p] − ∑_{k∈[0,v_{p}(j)−1]}(p − 1)(k + 1)p^{k}
− v_{p}(j)j[p]/p + ∑_{k∈[0,v_{p}(j)−2]}(p − 1)(k + 1)p^{k}
= j[p].

Lemma 2.4 Suppose given k ⩾ j ⩾ 0, with j, k ∉ (p). Write j = ∑_{l⩾0} a_{l}p^{l}, k = ∑_{l⩾0} a′_{l}p^{l}, k − j = ∑_{l⩾0} a′′_{l}p^{l}, where a_{l}, a′_{l}, a′′_{l} ∈ [0, p − 1]. Then

∑_{i∈[0,j−1]\(p)}(k − i)[p] = −1 + ∑_{l⩾0}(((a′_{l} − a′′_{l}) − (a′_{l+1} − a′′_{l+1})) (l + 1) − a_{l+1}) p^{l}.

In particular,

∑_{i∈[0,j−1]\(p)}(j − i)[p] = −1 + ∑_{l⩾0}(a_{l}(l + 1) − a_{l+1}(l + 2)) p^{l}.

This follows by (2.3) and the remark that ∑_{i∈[0,j−1]∩(p)}(k − i)[p] = #([0, j − 1] ∩ (p)) =

1 + ∑_{l⩾0}a_{l+1}p^{l}.

Lemma 2.5 Keep the notation of (2.4), but allow k ⩾ j ⩾ 0 to be arbitrary. We have

∑_{l⩾0}((a′_{l} − a′′_{l} − a_{l}) − (a′_{l+1} − a′′_{l+1} − a_{l+1})) (l + 1)p^{l} ⩾ 0.

Let U := {l ⩾ 0 | a_{l} + a′′_{l} ⩾ p} ⊆ Z_{⩾0}. We obtain

∑_{l⩾0}((a′_{l} − a′′_{l} − a_{l}) − (a′_{l+1} − a′′_{l+1} − a_{l+1})) (l + 1)p^{l}

= ∑_{l∈U}(−p)(l + 1)p^{l} + ∑_{l∈U+1}(l + 1)p^{l} − ∑_{l∈U−1}(−p)(l + 1)p^{l} − ∑_{l∈U}(l + 1)p^{l}
= ∑_{l∈U} (−(l + 1)p^{l+1} + (l + 2)p^{l+1} + lp^{l} − (l + 1)p^{l})
= ∑_{l∈U}(p − 1)p^{l}.

Lemma 2.6 The tuple τ = (1 − ζ_{p^{n}}^{j})_{j∈[0,p^{n}−1]\(p)} is minimally ordered.

Using (2.2), we need to see that

∑_{i∈[0,j−1]\(p)}(j − i)[p] ⩽ ∑_{i∈[0,j−1]\(p)}(k − i)[p]

for j ∈ [0, p^{n} − 1]\(p) and k ∈ [j + 1, p^{n} − 1]\(p). In the notation and using the assertion of (2.4) this amounts to the inequality

∑_{l⩾0}((a′_{l} − a′′_{l} − a_{l}) − (a′_{l+1} − a′′_{l+1} − a_{l+1})) (l + 1)p^{l} ⩾ 0

treated in (2.5).

Theorem 2.7 Suppose given j ∈ [0, p^{n} − 1]\(p). Let N(j) := # ([0, j]\(p)). Write j = ∑_{s⩾0} a_{s}p^{s} with a_{s} ∈ [0, p − 1]. The N(j)th Z_{(p)}[ζ_{p^{n}}]-linear elementary divisor of the cyclotomic Dedekind embedding

[figure: Z_{(p)}[ζ_{p^{n}}]; ⊗; Z_{(p)}[ζ_{p^{n}}]; δ_{p^{n}}; ∏_{i∈(Z/p^{n})^{∗}} Z_{(p)}[ζ_{p^{n}}]; ζ_{p^{n}}^{k}; ⊗; ζ_{p^{n}}^{l}; (ζ_{p^{n}}^{k+il})_{i∈(Z/p^{n})^{∗}},]

where k, l ∈ [0, (p − 1)p^{n−1} − 1], has valuation

−1 + ∑_{s⩾0}(a_{s}(s + 1) − a_{s+1}(s + 2)) p^{s}

at t.

This follows by (1.14 i) using (2.6, 2.4, 2.2).

Remark 2.8 Plesken gives a system of ties that describes the image of the Dedekind embedding case n = 1 [P 80, p. 60].

Remark 2.9 If n = 1, the valuation at t of the determinant of this embedding δ_{p} is −(p − 1) + ∑_{a∈[1,p−1]} a = (p − 1)(p − 2)/2. If n ⩾ 2, we obtain the valuation at t of the determinant of δ_{p^{n}} to be

−(p − 1)p^{n−1} + ∑_{(a_{l})_{l∈[0,n−1]}∈[0,p−1]^{n}, a_{n} = 0, a_{0}≠0} ∑_{l∈[0,n−1]}(a_{l}(l + 1) − a_{l+1}(l + 2))p^{l}
= −(p − 1)p^{n−1} + p^{n−2} · ∑_{a∈[1,p−1], b∈[0,p−1]}(a − 2b)
+ ∑_{l∈[1,n−2]}(p − 1)p^{n−3} · ∑_{a,b∈[0,p−1]}(a(l + 1) − b(l + 2))p^{l}
+ (p − 1)p^{n−2} · ∑_{a∈[0,p−1]}anp^{n−1}
= −(p − 1)p^{n−1} + p^{n−2} · (p − 1)(−p^{2} + 2p)/2
− (p − 1)p^{n−2} · p(p − 1)/2 · p(p^{n−2} − 1)/(p − 1)
+ (p − 1)p^{n−2} · p(p − 1)/2 · np^{n−1}
= p^{2n−2}(p − 1)((p − 1)n − 1)/2.

Hence for n ⩾ 1, we recalculated the valuation at p of the discriminant of Z_{(p)}[ζ_{p^{n}}] over Z_{(p)} to be p^{n−1}((p − 1)n − 1). Cf. [N 91, 10.1] (or 5.9).

Example 2.10 Let p = 3 and n = 2, so that

τ = (1 − ζ_{9}, 1 − ζ_{9}^{2}, 1 − ζ_{9}^{4}, 1 − ζ_{9}^{5}, 1 − ζ_{9}^{7}, 1 − ζ_{9}^{8}).

As elementary divisors of the Dedekind embedding

[figure: Z_{(3)}[ζ_{9}]; ⊗; Z_{(3)}[ζ_{9}]; δ_{9}; ∏_{j∈(Z/9)^{∗}} Z_{(3)}[ζ_{9}]; ζ_{9}^{k}; ⊗; ζ_{9}^{l}; (ζ_{9}^{k}ζ_{9}^{jl})_{j∈(Z/9)^{∗}},]

where k, l ∈ [0, 5], (2.7) yields

(t^{0}, t^{1}, t^{4}, t^{5}, t^{8}, t^{9}).

The determinant of δ_{9} has valuation 27 at t (cf. 2.9). By (1.14 iii), a triangular Z_{(3)}[ζ_{9}]-linear basis of the image of δ_{9} is given by the rows of the matrix

[ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 0 & −ζ_{9}+ζ_{9}^{2} & −ζ_{9}+ζ_{9}^{3} & −ζ_{9}+ζ_{9}^{4} & −2ζ_{9}−ζ_{9}^{4} & −ζ_{9}−ζ_{9}^{2}−ζ_{9}^{5} \\ 0 & 0 & −1−ζ_{9}^{4}−ζ_{9}^{5} & 1−ζ_{9}^{2}+2ζ_{9}^{3}−2ζ_{9}^{5} & −1+ζ_{9}^{2}+ζ_{9}^{3}+2ζ_{9}^{5} & −1−2ζ_{9}+ζ_{9}^{3}−ζ_{9}^{4} \\ 0 & 0 & 0 & 1−2ζ_{9}−2ζ_{9}^{2}+2ζ_{9}^{3}−ζ_{9}^{4}−ζ_{9}^{5} & 2+2ζ_{9}+2ζ_{9}^{2}+4ζ_{9}^{3}+ζ_{9}^{5} & −1−ζ_{9}+2ζ_{9}^{2}+ζ_{9}^{3}+ζ_{9}^{4}+ζ_{9}^{5} \\ 0 & 0 & 0 & 0 & 3+6ζ_{9}+3ζ_{9}^{3}−3ζ_{9}^{5} & 3ζ_{9}+3ζ_{9}^{2}+3ζ_{9}^{3}+3ζ_{9}^{4}+3ζ_{9}^{5} \\ 0 & 0 & 0 & 0 & 0 & 3ζ_{9}^{2}+3ζ_{9}^{4}+3ζ_{9}^{5} \end{matrix} ] .

3 The cyclic Wedderburn embedding

3.1 The q-Pascal method

We diagonalize the Vandermonde matrix that describes the Wedderburn embedding in the Z[ζ_{m}]-linear basis (c_{m}^{0}, …, c_{m}^{m−1}) of Z[ζ_{m}]C_{m}. More specifically speaking, we multiply with a lower triangular matrix G_{q} that records the q-Pascal triangle, consisting of Gaußian polynomials, from the right and with its transpose Gt_{q} from the left. The resulting diagonal matrix is of elementary divisor form, regardless whether or not the ground ring Z[ζ_{m}] is a principal ideal domain. We derive the necessary identities firstly in a pedestrian fashion, yielding in particular a formula for G_{q}^{−1}, and secondly, as a consequence of F. H. Jackson’s q-analogue (1910) of L. Saalschütz’ theorem (1890).

Notation 3.1 Let m ⩾ 1. The cyclic Wedderburn embedding is given by

[figure: Z[ζ_{m}]C_{m}; ω_{m}; ∏_{j∈[0,m−1]} Z[ζ_{m}]; c_{m}; (ζ_{m}^{j})_{j∈[0,m−1]}.]

3.1.1 Pedestrian

Notation 3.2 (Gaußian polynomials) Consider the field Q(q) of rational functions in the indeterminate q. Given i ⩾ 0 and j ∈ Z and another indeterminate t, we let

\begin{aligned}[i] & := & (q^{i} − 1)/(q − 1) \\ [i]! & := & ∏_{k∈[1,i]}[k] \\ [\begin{matrix}i \\ j \end{matrix}] & := & \{ \begin{matrix}\frac{[i]!}{[j]![i−j]!} & if j ∈ [0, i] \\ 0 & if j ∉ [0, i] \end{matrix} \\ (t; q)_{i} & := & ∏_{k∈[0,i−1]}(1 − q^{k}t). \end{aligned}

In particular, [0]! = 1. If necessary, we indicate q as [\begin{matrix}i \\ j \end{matrix}]_{q} := [\begin{matrix}i \\ j \end{matrix}]. Let the q-Pascal matrix of size m × m be defined by

G_{q} := ([\begin{matrix}i \\ j \end{matrix}])_{i,j∈[0,m−1]} .

Moreover, we let

\begin{aligned}I & := & (∂_{i,j})_{i,j∈[0,m−1]} \\ V_{q} & := & (q^{ij})_{i,j∈[0,m−1]} \\ D_{q} & := & (∂_{i,j}[i]!(q − 1)^{i}q^{(\begin{matrix}i \\ 2 \end{matrix})})_{i,j∈[0,m−1]} . \end{aligned}

Lemma 3.3 If i ⩾ 1 and j ⩾ 0, we have

\begin{aligned}[\begin{matrix}i \\ j \end{matrix}] & = & [\begin{matrix}i−1 \\ j−1 \end{matrix}] & + & q^{j} [\begin{matrix}i−1 \\ j \end{matrix}] \\ & = & q^{i−j} [\begin{matrix}i−1 \\ j−1 \end{matrix}] & + & [\begin{matrix}i−1 \\ j \end{matrix}] & . \end{aligned}

Moreover,

(t; q)_{i} = ∑_{k∈[0,i]}(−1)^{k}q^{(\begin{matrix}k \\ 2 \end{matrix} )} [\begin{matrix}i \\ k \end{matrix}] t^{k}.

Finally,

[\begin{matrix}i \\ j \end{matrix}]_{q^{−1}} = q^{−j(i−j)} [\begin{matrix}i \\ j \end{matrix}]_{q} .

Proposition 3.4 (Inversion of the q-Pascal matrix) We obtain the inverse of G_{q} to be

G_{q}^{−1} = ((−1)^{j+k}q^{(\begin{matrix}j−k \\ 2 \end{matrix} )} [\begin{matrix}j \\ k \end{matrix}])_{j,k∈[0,m−1]} .

Given 0 ⩽ k ⩽ i ⩽ m − 1, we need to show that

∑_{j∈[k,i]} [\begin{matrix}i \\ j \end{matrix}] [\begin{matrix}j \\ k \end{matrix}] (−1)^{j+k}q^{(\begin{matrix}j−k \\ 2 \end{matrix} )} = ∂_{i,k}.

We perform an induction on i − k. If i = k, we obtain 1 = 1. If i = k + 1, we obtain 0 = 0. If i ⩾ k + 2, we obtain

∑_{j⩾0}[\begin{matrix}i \\ j \end{matrix}] [\begin{matrix}j \\ k \end{matrix}] (−1)^{j+k}q^{(\begin{matrix}j−k \\ 2 \end{matrix} )} =^{(3.3)} ∑_{j⩾0}[\begin{matrix}i−1 \\ j \end{matrix}] [\begin{matrix}j \\ k \end{matrix}] (−1)^{j+k}q^{(\begin{matrix}j−k \\ 2 \end{matrix} )}
+ ∑_{j⩾0}[\begin{matrix}i−1 \\ j−1 \end{matrix}] [\begin{matrix}j−1 \\ k \end{matrix}] (−1)^{j+k}q^{(\begin{matrix}j−k \\ 2 \end{matrix} )+(i−j)}
+ ∑_{j⩾0}[\begin{matrix}i−1 \\ j−1 \end{matrix}] [\begin{matrix}j−1 \\ k−1 \end{matrix}] (−1)^{j+k}q^{(\begin{matrix}j−k \\ 2 \end{matrix} )+(i−j)+(j−k)}
=^{induction} ∑_{j⩾0}[\begin{matrix}i−1 \\ j−1 \end{matrix}] [\begin{matrix}j−1 \\ k−1 \end{matrix}] (−1)^{j+k}q^{(\begin{matrix}j−k \\ 2 \end{matrix} )+i−k}.

The last term vanishes by subinduction on k, since if k = 0 we obtain

∑_{j⩾0}[\begin{matrix}i \\ j \end{matrix}] (−1)^{j}q^{(\begin{matrix}j \\ 2 \end{matrix})} =^{(3.3)} (1; q)_{i} = 0.

Lemma 3.5 We have

V_{q}(G_{q}^{−1})^{t} = G_{q}D_{q}.

We need to show that for i, k ∈ [0, m − 1]

∑_{j∈[0,k]}q^{ij}(−1)^{j+k} [\begin{matrix}k \\ j \end{matrix}] q^{(\begin{matrix}k−j \\ 2 \end{matrix} )} = q^{(\begin{matrix}k \\ 2 \end{matrix} )}[k]!(q − 1)^{k} [\begin{matrix}i \\ k \end{matrix}] .

We perform an induction on k, starting in case k = 0 with 1 = 1. If k ⩾ 1, we calculate

∑_{j∈[0,k]}q^{ij}(−1)^{j+k} [\begin{matrix}k \\ j \end{matrix}] q^{(\begin{matrix}k−j \\ 2 \end{matrix} )} =^{(3.3)} ∑_{j∈[0,k−1]}q^{i(j+1)}(−1)^{j+1+k} [\begin{matrix}k−1 \\ j \end{matrix}] q^{(\begin{matrix}k−j−1 \\ 2 \end{matrix})}
+ ∑_{j∈[0,k−1]}q^{ij}(−1)^{j+k} [\begin{matrix}k−1 \\ j \end{matrix}] q^{(\begin{matrix}k−j \\ 2 \end{matrix} )+j}
=^{induction} (q^{i} − q^{k−1})q^{(\begin{matrix}k−1 \\ 2 \end{matrix})}[k − 1]!(q − 1)^{k−1} [\begin{matrix}i \\ k−1 \end{matrix}]
= q^{(\begin{matrix}k \\ 2 \end{matrix} )}[k]!(q − 1)^{k} [\begin{matrix}i \\ k \end{matrix}] .

Lemma 3.6 (Fourier inversion) We have V_{ζ_{m}}V_{ζ_{m}^{−1}} = mI. In particular, reordering the rows of V_{ζ_{m}^{−1}} shows that

(det V_{ζ_{m}})^{2} = \{ \begin{matrix}(−1)^{\frac{m−1}{2}} m^{m} & if m is odd \\ (−1)^{\frac{m−2}{2}} m^{m} & if m is even. \end{matrix}

In fact,

\begin{aligned}∑_{j∈[0,m−1]} ζ_{m}^{ij}ζ_{m}^{−jk} & = & ∑_{j∈[0,m−1]} ζ_{m}^{(i−k)j} \\ & = & m∂_{i,k}. \end{aligned}

Corollary 3.7 (cf. [N 91, 8.6]) If m is odd, then Q(ζ_{m}) contains \sqrt{(−1)^{\frac{m−1}{2}} m}.

Proposition 3.8

(i) We have the diagonalization

G_{ζ_{m}^{−1}}^{t}V_{ζ_{m}}G_{ζ_{m}^{−1}} = m(D_{ζ_{m}^{−1}})^{−1},

the right hand side being a diagonal matrix with (i + 1)st diagonal entry

\frac{mζ_{m}^{(i^{2})}}{∏_{j∈[1,i]}(1 − ζ_{m}^{j})},

where i ∈ [0, m − 1]. (ii) We have

(Z[ζ_{m}]C_{m})ω_{m}

= \{(y_{j})_{j∈[0,m−1]} | ∑_{j∈[i,m−1]}y_{j} [\begin{matrix}j \\ i \end{matrix}]_{ζ_{m}^{−1}} ∈ Z[ζ_{m}] · \frac{m}{∏_{j∈[1,i]}(1 − ζ_{m}^{j})} for i ∈ [0, m − 1]\}
⊆ ∏_{j∈[0,m−1]} Z[ζ_{m}].

(iii) A Z[ζ_{m}]-linear basis of (Z[ζ_{m}]C_{m})ω_{m} is given by the following tuple of elements of ∏_{j∈[0,m−1]} Z[ζ_{m}].

(((−1)^{k}ζ_{m}^{(\begin{matrix}k \\ 2 \end{matrix} )} \frac{m}{∏_{l∈[1,j]}(1 − ζ_{m}^{l})} [\begin{matrix}j \\ k \end{matrix}]_{ζ_{m}})_{k∈[0,m−1]})_{j∈[0,m−1]}

Ad (i). This formula follows from (3.5, 3.6).

Ad (ii). A vector y = (y_{j})_{j∈[0,m−1]} is contained in the image of ω_{m} if and only if yV_{ζ_{m}}^{−1} has entries in Z[ζ_{m}]. But V_{ζ_{m}}^{−1}(G_{ζ_{m}^{−1}}^{t})^{−1} = G_{ζ_{m}^{−1}}D_{ζ_{m}^{−1}}m^{−1}, so this condition translates into yG_{ζ_{m}^{−1}}D_{ζ_{m}^{−1}}m^{−1} to have entries in Z[ζ_{m}], which is the defining condition given above.

Ad (iii). This follows by G_{ζ_{m}^{−1}}^{t}V_{ζ_{m}} = m(D_{ζ_{m}^{−1}})^{−1}(G_{ζ_{m}^{−1}})^{−1} using (3.4, 3.3).

Remark 3.9 Because the derivative of

F (X) =^{1.} X^{m} − 1 =^{2.} ∏_{j∈[0,m−1]}(X − ζ_{m}^{j})

evaluated at 1 yields

F′(1) =^{1.} m =^{2.} ∏_{j∈[1,m−1]}(1 − ζ_{m}^{j}),

the mth diagonal entry in (3.8 i) equals ζ_{m}. This reproves in particular that all diagonal entries of (3.8 i) are contained in Z[ζ_{m}].

Remark 3.10 Using (3.9), for n ⩾ 1, p prime, m = p^{n}, the valuation at t = 1 − ζ_{p^{n}} of the (i + 1)th elementary divisor of ω_{p^{n}}, i ∈ [0, p^{n} − 1], is given by

\begin{aligned}v_{t} (\frac{p^{n}}{∏_{j∈[1,p^{n}−1−i]}(1 − ζ_{p^{n}}^{j})}) & = & ∑_{j∈[1,i]} v_{t} (1 − ζ_{p^{n}}^{−j}) \\ & =^{(2.2)} & ∑_{j∈[1,i]} j[p], \end{aligned}

in accordance with (3.15) below in view of (2.3).

3.1.2 Invoking q-Saalschütz

Following J. Stokman, we cite the

Theorem 3.11 (Saalschütz, Jackson [A 76, 3.3.12] (^{1})) Given n ⩾ 0, and further indeterminates a, b, c, we have

∑_{m⩾0} \frac{(a; q)_{m}(b; q)_{m}(q^{−n}; q)_{m}}{(q; q)_{m}(c; q)_{m}(abq^{1−n}c^{−1}; q)_{m}} · q^{m} = \frac{(c/a; q)_{n}(c/b; q)_{n}}{(c; q)_{n}(c/ab; q)_{n}} .

and specialize to the

Corollary 3.12 (⟺ 3.5) Given m ⩾ 1, we have

V_{q} = G_{q}D_{q}(G_{q})^{t}.

Given i, k ∈ [0, m − 1], we need to show that

q^{ik} = ∑_{j∈[0,m−1]} [\begin{matrix}i \\ j \end{matrix}] · [j]!(q − 1)^{j}q^{(\begin{matrix}j \\ 2 \end{matrix})} · [\begin{matrix}k \\ j \end{matrix}] .

Substituting q by q^{−1}, n by k and specializing c and then b to zero (after rewriting the right hand side), and specializing a to q^{i}, (3.11) becomes

q^{ik} = ∑_{j⩾0} \frac{(q^{i}; q^{−1})_{j}(q^{k}; q^{−1})_{j}}{(q^{−1}; q^{−1})_{j}} · q^{−j}
= ∑_{j⩾0} \frac{(1 − q^{i}) ⋯ (1 − q^{i−j+1}) · (1 − q^{k}) ⋯ (1 − q^{k−j+1})}{(1 − q^{−1}) ⋯ (1 − q^{−j})} · q^{−j}
= ∑_{j⩾0} \frac{(q^{i} − 1) ⋯ (q^{i−j+1} − 1)}{(q^{1} − 1) ⋯ (q^{j} − 1)} · \frac{(q^{k} − 1) ⋯ (q^{k−j+1} − 1)}{(q^{1} − 1) ⋯ (q^{j} − 1)} · q^{(\begin{matrix}j \\ 2 \end{matrix})}[j]!(q − 1)^{j}
= ∑_{j⩾0} [\begin{matrix}i \\ j \end{matrix}] · [\begin{matrix}k \\ j \end{matrix}] · q^{(\begin{matrix}j \\ 2 \end{matrix})}[j]!(q − 1)^{j}.

3.2 The general Vandermonde method

To compare, and to recalculate elementary divisors, we apply the method of §1.2 to the localized cyclic Wedderburn embedding.

Setup 3.13 Let p be a prime, let n ⩾ 1. In the notation of (1.13), we place ourselves in the situation S = Z_{(p)}, s = p, T = Z_{(p)}[ζ_{p^{n}}] and t = ϑ = 1 − ζ_{p^{n}}. We consider the localized cyclic Wedderburn embedding

[figure: Z_{(p)}[ζ_{p^{n}}]C_{p^{n}}; ω_{p^{n}}; ∼; ∏_{j∈Z/p^{n}} Z_{(p)}[ζ_{p^{n}}]; c_{p^{n}}^{i}; (ζ_{p^{n}}^{ij})_{j∈Z/p^{n}},]

where i ∈ [0, p^{n} − 1]. In the notation of (1.1), and with respect to the Z_{(p)}[ζ_{p^{n}}]-linear basis ((1 − c_{p^{n}})^{i})_{i∈[0,p^{n}−1]} of Z_{(p)}[ζ_{p^{n}}]C_{p^{n}} and the standard basis on the right hand side, this embedding is given by the Vandermonde matrix V_{τ} for τ := (1 − ζ_{p^{n}}^{j})_{j∈[0,p^{n}−1]}.

Lemma 3.14 The tuple τ := (1 − ζ_{p^{n}}^{j})_{j∈[0,p^{n}−1]} is minimally ordered.

Using (2.2), we need to see that

∑_{i∈[0,j−1]}(j − i)[p] ⩽ ∑_{i∈[0,j−1]}(k − i)[p]

1In the left hand side summand of [A 76, 3.3.14], a factor (ab/c)^{N−n} has been forgotten.

for j ∈ [0, p^{n} − 1] and k ∈ [j + 1, p^{n} − 1]. Write j = ∑_{l⩾0} a_{l}p^{l}, k = ∑_{l⩾0} a′_{l}p^{l}, k − j = ∑_{l⩾0} a′′_{l}p^{l}, where a_{l}, a′_{l}, a′′_{l} ∈ [0, p − 1]. By (2.3), we need to show that

∑_{l⩾0}((a′_{l} − a′′_{l} − a_{l}) − (a′_{l+1} − a′′_{l+1} − a_{l+1})) (l + 1)p^{l} ⩾ 0.

This follows from (2.5).

Proposition 3.15 Suppose given j ∈ [0, p^{n} − 1]. Write j = ∑_{k⩾0} a_{k}p^{k}, a_{k} ∈ [0, p − 1]. The (j + 1)st elementary divisor of the localized cyclic Wedderburn embedding

[figure: Z_{(p)}[ζ_{p^{n}}]C_{p^{n}}; ω_{p^{n}}; ∏_{j∈Z/p^{n}} Z_{(p)}[ζ_{p^{n}}]; c_{p^{n}}^{i}; (ζ_{p^{n}}^{ij})_{j∈Z/p^{n}}]

where i ∈ [0, p^{n} − 1], has valuation

∑_{k⩾0}(a_{k} − a_{k+1})(k + 1)p^{k}

at t.

This follows by (1.7, 1.12), using (3.14, 2.3), or by (3.10), using (2.3).

Remark 3.16 By (3.6), the square of the determinant of ω_{p^{n}} has valuation np^{n} at p. Alternatively, by (3.15), we obtain the valuation at t of the determinant of this embedding to be

∑_{(a_{l})_{l∈[0,n−1]}∈[0,p−1]^{n}, a_{n} = 0} ∑_{l∈[0,n−1]}(a_{l} − a_{l+1})(l + 1)p^{l}

= ( ∑_{l∈[0,n−2]}p^{n−2} · ∑_{a,b∈[0,p−1]}(a − b)(l + 1)p^{l}) + (p^{n−1} · ∑_{a∈[0,p−1]}anp^{n−1})
= p^{n−1} \frac{p(p−1)}{2}np^{n−1}.

3.3 The Pascal method

3.3.1 First order Pascal ties

There are some obvious ties, i.e. congruences of tuple entries, that are necessary for elements of ∏_{i∈[0,p^{n}−1]} Z[ζ_{p^{n}}] to lie in (Z[ζ_{p^{n}}]C_{p^{n}})ω_{p^{n}}. If n = 1, they are already sufficient, yielding a manageable basis of (Z[ζ_{p}]C_{p})ω_{p}. In general, they describe an intermediate ring between (Z[ζ_{p^{n}}]C_{p^{n}})ω_{p^{n}} and ∏_{i∈[0,p^{n}−1]} Z[ζ_{p^{n}}].

Notation 3.17 Suppose given s ⩾ 0, a polynomial f(X) := ∑_{i⩾0} a_{i}X^{i} ∈ C[X] and a tuple (y_{j})_{j∈[0,s]}, y_{j} ∈ C. We define the evaluation of f at (y_{j})_{j∈[0,s]} to be

(∑_{i⩾0}a_{i}X^{i}) [(y_{j})_{j∈[0,s]}] := f [(y_{j})_{j∈[0,s]}] := ∑_{j∈[0,s]} a_{j}y_{j}.

We note the difference between the polynomial power f^{i} [(y_{j})_{j∈[0,s]}] (power of f, taken in C[X], evaluated) and the ordinary power f [(y_{j})_{j∈[0,s]}]^{i} (power of the evaluation of f, taken in C), i ⩾ 0.

Let m ⩾ 1, let t = 1 − ζ_{m}. We consider the Z[ζ_{m}]-submodule W_{m}^{(1)} of

W_{m}^{(0)} := ∏_{j∈[0,m−1]} Z[ζ_{m}]

defined by the first order Pascal ties

W_{m}^{(1)} := {(y_{j})_{j∈[0,m−1]} | ((1 − X)^{i}) [(y_{j})_{j∈[0,m−1]}] ≡_{t^{i}} 0 for all i ∈ [0, m − 1]}
= \{(y_{j})_{j∈[0,m−1]} | ∑_{j∈[0,i]}(−1)^{j} (\begin{matrix}i \\ j \end{matrix}) y_{j} ≡_{t^{i}} 0 for all i ∈ [0, m − 1]\}
⊆ W_{m}^{(0)}.

Lemma 3.18

(i) The image (Z[ζ_{m}]C_{m})ω_{m} of the Wedderburn embedding (3.1) is contained in W_{m}^{(1)},

Z[ζ_{m}]C_{m} ^{\overline{}^{ω_{m}}} W_{m}^{(1)} [figure] W_{m}^{(0)}.

(ii) A Z[ζ_{m}]-linear basis of W_{m}^{(1)} is given by

(ξ_{m,i})_{i∈[0,m−1]} := (((−1)^{i}t^{i} (\begin{matrix}j \\ i \end{matrix}))_{j∈[0,m−1]})_{i∈[0,m−1]} .

(iii) The (i+1)st elementary divisor over Z[ζ_{m}] of the embedding W_{m}^{(1)} ⊆ W_{m}^{(0)} is given by t^{i}, i ∈ [0, m−1]. (In particular, Z[ζ_{m}] not being a principal ideal domain in general, there exist bases with respect to which the matrix that describes this embedding takes diagonal shape.)

(iv) The Z[ζ_{m}]-linear determinant of the embedding W_{m}^{(1)} ⊆ W_{m}^{(0)} is t^{m(m−1)/2}.

Ad (i). We have ((1 − X)^{i}) [(ζ_{m}^{lj})_{j∈[0,m−1]}] = (1 − ζ_{m}^{l})^{i} ≡_{t^{i}} 0 for all i ∈ [0, m − 1] and all l ∈ [0, m − 1].

Ad (ii). Inverting the matrix A := ((−1)^{i}t^{i} (\begin{matrix}j \\ i \end{matrix}))_{i∈[0,m−1], j∈[0,m−1]} arising from the tuple

of elements (ξ_{m,j})_{j∈[0,m−1]}, we obtain A^{−1} := ((−1)^{i}t^{−j} (\begin{matrix}j \\ i \end{matrix}))_{i∈[0,m−1], j∈[0,m−1]}. An element

y is contained in the Z[ζ_{m}]-linear span of our tuple if and only if, y considered as a row vector, yA^{−1} is entrywise contained in Z[ζ_{m}], i.e. if and only if y ∈ W_{m}^{(1)}.

In particular, we have the Proposition 3.19 Let p be a prime. We have a factorization of the Wedderburn embedding (3.1) into

Z[ζ_{p}]C_{p} \overline{}^{ω_{p}}_{∼} W_{p}^{(1)} ⊆ W_{p}^{(0)} = ∏_{j∈Z/p} Z[ζ_{p}].

The factorization follows by (3.18 i). The isomorphism follows by comparison of (3.18 iv) with (3.6), both yielding the valuation at t of the determinant of the respective embedding to be p(p − 1)/2, and zero elsewhere. We remark that the elementary divisors resulting from (3.18 iii) are in accordance with (3.15).

Remark 3.20 (coefficient criterion) The tuple (y_{i})_{i∈[0,p]} ∈ W_{p}^{(0)} is contained in W_{p}^{(1)}
if and only if, writing y_{i} =: ∑_{j∈[0,p−2]} y_{i,j}ζ^{j}, y_{i,j} ∈ Z,

∑_{i∈[0,p−1], j∈[0,p−2]}(−1)^{i} (\begin{matrix}u \\ i \end{matrix}) (\begin{matrix}j \\ v \end{matrix}) y_{i,j} ≡_{p} 0

holds for all 0 ⩽ v < u ⩽ p − 1.

An element ∑_{j∈[0,p−2]} x_{j}ζ_{p}^{j} = ∑_{v∈[0,p−2]} (∑_{j∈[0,p−2]} x_{j} (\begin{matrix}j \\ v \end{matrix})) (ζ_{p} − 1)^{v}, x_{j} ∈ Z, vanishes

modulo t^{u} for u ∈ [0, p − 1] if and only if ∑_{j∈[0,p−2]} x_{j} (\begin{matrix}j \\ v \end{matrix}) ≡_{p} 0 for all v ∈ [0, u − 1]. It

remains to plug in x_{j} = ∑_{i∈[0,p−1]}(−1)^{i} (\begin{matrix}u \\ i \end{matrix}) y_{i,j}.

Lemma 3.21 Given 0 ⩽ j ⩽ i, we have

ξ_{m,j}ξ_{m,i} = ∑_{k∈[0,j]} (\begin{matrix}j \\ k \end{matrix}) (\begin{matrix}i+k \\ j \end{matrix}) (−1)^{j−k}t^{j−k}ξ_{m,i+k},

where we let ξ_{m,l} := 0 for l ⩾ m. In particular, W_{m}^{(1)} is a subring of W_{m}^{(0)}.

We need to see that for l ∈ [0, m − 1]

(−1)^{j}t^{j} (\begin{matrix}l \\ j \end{matrix}) · (−1)^{i}t^{i} (\begin{matrix}l \\ i \end{matrix}) = ∑_{k∈[0,j]} (\begin{matrix}j \\ k \end{matrix}) (\begin{matrix}i+k \\ j \end{matrix}) (−1)^{j−k}t^{j−k} · (−1)^{i+k}t^{i+k} (\begin{matrix}l \\ i+k \end{matrix}) .

We may assume i ⩽ l and reformulate to

(\begin{matrix}l \\ i \end{matrix}) = ∑_{k⩾0}(\begin{matrix}j \\ k \end{matrix}) (\begin{matrix}l−j \\ l−i−k \end{matrix}) ,

which now follows from a comparison of coefficients in (1 + T)^{l} = (1 + T)^{j}(1 + T)^{l−j} at T^{l−i}.

Remark 3.22 Let p be a prime, let n ⩾ 1, let m = p^{n} (in particular, t = 1 − ζ_{p^{n}}). We dispose of ring automorphisms

[figure: Z[ζ_{p^{n}}]C_{p^{n}}; α_{p^{n}}; ∼; Z[ζ_{p^{n}}]C_{p^{n}}; c_{p^{n}}; ζ_{p^{n}}c_{p^{n}},]

and

[figure: W_{p^{n}}^{(0)}; α_{p^{n}}^{(0)}; ∼; W_{p^{n}}^{(0)}; (y_{j})_{j∈Z/p^{n}}; (y_{j+1})_{j∈Z/p^{n}}]

satisfying ω_{p^{n}}α_{p^{n}}^{(0)} = α_{p^{n}}ω_{p^{n}}. Moreover, α_{p^{n}}^{(0)} restricts to an automorphism α_{p^{n}}^{(1)} of W_{p^{n}}^{(1)}, with operation given by

ξ_{p^{n},j}α_{p^{n}}^{(1)} = ξ_{p^{n},j} − tξ_{p^{n},j−1} + (−1)^{p−j}t^{j−(p^{n}−1)} (\begin{matrix}p^{n} \\ j \end{matrix} ) ξ_{p^{n},p^{n}−1}

for j ∈ [1, p^{n} − 1], and by ξ_{p^{n},0}α_{p^{n}}^{(1)} = ξ_{p^{n},0}(= 1_{W_{p^{n}}^{(1)}}).

For j ∈ [1, p^{n} − 1] and i ∈ [0, p^{n} − 1], we have

((−1)^{j}t^{j} (\begin{matrix}i \\ j \end{matrix} )) − t ((−1)^{j−1}t^{j−1} (\begin{matrix}i \\ j−1 \end{matrix})) = (−1)^{j}t^{j} (\begin{matrix}i+1 \\ j \end{matrix}) ,

which we compare for i = p^{n} − 1 with

ξ_{p^{n},p^{n}−1} = (0, …, 0, (−1)^{p^{n}−1}t^{p^{n}−1}),

whence the formula describing the operation of α_{p^{n}}^{(1)}. The valuation at t of the third coefficient therein amounts to j + (n − v_{p}(j))(p − 1)p^{n−1} − (p^{n} − 1). In case n − v_{p}(j) ⩾ 2, we obtain

j + (n − v_{p}(j))(p − 1)p^{n−1} ⩾ 2(p − 1)p^{n−1} ⩾ p^{n} − 1.

If n − v_{p}(j) = 1, we obtain

j + (n − v_{p}(j))(p − 1)p^{n−1} ⩾ p^{n−1} + (p − 1)p^{n−1} ⩾ p^{n} − 1.

Hence α_{p^{n}}^{(0)} restricts to an automorphism α_{p^{n}}^{(1)}.

Lemma 3.23 The factorization of the Wedderburn embedding maps

[figure: Z[ζ_{m}]C_{m}; ω_{m}; W_{m}^{(1)}; c_{m}^{i}; ∑_{k∈[0,m−1]} (\frac{1−ζ_{m}^{i}}{t} )^{k} ξ_{m,k}]

for i ∈ [0, m − 1], where we let 0^{0} = 1.

In fact,

∑_{k∈[0,m−1]} (\frac{1−ζ_{m}^{i}}{t} )^{k} ξ_{m,k} = (∑_{k∈[0,m−1]} (\frac{1−ζ_{m}^{i}}{t} )^{k} (−1)^{k}t^{k} (\begin{matrix}j \\ k \end{matrix}))_{j∈[0,m−1]}
= ((1 − (1 − ζ_{m}^{i}))^{j})_{j∈[0,m−1]}
= c_{m}^{i}ω_{m},

which remains true for i = 0.

3.3.2 Second order Pascal ties

In this appendix to §3.3.1 we shall indicate a method of how to continue the approach via Pascal ties.

Let n ⩾ 2, let p be a prime and let t = 1 − ζ_{p^{n}}.

Lemma 3.24 Let s ⩾ 1. For any i ⩾ 1, we have in Q[X]

∑_{k∈[1,s]} \frac{1}{k} ((1 − (1 − X)^{i})^{k} − iX^{k}) ∈ (X^{s+1}).

This expression being contained in (X), it suffices to prove that its derivative is contained in (X^{s}). In fact,

\frac{d}{dX} ∑_{k∈[1,s]} \frac{1}{k} ((1 − (1 − X)^{i})^{k} − iX^{k})
= ∑_{k∈[1,s]} ((1 − (1 − X)^{i})^{k−1}i(1 − X)^{i−1} − iX^{k−1})
= i (\frac{1 − (1 − (1 − X)^{i})^{s}}{1 − (1 − (1 − X)^{i})} (1 − X)^{i−1} − \frac{1 − X^{s}}{1 − X} )
= \frac{i}{1 − X} (−(1 − (1 − X)^{i})^{s} + X^{s}) .

Lemma 3.25 For this lemma, we allow n ⩾ 1. Let

f_{p^{n}}(X) := ( ∑_{k∈[1,p−1]} \frac{t^{k−1}}{k} ) X^{p} − ∑_{k∈[1,p−1]} \frac{t^{k−1}}{k} X^{k} ∈ Z_{(p)}[ζ_{p^{n}}][X].

For any i ∈ [0, p^{n} − 1], we have

f_{p^{n}} (\frac{1 − ζ_{p^{n}}^{i}}{t} ) ≡_{t^{p−1}} 0.

We also write shorthand

γ := ∑_{k∈[1,p−1]} \frac{t^{k−1}}{k} .

In fact,

(∑_{k∈[1,p−1]} \frac{t^{k−1}}{k} ) (\frac{1−(1−t)^{i}}{t} )^{p} − ∑_{k∈[1,p−1]} \frac{t^{k−1}}{k} (\frac{1−(1−t)^{i}}{t} )^{k}
=  \frac{1}{t} ∑_{k∈[1,p−1]} \frac{1}{k} (t^{k} (\frac{1−(1−t)^{i}}{t} )^{p} − (1 − (1 − t)^{i})^{k})
≡_{p}  \frac{1}{t} ∑_{k∈[1,p−1]} \frac{1}{k} (t^{k} \frac{1−(1−t^{p})^{i}}{t^{p}} − (1 − (1 − t)^{i})^{k})
≡_{t^{p}}  \frac{1}{t} ∑_{k∈[1,p−1]} \frac{1}{k} (it^{k} − (1 − (1 − t)^{i})^{k})
≡^{(3.24)}_{t^{p−1}} 0.

We consider the Z[ζ_{p^{n}}]-submodule W_{p^{n}}^{(2)} of W_{p^{n}}^{(1)} defined by the second order Pascal ties

W_{p^{n}}^{(2)} := { ∑_{k∈[0,p^{n}−1]} z_{k}ξ_{p^{n},k} | z_{k} ∈ Z[ζ_{p^{n}}], (X^{i}f_{p^{n}}^{j}(X)) [(z_{k})_{k∈[0,p^{n}−1]}] ≡_{t^{j(p−1)}} 0
for all i ∈ [0, p − 1] and all j ∈ [0, p^{n−1} − 1] }
⊆ W_{p^{n}}^{(1)},

where the congruence is to be read in Z_{(p)}[ζ_{p^{n}}]. Lemma 3.26

(i) The image (Z[ζ_{p^{n}}]C_{p^{n}})ω_{p^{n}} of the Wedderburn embedding (3.1) is contained in W_{p^{n}}^{(2)}, Z[ζ_{p^{n}}]C_{p^{n}} ^{\overline{}^{ω_{p^{n}}}} W_{p^{n}}^{(2)} [figure] W_{p^{n}}^{(1)} [figure] W_{p^{n}}^{(0)}.

(ii) The (i + jp + 1)th elementary divisor over Z_{(p)}[ζ_{p^{n}}] of the embedding (W_{p^{n}}^{(2)})_{(p)} ⊆ (W_{p^{n}}^{(1)})_{(p)}, i ∈ [0, p − 1], j ∈ [0, p^{n−1} − 1], is given by t^{j(p−1)}. Moreover, the (i + jp + 1)th elementary divisor of the embedding (W_{p^{n}}^{(2)})_{(p)} ⊆ (W_{p^{n}}^{(0)})_{(p)} is given by t^{(i+jp)+j(p−1)}.

(iii) The valuation of the Z[ζ_{p^{n}}]-linear determinant of the embedding W_{p^{n}}^{(2)} ⊆ W_{p^{n}}^{(1)} is given by p^{n}(p^{n−1} − 1)(p − 1)/2 at t and by zero elsewhere.

Ad (i). We have to take care of the coefficients calculated in (3.23). For i ∈ [0, p − 1], j ∈ [0, p^{n−1} − 1] and h ∈ [0, p^{n} − 1] we obtain

\begin{aligned}(X^{i}f_{p^{n}}^{j}(X)) [((\frac{1−ζ_{p^{n}}^{h}}{t} )^{k})_{k∈[0,p^{n}−1]}] & = & (\frac{1−ζ_{p^{n}}^{h}}{t} )^{i} f_{p^{n}}^{j} (\frac{1−ζ_{p^{n}}^{h}}{t} ) \\ & ≡^{(3.25)}_{t^{j(p−1)}} & 0. \end{aligned}

Question 3.27 We do not know whether W_{p^{n}}^{(2)} is a subring of W_{p^{n}}^{(1)}.

Specializing to n = 2, we obtain the

Proposition 3.28 We have a factorization of the Wedderburn embedding into

Z[ζ_{p^{2}}]C_{p^{2}} \overline{}^{ω_{p^{2}}}_{∼} W_{p^{2}}^{(2)} ⊆ W_{p^{2}}^{(1)} ⊆ W_{p^{2}}^{(0)} = ∏_{j∈Z/p^{2}} Z[ζ_{p^{2}}].

The factorization follows by (3.26 i). The isomorphism follows by comparison of (3.26 iii) and (3.18 iv) with (3.6), both yielding the valuation at t of the determinant of the respective embedding to be p^{3}(p−1), and zero elsewhere. We remark that in this case, the elementary divisors resulting from (3.26 ii) are in accordance with (3.15).

Example 3.29 Consider the case p = 3, n = 2, thus t = 1 − ζ_{9}, γ = 1 + t/2. An element ∑_{k∈[0,8]} z_{k}ξ_{9,k} is contained in W_{9}^{(2)} if and only if, z = (z_{k})_{k∈[0,8]} considered as a row vector, it multiplies with

[ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & −1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & −t/2 & −1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & γ & −t/2 & −1 & t & 1 & 0 \\ 0 & 0 & 0 & 0 & γ & −t/2 & −2γ+t^{2}/4 & t & 1 \\ 0 & 0 & 0 & 0 & 0 & γ & −tγ & −2γ+t^{2}/4 & t \\ 0 & 0 & 0 & 0 & 0 & 0 & γ^{2} & −tγ & −2γ+t^{2}/4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & γ^{2} & −tγ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & γ^{2} \end{matrix} ] [ \begin{matrix}t^{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & t^{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & t^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & t^{−2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & t^{−2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & t^{−2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & t^{−4} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & t^{−4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t^{−4} \end{matrix} ]

to a vector entrywise contained in Z_{(3)}[ζ_{9}].

3.4 Comparison of methods in an example

Example 3.30 Consider the case m = 5, in which the cyclic Wedderburn embedding becomes

[figure: Z[ζ_{5}]C_{5}; ω_{5}; ∏_{j∈[0,4]} Z[ζ_{5}]; c_{5}; (ζ_{5}^{j})_{j∈[0,4]}.]

Let t = 1 − ζ_{5}.

(i) The q-Pascal method (3.8 iii) yields the Z[ζ_{5}]-linear basis of (Z[ζ_{5}]C_{5})ω_{5} given by the rows of

[ \begin{matrix}5 & 0 & 0 & 0 & 0 \\ 4 + 3ζ_{5} + 2ζ_{5}^{2} + ζ_{5}^{3} & −4 − 3ζ_{5} − 2ζ_{5}^{2} − ζ_{5}^{3} & 0 & 0 & 0 \\ 2 + ζ_{5} + 2ζ_{5}^{2} & −1 − ζ_{5} + 2ζ_{5}^{3} & 2ζ_{5} + ζ_{5}^{2} + 2ζ_{5}^{3} & 0 & 0 \\ 2 + ζ_{5} + ζ_{5}^{2} + ζ_{5}^{3} & −2 − ζ_{5}^{2} − 2ζ_{5}^{3} & 3 + 3ζ_{5} + 2ζ_{5}^{2} + 2ζ_{5}^{3} & ζ_{5}^{2} − ζ_{5}^{3} & 0 \\ 1 & −1 + ζ_{5} + ζ_{5}^{3} & −2 − ζ_{5} − 2ζ_{5}^{2} + ζ_{5}^{3} & −3ζ_{5}^{2} − ζ_{5}^{3} & ζ_{5} \end{matrix} ] .

(ii) The general Vandermonde method (1.6) yields the Z[ζ_{5}]-linear basis of (Z[ζ_{5}]C_{5})ω_{5} given by the rows of

[\begin{matrix}1 & 1 & 1 & 1 & 1 \\ 0 & (ζ_{5} − 1) & (ζ_{5}^{2} − 1) & (ζ_{5}^{3} − 1) & (ζ_{5}^{4} − 1) \\ 0 & 0 & (ζ_{5}^{2} − 1)(ζ_{5}^{2} − ζ_{5}) & (ζ_{5}^{3} − 1)(ζ_{5}^{3} − ζ_{5}) & (ζ_{5}^{4} − 1)(ζ_{5}^{4} − ζ_{5}) \\ 0 & 0 & 0 & (ζ_{5}^{3} − 1)(ζ_{5}^{3} − ζ_{5})(ζ_{5}^{3} − ζ_{5}^{2}) & (ζ_{5}^{4} − 1)(ζ_{5}^{4} − ζ_{5})(ζ_{5}^{4} − ζ_{5}^{2}) \\ 0 & 0 & 0 & 0 & (ζ_{5}^{4} − 1)(ζ_{5}^{4} − ζ_{5})(ζ_{5}^{4} − ζ_{5}^{2})(ζ_{5}^{4} − ζ_{5}^{3}) \end{matrix}] .

(iii) The Pascal method (3.18 ii, 3.19) yields the Z[ζ_{5}]-linear basis of (Z[ζ_{5}]C_{5})ω_{5} given by the rows of

[ \begin{matrix}ξ_{5,0} & 1 & 1 & 1 & 1 & 1 \\ ξ_{5,1} & 0 & −t & −2t & −3t & −4t \\ ξ_{5,2} & 0 & 0 & t^{2} & 3t^{2} & 6t^{2} \\ ξ_{5,3} & 0 & 0 & 0 & −t^{3} & −4t^{3} \\ ξ_{5,4} & 0 & 0 & 0 & 0 & t^{4} \end{matrix} ] .

But this method works only for m prime (and for m = p^{2}, but less simply).

4 The radical series of (W_{p^{n}}^{(1)})_{(p)}

Let T be a discrete valuation ring with maximal ideal generated by t and residue field k = T/tT. Let Λ be a subalgebra over T of a direct product of m copies of T such that its embedding into this product has torsion cokernel. The dimension over k of the radical layers 𝔯^{i}Λ/𝔯^{i+1}Λ stabilizes for large i at m [Kü 99, E.2.3]. On the other hand, the surjection

𝔯Λ/𝔯^{2}Λ ⊗_{k} 𝔯^{i}Λ/𝔯^{i+1}Λ \overline{}𝔯^{i+1}Λ/𝔯^{i+2}Λ

induced by multiplication yields the bound

(dim_{k} 𝔯Λ/𝔯^{2}Λ)(dim_{k} 𝔯^{i}Λ/𝔯^{i+1}Λ) ⩾ dim_{k}(𝔯^{i+1}Λ/𝔯^{i+2}Λ).

The question is the behaviour of this sequence of dimensions dim_{k}(𝔯^{i}Λ/𝔯^{i+1}Λ). The ring W_{p^{n}}^{(1)} introduced in (3.17), localized at (p), yields some example material.

Notation 4.1 Let p be a prime, let n ⩾ 1 and write t = 1 − ζ_{p^{n}}. For i ∈ Z, denote \underline{i} := max(0, i). Given an integer i ⩾ 0, we write it as i = ∑_{j⩾0} a_{j}p^{j} with a_{j} ∈ [0, p − 1] and denote its p-adic Quersumme by q_{p}(i) := ∑_{j⩾0} a_{j}.

For a, b, m ⩾ 0, we let

( ∑_{i∈[0,m]} T^{i})^{a} =: ∑_{b⩾0}(\begin{matrix}a \\ b \end{matrix})_{m} T^{b} ∈ Z[T].

That is, (\begin{matrix}a \\ b \end{matrix})_{m} denotes the number of of choices of b balls out of a different balls with each ball chosen at most m times, disregarding the order in which we choose.

In particular, (\begin{matrix}a \\ b \end{matrix})_{1} = (\begin{matrix}a \\ b \end{matrix}). The sequence ((\begin{matrix}a \\ b \end{matrix})_{m})_{m⩾0} becomes stationary for m ⩾ b at the value (\begin{matrix}a+b−1 \\ b \end{matrix}). We have the recursion formula

(\begin{matrix}a+1 \\ b \end{matrix})_{m} = ∑_{j∈[0,m]} (\begin{matrix}a \\ b−j \end{matrix})_{m} ,

where we let (\begin{matrix}a \\ c \end{matrix})_{m} := 0 for c < 0. Furthermore, plugging in T = 1 in the definition yields

∑_{b∈[0,am]} (\begin{matrix}a \\ b \end{matrix})_{m} = (m + 1)^{a}.

Proposition 4.2

(i) The Z_{(p)}[ζ_{p^{n}}]-order (W_{p^{n}}^{(1)})_{(p)} is a local ring, with simple module F_{p} acted upon identically by ξ_{p^{n},0} and trivially by ξ_{p^{n},i} for i ∈ [1, p − 1] (cf. 3.18 ii). I.e.

𝔯(W_{p^{n}}^{(1)})_{(p)} = (W_{p^{n}}^{(1)})_{(p)} ∩ t(W_{p^{n}}^{(0)})_{(p)}

(ii) For i ⩾ 0, the ideal 𝔯^{i}(W_{p^{n}}^{(1)})_{(p)} is Z_{(p)}[ζ_{p^{n}}]-linearly generated by the tuple

(t\underline{i−q_{p}(j)}ξ_{p^{n},j})_{j∈[0,p^{n}−1]} .

In particular, for i ⩾ n(p − 1) we have 𝔯^{i+1}(W_{p^{n}}^{(1)})_{(p)} = t𝔯^{i}(W_{p^{n}}^{(1)})_{(p)} (cf. [Kü 99, E.2.3]).

(iii) On the quantitative side, we obtain for i ⩾ 0

l_{p^{n},i} := dim_{F_{p}} 𝔯^{i}(W_{p^{n}}^{(1)})_{(p)}/𝔯^{i+1}(W_{p^{n}}^{(1)})_{(p)} = ∑_{j∈[0,i]} (\begin{matrix}n \\ j \end{matrix})_{p−1} .

Ad (i). On the one hand, the quotient of the respective Z_{(p)}[ζ_{p^{n}}]-order modulo the ideal claimed to be its radical is a simple module. On the other hand, this ideal is nilpotent modulo t.

Ad (ii). We abbreviate 𝔯^{i} := 𝔯^{i}(W_{p^{n}}^{(1)})_{(p)} for i ⩾ 0 and perform an induction on i, the case of i = 1 being true by (i), and a basis of 𝔯Λ being given by (tξ_{p^{n},0}, ξ_{p^{n},1}, …, ξ_{p^{n},p^{n}−1}). Note that for 0 ⩽ b ⩽ a, the binomial coefficient (\begin{matrix}a \\ b \end{matrix}) has valuation p^{n−1}(q_{p}(a−b)+q_{p}(b)−q_{p}(a)) at t.

We claim that 𝔯 · 𝔯^{i−1} is contained in the span of the given tuple.

The set tξ_{p^{n},0} ·𝔯^{i−1} is contained in this span. Moreover, by (3.21), we obtain in case 1 ⩽ j′, 0 ⩽ j ⩽ j′,

(∗)  ξ_{p^{n},j′}(t\underline{i−1−q_{p}(j)}ξ_{p^{n},j}) = t\underline{i−1−q_{p}(j)} ∑_{s∈[0,j]} (\begin{matrix}j \\ s \end{matrix}) (\begin{matrix}j+j′−s \\ j \end{matrix}) (−1)^{s}t^{s}ξ_{p^{n},j+j′−s}.

So it suffices to show the inequality

\begin{aligned}v_{t}((\begin{matrix}j+j′−s \\ j \end{matrix})) + \underline{i − 1 − q_{p}(j)} + s & ⩾ & \underline{i − q_{p}(j + j′ − s)} \\ & = & \underline{i − q_{p}(j) − q_{p}(j′ − s) + p^{1−n}v_{t}((\begin{matrix}j+j′−s \\ j \end{matrix}))} \end{aligned}

for s ∈ [0, j], whence in turn it suffices to show that

(1 − p^{1−n})v_{t}((\begin{matrix}j+j′−s \\ j \end{matrix})) − 1 + s + q_{p}(j′ − s) ⩾ 0.

But we have s ⩾ 1 or (s = 0 and q_{p}(j′) ⩾ 1).

Furthermore, (3.21) yields

(∗∗)  ξ_{p^{n},j′}(t\underline{i−1−q_{p}(j)}ξ_{p^{n},j}) = t\underline{i−1−q_{p}(j)} ∑_{s∈[0,j′]} (\begin{matrix}j′ \\ s \end{matrix}) (\begin{matrix}j+j′−s \\ j′ \end{matrix}) (−1)^{s}t^{s}ξ_{p^{n},j+j′−s}.

in case 1 ⩽ j′ ⩽ j. So it suffices to show that

v_{t}((\begin{matrix}j′ \\ s \end{matrix}) (\begin{matrix}j+j′−s \\ j′ \end{matrix})) + \underline{i − 1 − q_{p}(j)} + s ⩾ \underline{i − q_{p}(j + j′ − s)}

for s ∈ [0, j′], whence in turn, dropping a factor p^{n−1}, it suffices to see that

((q_{p}(j′−s)+q_{p}(s)−q_{p}(j′))+(q_{p}(j−s)+q_{p}(j′)−q_{p}(j+j′−s)))−q_{p}(j)+q_{p}(j+j′−s)+s−1 ⩾ 0.

For s ⩾ 1, we see that both in case (∗) and in case (∗∗), the corresponding summand is even contained in t𝔯^{i−1}.

We claim that 𝔯 · 𝔯^{i−1} contains the given tuple.

First, we note that t\underline{i−q_{p}(0)}ξ_{p^{n},0} = (tξ_{p^{n},0})^{i}. Moreover, by the summandwise argument above concerning (∗∗), we dispose of the congruence

ξ_{p^{n},j′}(t\underline{i−1−q_{p}(j)}ξ_{p^{n},j}) ≡_{t𝔯^{i−1}} t\underline{i−1−q_{p}(j)} (\begin{matrix}j+j′ \\ j′ \end{matrix}) ξ_{p^{n},j+j′}.

in case 1 ⩽ j′ ⩽ j. In particular, given j ∈ [1, p^{n} − 1] such that j[p] ⩽ j − j[p], i.e. such that j is not a power of p, we obtain

ξ_{p^{n},j[p]}(t\underline{i−1−q_{p}(j−j[p])}ξ_{p^{n},j−j[p]}) ≡_{t𝔯^{i−1}} t\underline{i−q_{p}(j)} (\begin{matrix}j \\ j[p] \end{matrix}) ξ_{p^{n},j} ,

and it remains to note that v_{p}((\begin{matrix}j \\ j[p] \end{matrix})) = 0.

Furthermore, for m ∈ [0, n − 1] we have

ξ_{p^{n},p^{m}}(t\underline{i−1−q_{p}(0)}ξ_{p^{n},0}) = t\underline{i−q_{p}(p^{m})}ξ_{p^{n},p^{m}}.

Ad (iii). Concerning the dimension of the ith radical layer, we calculate

\begin{aligned}#\{k ∈ [0, p^{n} − 1] | q_{p}(k) = j\} & = & #\{(a_{l})_{l∈[0,n−1]} ∈ [0, p − 1]^{n} | ∑_{l∈[0,n−1]} a_{l} = j\} \\ & = & (\begin{matrix}n \\ j \end{matrix})_{p−1} . \end{aligned}

Remark 4.3 The inequality mentioned in the introduction to this section reads l_{p^{n},1}·l_{p^{n},i} ⩾ l_{p^{n},i+1} for i ⩾ 1. By (4.2 iii), this translates into the assertion that

(n + 1) ( ∑_{j∈[0,i]} (\begin{matrix}n \\ j \end{matrix})_{p−1}) ⩾ ∑_{j∈[0,i+1]} (\begin{matrix}n \\ j \end{matrix})_{p−1} .

For p large, this follows from

(n + 1) (∑_{j∈[0,i]} (\begin{matrix}n \\ j \end{matrix})_{p−1}) = (n + 1) (∑_{j∈[0,i]} (\begin{matrix}n+j−1 \\ n−1 \end{matrix}))
= (n + 1) (\begin{matrix}n+i \\ n \end{matrix})
⩾ \frac{n+i+1}{i+1} · \frac{(n+i)!}{n!i!}
= (\begin{matrix}n+i+1 \\ n \end{matrix})
= ∑_{j∈[0,i+1]} (\begin{matrix}n \\ j \end{matrix})_{p−1} .

Example 4.4 The sequence (l_{81,i})_{i⩾0} is given by

1, 5, 15, 31, 50, 66, 76, 80, 81, 81, …

5 The absolute cyclic Wedderburn embedding

Once given the Kervaire-Murthy pullback, a closed formula describing the image of the absolute Wedderburn embedding can be derived in a straightforward manner (5.14, 5.18).

5.1 An inversion formula for QC_{p^{n}}

Let p be a prime and let n ⩾ 1.

Lemma 5.1 The inverse of the absolute Wedderburn isomorphism

[figure: QC_{p^{n}}; ω_{Q,p^{n}}; ∼; ∏_{k∈[0,n]}Q[ζ_{p^{k}}]; c_{p^{n}}; (ζ_{p^{k}})_{k∈[0,n]}]

is given by

∏_{k∈[0,n]}Q[ζ_{p^{k}}] \overline{}^{ω_{Q,p^{n}}^{−1}}_{∼} QC_{p^{n}}

(∑_{i∈Z}y_{k,i}ζ_{p^{k}}^{i})_{k∈[0,n]}  \overline{}p^{−n}∑_{i∈Z} ( ∑_{k∈[0,n]}p^{k}y_{k,i} ∑_{j∈[1,p^{n−k}]}c_{p^{n}}^{i+jp^{k}} − ∑_{k∈[1,n]}p^{k−1}y_{k,i} ∑_{j∈[1,p^{n−k+1}]}c_{p^{n}}^{i+jp^{k−1}} )
= p^{−n}∑_{i∈Z}c_{p^{n}}^{i} ∑_{k∈[0,n]}p^{k} ∑_{j∈[1,p^{n−k}]} (y_{k,i−jp^{k}} − y_{k+1,i−jp^{k}}) ,

where y_{k,i} ∈ Q and where y_{n+1,i} := 0 for i ∈ Z. We allow ‘non-reduced’ expressions for elements in Q(ζ_{p^{k}}), merely requiring y_{k,i} = 0 for all but finitely many i ∈ Z.

We denote the absolute Wedderburn embedding, i.e. the restriction of ω_{Q,p^{n}} to ZC_{p^{n}}, by

ZC_{p^{n}}^{\overline{}^{ω_{Z,p^{n}}}} ∏_{k∈[0,n]}Z[ζ_{p^{k}}].

By (3.6), the inversion formula over Q(ζ_{p^{n}}) reads

[figure: Q(ζ_{p^{n}})C_{p^{n}}; ∼; ∏_{i∈[1,p^{n}]}Q(ζ_{p^{n}}); c_{p^{n}}; (ζ_{p^{n}}^{i})_{i∈[1,p^{n}]}; p^{−n} ∑_{j∈[1,p^{n}]}c_{p^{n}}^{j} ∑_{i∈[1,p^{n}]}y_{i}ζ_{p^{n}}^{−ij}; (y_{i})_{i}.]

We precompose the inverse direction with the direct product of the maps

[figure: Q(ζ_{p^{k}}); ∏_{σ∈Gal(Q(ζ_{p^{k}})/Q)}Q(ζ_{p^{n}}); x; (xσ)_{σ}]

over k ∈ [0, n], where the position of the Galois automorphism corresponding to i ∈ (Z/p^{k})^{∗}, represented by i ∈ [1, p^{k}], is to be identified with the position ip^{n−k} ∈ [1, p^{n}].

If l ∈ [1, n], the image of (∂_{l,k})_{k∈[0,n]} ∈ ∏_{k∈[0,n]} Z[ζ_{p^{k}}] under this composition is

p^{−n} ∑_{j∈[1,p^{n}]}c_{p^{n}}^{j} ∑_{i∈[1,p^{n}], v_{p}(i) = n−l}ζ_{p^{n}}^{−ij}

= p^{−n} ∑_{j∈[1,p^{n}]}c_{p^{n}}^{j} ∑_{i∈(Z/p^{l})^{∗}}ζ_{p^{n}}^{−ijp^{n−l}}

= p^{−n} ( ∑_{j∈[1,p^{n}], v_{p}(j)⩾l}c_{p^{n}}^{j}(p − 1)p^{l−1} + ∑_{j∈[1,p^{n}], v_{p}(j) = l−1}c_{p^{n}}^{j}(−1)p^{l−1})

= p^{−n} (p^{l} ( ∑_{j∈[1,p^{n−l}]}c_{p^{n}}^{jp^{l}} ) − p^{l−1} ( ∑_{j∈[1,p^{n−l+1}]}c_{p^{n}}^{jp^{l−1}} )) .

If l = 0, it is p^{−n} ∑_{j∈[1,p^{n}]}c_{p^{n}}^{j}. The inversion formula follows by QC_{p^{n}}-linearity of ω_{Q,p^{n}}.

5.2 An inversion formula for QC_{m}

Let m ⩾ 1. Writing d|m stands for d ⩾ 1 and m ∈ (d). The letters p and q are reserved to denote prime numbers. We denote by k[p′] := k/k[p] the p′-part of an integer k ⩾ 1. Given a finite family of groups (G_{i})_{i∈[1,l]}, l ⩾ 1, there is a ring isomorphism

[figure: Z[ ∏_{i∈[1,l]}G_{i}]; ∼; ⨂_{i∈[1,l]}ZG_{i}; (g_{i})_{i}; (g_{i})_{i}^{⊗} := g_{1} ⊗ ⋯ ⊗ g_{l}.]

For p|k we denote by s_{k,p} a representative in Z of the inverse of k[p′] in (Z/k[p])^{∗} to obtain

[figure: C_{k}; ∼; ∏_{p|k}C_{k[p]}; c_{k}; (c_{k[p]})_{p|k}; c_{k}^{∑_{p|k} j_{p}s_{k,p}k[p′]}; (c_{k[p]}^{j_{p}})_{p|k},]

and

[figure: Z[ζ_{k}]; ∼; ⨂_{p|k}Z[ζ_{k[p]}]; ζ_{k}^{∑_{p|k} s_{k,p}j_{p}k[p′]}; (ζ_{k[p]}^{j_{p}})_{p|k}^{⊗}; ζ_{k}; (ζ_{k[p]})_{p|k}^{⊗}.]

We use these isomorphisms as identifications.

We consider the absolute Wedderburn isomorphism

[figure: QC_{m}; ω_{Q,m}; ∼; ∏_{d|m}Q(ζ_{d}); c_{m}; (ζ_{d})_{d|m}]

and its restriction, the absolute Wedderburn embedding

ZC_{m}^{\overline{}^{ω_{Z,m}}} ∏_{d|m}Z[ζ_{d}].

Remark 5.2 The embedding ω_{Z,m} identifies with ⨂_{p|m} ω_{Z,m[p]} in the sense that the identifications yield an isomorphism of embeddings

[figure: ZC_{m}; ω_{Z,m}; ∏_{d|m}Z[ζ_{d}]; ⨂_{p|m} ZC_{m[p]}; ⨂_{p|m} ω_{Z,m[p]}; ⨂_{p|m} ∏_{e|m[p]}Z[ζ_{e}].]

On both ways, c_{m} is mappped to (ζ_{d})_{d|m}.

We factorize ω_{Q,m}^{−1} along identifications u and w into

∏_{d|m}Q(ζ_{d}) \overline{}^{u}_{∼} ⨂_{p|m} ( ∏_{d|m[p]}Q[ζ_{d}]) \overline{}^{v}_{∼} ⨂_{p|m} QC_{m[p]} \overline{}^{w}_{∼} QC_{m},

v being the tensor product of the inverses of the absolute Wedderburn isomorphism belonging to the respective prime part (cf. 5.2).

For e|m and a prime p|m, we abbreviate the element

QC_{m} ∋ f_{e,p}(c_{m}) := \{ \begin{matrix}\frac{1}{m[p]} ∑_{i∈Z/m[p]}c_{m}^{i} & for v_{p}(e) = 0 \\ \frac{e[p]}{m[p]} ( ∑_{i∈Z/( \frac{m[p]}{e[p]} )}c_{m}^{ie[p]} − \frac{1}{p} ∑_{i∈Z/(p \frac{m[p]}{e[p]} )}c_{m}^{ie[p]/p}) & for v_{p}(e) ⩾ 1. \end{matrix}

Suppose given an element a_{d}(c_{m}) = ∑_{i∈Z/m}a_{d,i}c_{m}^{i} ∈ QC_{m} for each d|m, representing a_{d}(ζ_{d}) =

∑_{i∈Z/m} a_{d,i}ζ_{d}^{i}. Using QC_{m}-linearity, we obtain

(a_{d}(ζ_{d}))_{d|m}uvw = ∑_{e|m} a_{e}(c_{m}) · (∂_{e,d})_{d|m}uvw
= ∑_{e|m} a_{e}(c_{m}) · ((∂_{e[p],d})_{d|m[p]})_{p|m}^{⊗} vw
=^{(5.1)} ∑_{e|m} a_{e}(c_{m}) · (f_{e,p}(c_{m[p]}))_{p|m}^{⊗} w
= ∑_{e|m} a_{e}(c_{m}) ∏_{p|m} f_{e,p}(c_{m}^{s_{m,p}m[p′]}).

We write

f_{e,p}(c_{m}) =: ∑_{j∈Z/m[p]}f_{e,p,j}c_{m}^{j}

and continue to calculate

∑_{e|m} a_{e}(c_{m}) ∏_{p|m} f_{e,p}(c_{m}^{s_{m,p}m[p′]}) = ∑_{e|m} ∑_{i∈Z/m} a_{e,i}c_{m}^{i} ∏_{p|m} ∑_{j∈Z/m[p]} f_{e,p,j}c_{m}^{js_{m,p}m[p′]}
= ∑_{e|m} ∑_{i∈Z/m} a_{e,i}c_{m}^{i} ∑_{k∈Z/m} c_{m}^{k} ∏_{p|m} f_{e,p,k}
= ∑_{l∈Z/m} c_{m}^{l} [∑_{e|m} ∑_{k∈Z/m} a_{e,l−k} ∏_{p|m} f_{e,p,k}]

If A is a condition which might be true or not, we let the expression {if A} take the value 1 if A holds, and 0 if A does not hold. Given d|m, we let d′|m be defined by

v_{p}(d′) := max(v_{p}(d) − 1, 0)

for p|m. We rewrite

f_{e,p,k} = \frac{e[p]}{m[p]} [\{if e[p]|k\} − \frac{1}{p}\{if p|e[p]\}\{if e[p]|pk\}]

to obtain

∏_{p|m} f_{e,p,k} = \frac{e}{m} ∏_{p|m} [\{if e[p]|k\} − \frac{1}{p}\{if p|e[p]\}\{if e[p]|pk\}]
= \{if e′|k\} · \frac{e}{m} [ ∏_{p|e, e[p]=pk[p]} (−\frac{1}{p})] [ ∏_{p|e, e[p]|k[p]} (1 − \frac{1}{p})] .

Proposition 5.3 (inversion formula) The inverse of the absolute Wedderburn isomorphism

[figure: QC_{m}; ω_{Q,m}; ∼; ∏_{d|m}Q(ζ_{d}); c_{m}; (ζ_{d})_{d|m}]

maps (a_{d}(ζ_{d}))_{d|m}, written using representing elements a_{d}(c_{m}) = ∑_{i∈Z/m}a_{d,i}c_{m}^{i} ∈ QC_{m}, to

\frac{1}{m} ∑_{l∈Z/m} c_{m}^{l} [∑_{d|m} d ∑_{k∈Z/( \frac{m}{d′})} a_{d,l−kd′} [ ∏_{p|d, p|̸ k} (−\frac{1}{p})] [ ∏_{p|d, p|k} (1 − \frac{1}{p})]] ,

where v_{p}(d′) := max(v_{p}(d) − 1, 0).

But these inversion formulas (5.1, 5.3) are not suited for giving convenient ties that describe the image of the absolute Wedderburn embedding, for in neither case they yield a triangular system of ties, i.e. of congruences of tuple entries.

5.3 The Kervaire-Murthy pullback

Let p be a prime and let n ⩾ 1. Lemma 5.4 Let k ∈ [1, n]. Writing

\begin{aligned}f_{k}(X) & := & − ∑_{i∈[0,p−2]}(p − 1 − i)X^{ip^{k−1}} \\ g_{k}(X) & := & ∑_{i∈[0,p^{n−k}−1]} (−(p^{n−k+1} − p(i + 1))X^{ip^{k}+p^{k−1}} + (p^{n−k+1} − p(i + 1) + 1)X^{ip^{k}}) , \end{aligned}

we obtain

f_{k}(X) · ( ∏_{i∈[0,n]\{k}}Φ_{p^{i}}(X)) + g_{k}(X) · Φ_{p^{k}}(X) = p^{n−k+1}.

We expand

f_{k}(X) · ( ∏_{i∈[0,n]\{k}}Φ_{p^{i}}(X))

= − (∑_{i∈[0,p−2]}(p − 1 − i)X^{ip^{k−1}}) ((X^{p^{k−1}} − 1) ∑_{i∈[0,p^{n−k}−1]} X^{ip^{k}})
= − (∑_{i∈[1,p−1]}(p − i)X^{ip^{k−1}} − ∑_{i∈[0,p−2]}(p − 1 − i)X^{ip^{k−1}}) (∑_{i∈[0,p^{n−k}−1]} X^{ip^{k}})
= − (−p + ∑_{i∈[0,p−1]} X^{ip^{k−1}}) (∑_{i∈[0,p^{n−k}−1]} X^{ip^{k}})
= p (∑_{i∈[0,p^{n−k}−1]} X^{ip^{k}}) − (∑_{i∈[0,p^{n−k+1}−1]} X^{ip^{k−1}}) .

Then we calculate

g_{k}(X) · Φ_{p^{k}}(X)

= (∑_{i∈[0,p^{n−k}−1]} (−(p^{n−k+1} − p(i + 1))X^{ip^{k}+p^{k−1}} + (p^{n−k+1} − p(i + 1) + 1)X^{ip^{k}})) ·
· (∑_{j∈[0,p−1]} X^{jp^{k−1}})
= − ∑_{i∈[0,p^{n−k}−1]} ∑_{j∈[1,p]}(p^{n−k+1} − p(i + 1))X^{ip^{k}+jp^{k−1}}
+ ∑_{i∈[0,p^{n−k}−1]} ∑_{j∈[0,p−1]}(p^{n−k+1} − p(i + 1) + 1)X^{ip^{k}+jp^{k−1}}
= − ∑_{i∈[0,p^{n−k}−1]}(p^{n−k+1} − p(i + 1))X^{(i+1)p^{k}} + ∑_{i∈[0,p^{n−k}−1]} ∑_{j∈[1,p−1]} X^{ip^{k}+jp^{k−1}}
+ ∑_{i∈[0,p^{n−k}−1]}(p^{n−k+1} − p(i + 1) + 1)X^{ip^{k}}
= − ∑_{i∈[1,p^{n−k}−1]}(p^{n−k+1} − pi)X^{ip^{k}} + ∑_{i∈[0,p^{n−k+1}−1]} X^{ip^{k−1}}
+ ∑_{i∈[0,p^{n−k}−1]}(p^{n−k+1} − p(i + 1))X^{ip^{k}}
= −p (∑_{i∈[1,p^{n−k}−1]} X^{ip^{k}}) + (∑_{i∈[0,p^{n−k+1}−1]} X^{ip^{k−1}}) + (p^{n−k+1} − p).

Lemma 5.5 Writing

g_{0}(X) := − ∑_{i∈[0,p^{n}−2]}(p^{n} − 1 − i)X^{i},

we obtain

( ∏_{i∈[0,n]\{0}}Φ_{p^{i}}(X)) + g_{0}(X) · Φ_{p^{0}}(X) = p^{n}.

We expand

g_{0}(X) · Φ_{p^{0}}(X) = − (∑_{i∈[0,p^{n}−2]}(p^{n} − 1 − i)X^{i}) (X − 1)
= − (∑_{i∈[1,p^{n}−1]}(p^{n} − i)X^{i}) + (∑_{i∈[0,p^{n}−1]}(p^{n} − 1 − i)X^{i})
= − (∑_{i∈[0,p^{n}−1]} X^{i}) + p^{n}.

Proposition 5.6 (Kervaire, Murthy [KM 77, §1]) Let k ⩾ 1. There is a pullback diagram of rings

[figure: ZC_{p^{k}}; ZC_{p^{k−1}}; α_{k}; Z[ζ_{p^{k}}]; F_{p}C_{p^{k−1}}; c_{p^{k}}; c_{p^{k−1}}; ζ_{p^{k}}; c_{p^{k−1}}.]

Given a commutative ring A containing ideals 𝔞 and 𝔟, there is a pullback diagram

[figure: A/𝔞 ∩ 𝔟; A/𝔟; A/𝔞; A/𝔞 + 𝔟.]

Hence, letting A = Z[X], 𝔞 = (Φ_{p^{k}}(X)), 𝔟 = (X^{p^{k−1}} −1), we need to show that (Φ_{p^{k}}(X))∩ (X^{p^{k−1}} − 1) = (X^{p^{k}} − 1) and that (Φ_{p^{k}}(X), X^{p^{k−1}} − 1) = (p, X^{p^{k−1}} − 1). The intersection follows from the analoguous assertion in Q[X], using Z[X]f(X) = Z[X] ∩ Q[X]f(X) for a monic polynomial f(X) ∈ Z[X]. The sum follows by

(Φ_{p^{k}}(X), X^{p^{k−1}} − 1) =^{(5.4)} (p, Φ_{p^{k}}(X), X^{p^{k−1}} − 1) = (p, (X − 1)^{p^{k}−p^{k−1}}, (X − 1)^{p^{k−1}}).

Corollary 5.7 The index of the absolute Wedderburn embedding ω_{Z,p^{n}} is given by

# ∏_{k∈[1,n]}F_{p}C_{p^{k−1}} = p^{\frac{p^{n}−1}{p−1}} .

Corollary 5.8 For m ⩾ 1, the index of the absolute Wedderburn embedding ω_{Z,m} is given by

∏_{p|m} p^{\frac{m[p]−1}{p−1} ·m[p′]}.

This follows from (5.7) using (5.2).

Remark 5.9 For k ⩾ 1, we let Δ_{k} denote the absolute value of the discriminant of Z[ζ_{k}] over Z. We take from [Kü 99, S 1.1.2] that the index of ω_{Z,m} is given by

\sqrt{\frac{m^{m}}{∏_{d|m} Δ_{d}}} .

A comparison with (5.8) allows to re-calculate Δ_{m}. First we remark that the inverse of the Dedekind isomorphism

[figure: Q(ζ_{m}); ⊗; Q(ζ_{m}); δ_{m}; ∼; ∏_{j∈(Z/m)^{∗}} Q(ζ_{m}); ζ_{m}^{k}; ⊗; ζ_{m}^{l}; (ζ_{m}^{k}ζ_{m}^{jl})_{j∈(Z/m)^{∗}},]

where k, l ∈ [0, φ(m) − 1], is given by

[figure: ∏_{j∈(Z/m)^{∗}} Q(ζ_{m}); δ_{m}^{−1}; Q(ζ_{m}) ⊗ Q(ζ_{m}); (y_{j})_{j∈(Z/m)^{∗}}; m^{−1} ∑_{i∈Z/m} ∑_{j∈(Z/m)^{∗}}y_{j}ζ_{m}^{−ij} ⊗ ζ_{m}^{i}.]

In particular, its restriction to the Dedekind embedding

Z[ζ_{m}] ⊗ Z[ζ_{m}] [figure: δ_{m}] ∏_{j∈(Z/m)^{∗}} Z[ζ_{m}]

has a cokernel annihilated by m. In particular, the prime divisors of Δ_{m} divide m. We infer that for p|m

∑_{d|m, v_{p}(d)⩾1}v_{p}(Δ_{d}) = ∑_{d|m}v_{p}(Δ_{d}) =^{(5.8)} m[p′] (v_{p}(m)m[p] − 2 · \frac{m[p] − 1}{p − 1}) .

Given d ⩾ 1, p|d, we denote

u(p, d) := \frac{d[p]}{p} (v_{p}(d)(p − 1) − 1) · φ(d[p′])

and obtain

∑_{d|m, v_{p}(d)⩾1}u(p, d) = ∑_{d|m, v_{p}(d)⩾1} \frac{d[p]}{p} (v_{p}(d)(p − 1) − 1) · φ(d[p′])
= ∑_{i∈[1,v_{p}(m)]}p^{i−1} (i(p − 1) − 1) · ∑_{d|m[p′]}φ(d)
= (v_{p}(m)m[p] − 2\frac{m[p]−1}{p−1}) · m[p′] ,

whence by induction on m, we get v_{p}(Δ_{m}) = u(p, m), i.e. the absolute value of the discriminant is

Δ_{m} = ∏_{p|m} p^{\frac{m[p]}{p} (v_{p}(m)(p−1)−1)φ(m[p′])} .

5.4 Kervaire-Murthy ties for ZC_{p^{n}}

Let p be a prime and let n ⩾ 0. Given j ∈ Z and k ⩾ 1, we let [j]_{k} ∈ [0, k − 1] be defined by [j]_{k} ≡_{k} j. For a, b, c ∈ Z and a ⩽ b, we let χ_{[a,b]}(c) := 1 if c ∈ [a, b] and χ_{[a,b]}(c) := 0 if c ∉ [a, b].

Let 𝔩^{f}(Z) be the abelian group consisting of sequences x = (x_{j})_{j∈Z} with entries x_{j} ∈ Z such that the support

\underline{x} := \{i ∈ Z | x_{i} ≠ 0\} ⊆ Z

of positions carrying nonzero entries is finite. For i ⩾ 0, m ⩾ 0 and s ⩾ 0 we shall define a Z-linear operator T_{i}^{m,s} : 𝔩^{f}(Z) \overline{}𝔩^{f}(Z) (which we write on the left).

Suppose given j ∈ Z. If i ⩾ m, we let

(T_{i}^{m,s}x)_{j} := χ_{[0,p^{i}−1]}(j) · ∑_{k∈Z}x_{[j]_{p^{i−m}}−p^{i−m}+kp^{i+s}}.

If i < m, we let

(T_{i}^{m,s}x)_{j} := 0.

We note that

(T_{i}^{0,0}x)_{j} := χ_{[0,p^{i}−1]}(j) · ∑_{k∈Z}x_{j+kp^{i}}.

Lemma 5.10

(i) Given a, b, c, d, s, t ⩾ 0 such that b − a ⩽ d − c ⩽ b + s ⩽ d. Then T_{b}^{a,s} ∘ T_{d}^{c,t} = p^{d−b−s}T_{b}^{a,d−b+t}.

(ii) Given a, b, d, s ⩾ 0 such that b + s ⩽ d. Then

T_{b}^{a,s} ∘ T_{d}^{0,0} = T_{b}^{a,s}.

Ad (i). We may assume 0 ⩽ b − a. Given j ∈ Z, we obtain

(T_{b}^{a,s}T_{d}^{c,t}x)_{j} = (T_{b}^{a,s} (χ_{[0,p^{d}−1]}(j′)∑_{k∈Z}x_{[j′]_{p^{d−c}}−p^{d−c}+kp^{d+t}})_{j′∈Z})_{j}
= χ_{[0,p^{b}−1]}(j) · ∑_{k′∈Z}χ_{[0,p^{d}−1]}([j]_{p^{b−a}} − p^{b−a} + k′p^{b+s})·

·∑_{k∈Z}x_{[[j]_{p^{b−a}}−p^{b−a}+k′p^{b+s}]_{p^{d−c}}−p^{d−c}+kp^{d+t}}

= χ_{[0,p^{b}−1]}(j)p^{d−b−s}∑_{k∈Z}x_{[j]_{p^{b−a}}−p^{b−a}+kp^{d+t}}
= (p^{d−b−s}T_{b}^{a,d−b+t}x)_{j} .

Ad (ii). The operator T_{b}^{a,s} is invariant under sequence shifts by p^{b+s}, hence under sequence shifts by p^{d}. Lemma 5.11 Given m ⩾ 1, l ∈ [1, m] and a ⩾ 0, we obtain

∑_{i∈[0,l−1]}p^{l−1−i}T_{m−l}^{a,l−1−i} ∘ (T_{m−i}^{0,0} − T_{m−i}^{1,0}) = T_{m−l}^{a,0} − p^{l}T_{m−l}^{a,l}.

In fact,

∑_{i∈[0,l−1]}p^{l−1−i}T_{m−l}^{a,l−1−i} ∘ (T_{m−i}^{0,0} − T_{m−i}^{1,0}) =^{(5.10 i, ii)} ∑_{i∈[0,l−1]}p^{l−1−i}T_{m−l}^{a,l−1−i} − ∑_{i∈[0,l−1]}p^{l−i}T_{m−l}^{a,l−i}
= ∑_{i∈[1,l]}p^{l−i}T_{m−l}^{a,l−i} − ∑_{i∈[0,l−1]}p^{l−i}T_{m−l}^{a,l−i}
= T_{m−l}^{a,0} − p^{l}T_{m−l}^{a,l}.

Lemma 5.12 For x = (x_{j})_{j∈Z} ∈ 𝔩^{f}(Z) and l ⩾ 0, we denote

x ∗ ζ_{p^{l}} := ∑_{j∈Z} x_{j}ζ_{p^{l}}^{j}

and obtain

x ∗ ζ_{p^{l}} = ((T_{l}^{0,0} − T_{l}^{1,0})x) ∗ ζ_{p^{l}}.

We claim that (T_{l}^{1,0}x) ∗ ζ_{p^{l}} = 0. We may suppose l ⩾ 1 to calculate

(T_{l}^{1,0}x) ∗ ζ_{p^{l}} = ∑_{j∈Z} χ_{[0,p^{l}−1]}(j) ∑_{k∈Z} x_{[j]_{p^{l−1}}−p^{l−1}+kp^{l}}ζ_{p^{l}}^{j}
= ∑_{k∈Z} ∑_{h∈[0,p^{l−1}−1]} ∑_{i∈[0,p−1]} x_{[h+ip^{l−1}]_{p^{l−1}}−p^{l−1}+kp^{l}}ζ_{p^{l}}^{h+ip^{l−1}}
= ∑_{k∈Z} ∑_{h∈[0,p^{l−1}−1]} x_{h−p^{l−1}+kp^{l}}ζ_{p^{l}}^{h} ∑_{i∈[0,p−1]} ζ_{p^{l}}^{ip^{l−1}}
= 0.

Lemma 5.13 For l ⩾ 1, s ⩾ 0 and j ∈ Z\[0, φ(p^{l}) − 1], we have ((T_{l}^{0,s} − T_{l}^{1,s})x)_{j} = 0. In fact,

(T_{l}^{1,s}x)_{(p−1)p^{l−1}+j} = ∑_{k∈Z} x_{[(p−1)p^{l−1}+j]_{p^{l−1}}−p^{l−1}+kp^{l+s}}
= ∑_{k∈Z} x_{j−p^{l−1}+kp^{l+s}}
= (T_{l}^{0,s}x)_{(p−1)p^{l−1}+j}

for j ∈ [0, p^{l−1} − 1].

Theorem 5.14 The image of the absolute Wedderburn embedding is given by

(ZC_{p^{n}})ω_{Z,p^{n}} = { ( ∑_{j∈[0,φ(p^{i})−1]}x_{i,j}ζ_{p^{i}}^{j})_{i∈[0,n]} , x_{i,j} ∈ Z |

for l ∈ [1, n] and j ∈ [0, φ(p^{n−l}) − 1] we have x_{n−l,j} ≡_{p^{l}} ∑_{i∈[0,l−1]}p^{l−1−i}·

· ∑_{k∈[1,p−1]} (x_{n−i,j−p^{n−l}+kp^{n−1−i}} − (1 − ∂_{l,n})x_{n−i,[j]_{p^{n−l−1}}−p^{n−l−1}+kp^{n−1−i}}) }

= \{ (x_{i} ∗ ζ_{p^{i}})_{i∈[0,n]} , x_{i} ∈ 𝔩^{f}(Z), \underline{x_{i}} ⊆ [0, φ(p^{i}) − 1] |

for l ∈ [1, n] we have x_{n−l} ≡_{p^{l}} ∑_{i∈[0,l−1]}p^{l−1−i} (T_{n−l}^{0,l−1−i} − T_{n−l}^{1,l−1−i}) x_{n−i} }

⊆ ∏_{i∈[0,n]}Z[ζ_{p^{i}}].

This system of ties is of triangular shape. In particular, the elementary divisors of ω_{Z,p^{n}} over Z are given by p^{i} with multiplicity φ(p^{n−i}) for i ∈ [0, n]. The second description being true if n = 0, we perform an induction on n. We see by (5.6) that (x_{i} ∗ ζ_{p^{i}})_{i∈[0,n]} ∈ ∏_{i∈[0,n]}Z[ζ_{p^{i}}] is contained in (ZC_{p^{n}})ω_{Z,p^{n}} if and only if the conditions

(a) and (b) below hold. To formulate (b), we shall make use of (5.12).

(a) The tuple (x_{i} ∗ ζ_{p^{i}})_{i∈[0,n−1]} is contained in (ZC_{p^{n−1}})ω_{Z,p^{n−1}}.

(b) We have (((T_{i}^{0,0} − T_{i}^{1,0})x_{n}) ∗ ζ_{p^{i}})_{i∈[0,n−1]} − (x_{i} ∗ ζ_{p^{i}})_{i∈[0,n−1]} ∈ p(ZC_{p^{n−1}})ω_{Z,p^{n−1}}.

Condition (b) in turn holds if and only if the conditions (ba) and (bb) below hold. To formulate (ba), we use (5.13). To formulate (bb), we shall use that by induction, the description is valid in case n − 1.

(ba) We have (T_{i}^{0,0} − T_{i}^{1,0})x_{n} ≡_{p} x_{i} for i ∈ [0, n − 1].

(bb) We have

− (T_{(n−1)−l}^{0,0} − T_{(n−1)−l}^{1,0}) x_{n}

+ ∑_{i∈[0,l−1]}p^{l−1−i} (T_{(n−1)−l}^{0,l−1−i} − T_{(n−1)−l}^{1,l−1−i}) (T_{(n−1)−i}^{0,0} − T_{(n−1)−i}^{1,0}) x_{n}

≡_{p^{l+1}} −x_{(n−1)−l} + ∑_{i∈[0,l−1]}p^{l−1−i} (T_{(n−1)−l}^{0,l−1−i} − T_{(n−1)−l}^{1,l−1−i}) x_{(n−1)−i}
for l ∈ [1, n − 1].

By (5.11), we may equivalently reformulate to

(bb) We have

\begin{aligned}0 ≡_{p^{l+1}} & −x_{(n−1)−l} + p^{l} (T_{(n−1)−l}^{0,l} − T_{(n−1)−l}^{1,l}) x_{n} \\ & + ∑_{i∈[0,l−1]}p^{l−1−i} (T_{(n−1)−l}^{0,l−1−i} − T_{(n−1)−l}^{1,l−1−i}) x_{(n−1)−i} \end{aligned}

for l ∈ [1, n − 1].

Shifting the indices i and l by one, we may in turn equivalently reformulate this assertion to

(bb) We have

x_{n−l} ≡_{p^{l}} ∑_{i∈[0,l−1]}p^{l−1−i} (T_{n−l}^{0,l−1−i} − T_{n−l}^{1,l−1−i}) x_{n−i}

for l ∈ [2, n].

Adjoining condition (ba) in case i = n − 1, we obtain the necessity of the claimed set of ties.

Conversely, we may use (5.11) to see that

(T_{i−1}^{0,0} − T_{i−1}^{1,0}) ∘ (T_{i}^{0,0} − T_{i}^{1,0}) ≡_{p} (T_{i−1}^{0,0} − T_{i−1}^{1,0})

for i ⩾ 1, whence the reduction of (bb) to the modulus p suffices to conclude that (ba) holds in all its cases. Moreover, condition (a) follows by induction assumption and by reading (bb) modulo p^{l−1} for l ∈ [2, n], thus discarding the summand for i = 0. Thus our claimed set of ties is also sufficient.

To see that the second description agrees with the first one, it remains to show that

((T_{n−l}^{0,l−1−i} − T_{n−l}^{1,l−1−i}) x_{n−i})_{j} = 0

for l ∈ [1, n − 1], i ∈ [0, l − 1] and j ∈ Z\[0, φ(p^{n−l}) − 1]. But this follows from (5.13).

Remark 5.15 It might be worthwhile to try to give an ad hoc proof of (5.14) by verification of the ties on the image of the canonical group basis of ZC_{p^{n}} and by a comparison of indices – which we have not attempted to do. We preferred to proceed in the straightforward manner as above since, in this way, the role of the Kervaire-Murthy pullback (5.6) remains visible.

Example 5.16 Let p = 3. We obtain

ZC_{3}  \overline{}^{ω_{Z,3}}_{∼} \{ (∑_{k∈[0,1]} x_{1,k}ζ_{3}^{k}) × x_{0,0} | x_{0,0} ≡_{3} x_{1,0} + x_{1,1}\} ⊆ Z[ζ_{3}] × Z
ZC_{9}  \overline{}^{ω_{Z,9}}_{∼} \{ (∑_{k∈[0,5]} x_{2,k}ζ_{9}^{k}) × (∑_{k∈[0,1]} x_{1,k}ζ_{3}^{k}) × x_{0,0} |

x_{1,0} ≡_{3} x_{2,0} + x_{2,3} − x_{2,2} − x_{2,5} x_{1,1} ≡_{3} x_{2,1} + x_{2,4} − x_{2,2} − x_{2,5} x_{0,0} ≡_{9} 3(x_{2,2} + x_{2,5}) + (x_{1,0} + x_{1,1})} ⊆ Z[ζ_{9}] × Z[ζ_{3}] × Z

ZC_{27}  \overline{}^{ω_{Z,27}}_{∼} \{ (∑_{k∈[0,17]} x_{3,k}ζ_{27}^{k}) × (∑_{k∈[0,5]} x_{2,k}ζ_{9}^{k}) × (∑_{k∈[0,1]} x_{1,k}ζ_{3}^{k}) × x_{0,0} |

x_{2,0} ≡_{3} x_{3,0} + x_{3,9} − x_{3,6} − x_{3,15} x_{2,1} ≡_{3} x_{3,1} + x_{3,10} − x_{3,7} − x_{3,16} x_{2,2} ≡_{3} x_{3,2} + x_{3,11} − x_{3,8} − x_{3,17} x_{2,3} ≡_{3} x_{3,3} + x_{3,12} − x_{3,6} − x_{3,15} x_{2,4} ≡_{3} x_{3,4} + x_{3,13} − x_{3,7} − x_{3,16} x_{2,5} ≡_{3} x_{3,5} + x_{3,14} − x_{3,8} − x_{3,17} x_{1,0} ≡_{9} 3(x_{3,6} + x_{3,15} − x_{3,8} − x_{3,17}) + (x_{2,0} + x_{2,3} − x_{2,2} − x_{2,5}) x_{1,1} ≡_{9} 3(x_{3,7} + x_{3,16} − x_{3,8} − x_{3,17}) + (x_{2,1} + x_{2,4} − x_{2,2} − x_{2,5}) x_{0,0} ≡_{27} 9(x_{3,8} + x_{3,17}) + 3(x_{2,2} + x_{2,5}) + (x_{1,0} + x_{1,1})} ⊆ Z[ζ_{27}] × Z[ζ_{9}] × Z[ζ_{3}] × Z.

5.5 Kervaire-Murthy ties for ZC_{m}

Let m ⩾ 1. We maintain the notation of §5.2. Suppose given an inclusion of commutative Z-orders Λ ⊆ Γ that has, as an inclusion of abelian groups, a cokernel Γ/Λ which is finite as a set. Let the naive localization Λ_{[p],Γ} be the kernel of the composition Γ \overline{}Γ/Λ \overline{}(Γ/Λ)_{(p)} (cf. [Kü 99, D.2]). Since Λ_{[p],Γ} = Γ ∩ Λ_{(p)}, intersected as subsets of Γ_{(p)}, the naive localization Λ_{[p],Γ} is a suborder of Γ of index (#(Γ/Λ))[p]. We have Λ_{(p)} = (Λ_{[p],Γ})_{(p)}. Moreover,

Λ = ⋂_{p} Λ_{[p],Γ}.

Note that if Λ_{(p)} = Γ_{(p)}, then Λ_{[p],Γ} = Γ.

Lemma 5.17 Given inclusions of Z-orders Λ ⊆ Γ and Λ′ ⊆ Γ′, we obtain an equality

(Λ ⊗ Λ′)_{[p],Γ⊗Γ′} = Λ_{[p],Γ} ⊗ Λ′_{[p],Γ′}.

as subsets of Γ ⊗ Γ′.

In fact, flatness yields

Λ_{[p],Γ} ⊗ Λ′_{[p],Γ′} = (Λ_{[p],Γ} ⊗ Λ′_{(p)}) ∩ (Λ_{[p],Γ} ⊗ Γ′)
= (Λ ⊗ Λ′)_{(p)} ∩ (Λ_{(p)} ⊗ Γ′) ∩ (Γ ⊗ Γ′)
= (Λ ⊗ Λ′)_{[p],Γ⊗Γ′}.

Letting

[figure: (Λ ⊆ Γ); :=; ((ZC_{m[p]})ω_{Z,m[p]}; ∏_{e|m[p]}Z[ζ_{e}])]
[figure: (Λ′ ⊆ Γ′); :=; ((ZC_{m[p′]})ω_{Z,m[p′]}; ∏_{f|m[p′]}Z[ζ_{f}]),]

we obtain

(Λ ⊗ Λ′)_{[p],Γ⊗Γ′} =^{(5.17)} Λ_{[p],Γ} ⊗ Λ′_{[p],Γ′}
=^{(5.8)} Λ ⊗ Γ′
= ∏_{f|m[p′]}Λ ⊗ Z[ζ_{f}]

⊆ ∏_{f|m[p′]} ( ∏_{e|m[p]}Z[ζ_{e}]) ⊗ Z[ζ_{f}]
= ∏_{f|m[p′]} ∏_{e|m[p]}Z[ζ_{ef}].

We could not do better than to argue with a Z-linear basis of Z[ζ_{f}]. This lead to the ‘non-canonical’ representation of an element of Z[ζ_{d}], d|m, that is used in the following

Proposition 5.18 Suppose given an element

( ∑ ; (j_{p})_{p|d}∈∏_{p|d}[0,φ(d[p])−1]a_{d,(j_{p})_{p|d}}ζ_{d}^{∑_{p|d} s_{d,p}j_{p}d[p′]})_{d|m} ∈ ∏_{d|m}Z[ζ_{d}],

where a_{d,(j_{p})_{p|d}} ∈ Z. This element is contained in the image of ω_{Z,m} if and only if for each p|m, for each f|m[p′] and for each tuple (j_{q})_{q|f}, j_{q} ∈ [0, φ(d[q]) − 1], we have

( ∑_{j_{p}∈[0,φ(e)−1]}a_{ef, j_{p}×(j_{q})_{q|f}} ζ_{e}^{j_{p}} )_{e|m[p]} ∈ (ZC_{m[p]})ω_{Z,m[p]},

where j_{p} × (j_{q})_{q|f} denotes the tuple that has entry j_{p} at position p and j_{q} at position q for q|f. Thus we may employ (5.14).

Example 5.19 We have s_{6,2} = 1, s_{6,3} = 2 and s_{12,2} = 3, s_{12,3} = 1. The element

\begin{matrix}(a_{1,(•,•)}) × (a_{2,(0,•)}) × (a_{3,(•,0)} + a_{3,(•,1)}ζ_{3}) × (a_{4,(0,•)} + a_{4,(1,•)}ζ_{4}) × (a_{6,(0,0)} + a_{6,(0,1)}ζ_{6}^{4}) \\ ×(a_{12,(0,0)} + a_{12,(0,1)}ζ_{12}^{4} + a_{12,(1,0)}ζ_{12}^{9} + a_{12,(1,1)}ζ_{12}), \end{matrix}

the symbol • indicating a non-existing entry, is contained in (ZC_{12})ω_{Z,12} if and only if

(a_{1,(•,•)}) × (a_{2,(0,•)}) × (a_{4,(0,•)} + a_{4,(1,•)}ζ_{4}) ∈ (ZC_{4})ω_{Z,4}
(a_{3,(•,0)}) × (a_{6,(0,0)}) × (a_{12,(0,0)} + a_{12,(1,0)}ζ_{4}) ∈ (ZC_{4})ω_{Z,4}
(a_{3,(•,1)}) × (a_{6,(0,1)}) × (a_{12,(0,1)} + a_{12,(1,1)}ζ_{4}) ∈ (ZC_{4})ω_{Z,4}
(a_{1,(•,•)}) × (a_{3,(•,0)} + a_{3,(•,1)}ζ_{3}) ∈ (ZC_{3})ω_{Z,3}
(a_{2,(0,•)}) × (a_{6,(0,0)} + a_{6,(0,1)}ζ_{3}) ∈ (ZC_{3})ω_{Z,3}
(a_{4,(0,•)}) × (a_{12,(0,0)} + a_{12,(0,1)}ζ_{3}) ∈ (ZC_{3})ω_{Z,3}
(a_{4,(1,•)}) × (a_{12,(1,0)} + a_{12,(1,1)}ζ_{3}) ∈ (ZC_{3})ω_{Z,3}.

By (5.14), an element (b_{1,0}) × (b_{2,0}) × (b_{4,0} + b_{4,1}ζ_{4}) is contained in (ZC_{4})ω_{Z,4} if and only if

\begin{aligned}b_{2,0} & ≡_{2} & b_{4,0} − b_{4,1} \\ b_{1,0} & ≡_{4} & 2b_{4,1} + b_{2,0}, \end{aligned}

and an element (b_{1,0}) × (b_{3,0} + b_{3,1}ζ_{3}) is contained in (ZC_{3})ω_{Z,3} if and only if

b_{1,0} ≡_{3} b_{3,0} + b_{3,1}.

Remark 5.20 Kleinert gives a system of ties that describes the image of the absolute Wedderburn embedding ZC_{m}^{\overline{}^{ω_{Z,m}}} ∏_{d|m} Z[ζ_{d}] in terms of certain prime ideals of Z[ζ_{d}], d|m, in case m is squarefree [Kl 81, p. 550].

6 References

G. E. Andrews [A 76] The Theory of Partitions, Encyclopedia of Math. and Appl., Vol. 2, Addison-Wesley, 1976. Buenos Aires Cyclic Homology Group (J. A. Guccione, J. J. Guccione, M. J. Redondo, A. Solotar, O. E. Villamayor)

[BACH 91] Cyclic Homology of Algebras with One Generator, K-Theory 5, p. 51-69, 1991.

M.A. Kervaire, M.P. Murthy [KM 77] On the projective class group of cyclic groups of prime power order, Comm. Math. Helv. 52, p.

415-452, 1977. M. Künzer

[Kü 99] Ties for the integral group ring of the symmetric group, thesis,

http://www.mathematik.uni-bielefeld.de/∼kuenzer, Bielefeld, 1999. Kleinert, E. [Kl 81] Einheiten in Z[D_{2m}], J. Num. Th. 13, p. 541-561, 1981. M. Larsen, A. Lindenstrauss [LL 92] Cyclic homology of Dedekind domains, K-Theory 6, p. 301-334, 1992. J. Neukirch [N 91] Algebraische Zahlentheorie, Springer, 1991. Plesken, W. [P 80] Gruppenringe über lokalen Dedekindbereichen, Habilitationsschrift, Aachen, 1980.

Harald Weber Mathematisches Institut B

3. Lehrstuhl Pfaffenwaldring 57 70569 Stuttgart harald@poolb.mathematik.uni-stuttgart.de

Matthias Künzer Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld kuenzer@mathematik.uni-bielefeld.de