Continuation of hereditary orders in local central simple

algebras

by Martin Grabitz, Berlin

Summer, 1998

Introduction

The subject of this work can be viewed as a part of the continuation theory of special pseudo valuations in the sense of Benz [Be1].

Let A be a finite-dimensional central simple algebra over a non-archimedean local field F. A function ω : A → ℝ ∪ {∞} is called a (non-archimedian) pseudo valuation of the algebra A if it

fulfils the following properties:

(i) ω(x) = ∞ iff x = 0; (i)′ 0 < ω(x_{0}) < ∞ for at least one x_{0}; (ii) ω(ab) ≥ ω(a) + ω(b); (ii)′ for each value w := ω(x) for x ∈ A, exists an element x_{w} ∈ A mit ω(x_{w}) = w and ω(x_{w}y) =

ω(yx_{w}) = ω(x_{w}) + ω(y) for all y ∈ A; (iii) ∀x, y ∈ A : ω(x − y) ≥ min(ω(x), ω(y)). A pseudo valuation ω is called an almost valuation if we have in addition: (iv) The residual class ring 𝔄/𝔓 of the valuation ring 𝔄 := {x ∈ A | ω(x) ≥ 0} with the valuation

ideal 𝔓 := {x ∈ A | ω(x) > 0} is a semisimple (finite) ring.

The elements x_{w} fulfilling the condition (ii)′ are called the ω-elements of A. (If, for an almost valuation ω, each x ∈ A is an ω-element, then ω is a valuation in the usual sense).

In our case of a central simple algebra A, the almost valuations ω, which continue the valuation of the ground field, biuniquely correspond to the so-called principal orders 𝔄 of A, and the set of non-zero ω-elements corresponds to the normalizers K_{𝔄} of the 𝔄′s. Benz [Be2] has shown that the valuation of a maximal extension M/F in A has always a unique continuation to an almost valuation of A; hence, there exists a unique principal order normalized by M^{×}. Fröhlich [F] has given a shorter proof and examined the associated embeddings of maximal fields. More generally to each hereditary order 𝔄 in A, we can associate a pseudo valuation ω_{𝔄} : A → ℤ∪\{∞\} by ω_{𝔄}(x) := \frac{t}{rd}max\{j ∈ ℤ∪\{∞\} | x ∈ 𝔓^{tj}\}, where 𝔓^{∞} := {0}, d is the index of A, r the period of 𝔄 as in 0.3 and t is as in 0.5 or 1.2. In fact, t is the smallest positive power of the Jacobson radical 𝔓 of 𝔄 that is a principal ideal and if t = 1, then 𝔄 is called a principal order. Again the A^{×}-normalizer K_{𝔄} is the set of non-zero ω_{𝔄}-elements.

The most important aim of our work is to show that there always exists a continuation ω_{𝔄} associated to a hereditary order 𝔄 in A of a pseudo valuation ω_{𝔅} associated to a hereditary order 𝔅 in a simple subalgebra B (central over a field extension E/F of our main local ground field F) to our main algebra A.

The proof requires the handling of several special cases and uses the results of [Be2], [F] and [Z] about principal orders. So first, in Chapter 1, we consider the relationship between principal and arbitrary hereditary orders. We show in 1.2 that each hereditary order is the intersection of a canonical set of principal orders, such that also its normalizer is the intersection of the normalizers of that set of principal orders. If 𝔄_{1}, …, 𝔄_{t} is that canonical set, then 1.2(ii)(c) means in terms of pseudo valuations ∀x ∈ A : ω_{𝔄}(x) = max(ω_{𝔄_{1}}(x), …, ω_{𝔄_{t}}(x)); hence, in terms of [Be1](8.1), ω_{𝔄} is the direct sum of the ω_{𝔄_{1}}, …, ω_{𝔄_{t}}. We introduce the notion of pairs and will explain our main result 1.9 in all details.

In Chapter 2 we treat the special case where the inertial degree f(E/F) and the index of the algebra A are prime to each other, which we call the splittable case, because, for an unramified extension L/F having the same degree as the index of the algebra A, we obtain that E ⊗_{F} L/L is an extension in the L-algebra A ⊗_{F} L and by [CR](26.29) 𝔄 ⊗_{o_{F}} o_{L} remains hereditary. Therefore, all results can be obtained here by splitting the algebra A by an unramified splitting field and using the results of [BK3]. In 2.2 we obtain that such a continuation exists and is always unique. From that we reduce to the case where E/F is unramified and [E : F] divides the index of the algebra A, which means that E/F can be embedded into the central division algebra D over F, whereof A is a matrix algebra. This is the special case considered in Chapter 3.

In the splittable case and in Lemma 3.2 we have the unramified case of our construction, where an unramified continuation means that the associated continuation of pseudo valuations is unramified in the usual sense; hence, the pseudo valuation of B and its continuation to A have a common uniformizer in B (also see Theorem 1.9(i)(b) and notice that the continuation is unramified iff f(E/F) = l_{1}). For the existence of a continuation we can use 2.1. The key to the general case is then the totally ramified case of Lemma 3.3. One of the main ideas of its proof is that each pair (𝔄, 𝔄_{1}) of a hereditary order 𝔄 and an associated principal order 𝔄_{1} as in 1.2 admits a common normal form, as we have realized in 1.5. This suggests the introduction of the notion of pairs and their continuations. For a given hereditary order 𝔄 a partner is given by a principal order 𝔄_{1} containing 𝔄 such that the associated pseudo valuations are admit a common uniformizer and, for a given pair (𝔅, 𝔅_{1}) in B, (𝔄, 𝔄_{1}), are a continuation iff the pseudo valuation associated to 𝔅 is continued by the pseudo valuation associated to 𝔄, and the almost valuation associated to 𝔅_{1} is continued by the almost valuation associated to 𝔄_{1}. In 3.4 it turns out that this characterizes our construction of a continuation uniquely.

In Chapter 4 we combine our results to obtain the general case of Theorem 1.9, where we state the existence of a unique continuation of a pair (𝔅, 𝔅_{1}) in B to a pair (𝔄, 𝔄_{1}) to A.

This work is the compilation of the most important results of a diploma work [G] completed under the supervision of my teacher Ernst-Wilhelm Zink of the Humboldt-University Berlin. I would like to thank him for teaching me the foundations of the theory of hereditary orders and for developing the fundamental ideas of this work in mutual discussions. We generalize here the main results of the work [Z] to the notion of pairs (see the next section). Special thanks are also due to Paul Broussous for his hint to Lemma 3.1 and Bernhard Runge for valuable discussions.

Concerning standard books about the structure theory of hereditary orders, we refer to [R] and [CR]. Additional information can be found in [BF]. Now let us make our notation more concrete.

0 Notation and preparations

Let F be a non-archimedean local field and A a finite-dimensional central simple algebra over F. Let V be a left simple A-module. V is a right D = End_{A}(V)-vector space and

A ≃ End_{D}(V) ≃ M_{m}(D)

(where M_{m}(D) denotes the ring of m × m matrices in D). Denote by d the index of the division algebra D (that is, dim_{F} (D) = d^{2}), fix an isomorphism A ≃ M_{m}(D), and identify A with M_{m}(D). Denote by 𝔬_{F}, 𝔬_{D}, … the rings of integers of F, D, …, by 𝔭_{F}, 𝔭_{D}, … its unique prime ideals and by 𝔨_{F}, 𝔨_{D}, … the associated residual fields.

0.1 An 𝔬_{F} -order in A is a subring with the same unit element as A which is a complete 𝔬_{F} -lattice in A. Let 𝔄 be a hereditary 𝔬_{F} order in A, i. e., an 𝔬_{F} -order which is (left- or right-) hereditary as a ring. We denote by 𝔓 its Jacobson radical and by K_{𝔄} its normalizer. It is well known (see [R](39.12)) that 𝔄 is hereditary iff 𝔓 is a projective left or right 𝔄-module, which is also the case iff 𝔓 is a fractional invertible ideal of 𝔄. Hence, there exists an (𝔄,𝔄)-bisubmodule 𝔓^{−1} of A which is a complete 𝔬_{F} -lattice in A such that 𝔓𝔓^{−1} = 𝔓^{−1}𝔓 = 𝔄.

An 𝔬_{D}-lattice chain in V is a family {X_{j}}_{j∈ℤ} of complete 𝔬_{F} -lattices in V that are right 𝔬_{D}-modules such that

a) ∀j ∈ ℤ, X_{j} ⊋ X_{j+1},

b) ∃r ∈ ℕ, ∀j ∈ ℤ, X_{j}𝔭_{D} = X_{j+r}.

0.2 One knows from [BF] that 𝔄 is also hereditary iff there exists a (unique up to a translation of the index) 𝔬_{D}-lattice chain {X_{k}}_{k∈ℤ} in V such that

𝔄 = End_{𝔬_{D}}{X_{k}}_{k∈ℤ} = {x ∈ A | ∀k ∈ ℤ, xX_{k} ⊂ X_{k}}.

0.3 We have

𝔄/𝔓 ≃ ⨁_{i=1}^{r} End_{𝔨_{D}}(X_{i−1}/X_{i}) ≃ M_{n_{1}}(𝔨_{D}) × … × M_{n_{r}}(𝔨_{D})

such that the ordered numbers n_{1},…,n_{r} provide numerical invariants, up to a cyclic permutation, of the conjugacy class of 𝔄. These numbers also come from a normal form (see [R] 39.14)

g𝔄g^{−1} = ( \begin{matrix}(𝔬_{D}) , & (𝔭_{D}) & ⋯, & (𝔭_{D}) \\ (𝔬_{D}) , & (𝔬_{D}) & ⋯, & ⋯ \\ ⋯, & ⋯ & ⋯, & (𝔭_{D}) \\ (𝔬_{D}) , & ⋯ & ⋯, & (𝔬_{D}) \end{matrix} )^{\{n_{1},...,n_{r}\}} ,

where (𝔬_{D}) (and (𝔭_{D}) respectively) mean n_{i} × n_{j}-matrices, where the lower-diagonal and diagonal blocks have entries in 𝔬_{D} and the upper-diagonal blocks have entries in 𝔭_{D} (note that n_{1}+…+n_{r} = m) and g is a suitable element of A^{×}. Here the number r is uniquely determined and called the period of 𝔄 and {X_{j}}_{j∈ℤ}, respectively. The numbers n_{1}, …, n_{r} are unique up to a cyclic permutation.

The modules X_{0}, …, X_{r−1} of the lattice chain of 𝔄 in V (as above) are a system of representatives of the isomorphy classes of direct indecomposable right 𝔄-modules

(1)

(see [BF](1.2.8)(ii) and [R](39.23)).

We also have

g𝔓g^{−1} = ( \begin{matrix}(𝔭_{D}) , & (𝔭_{D}) , & ⋯, & (𝔭_{D}) , & (𝔭_{D}) \\ (𝔬_{D}) , & (𝔭_{D}) , & ⋯, & ⋯, & ⋯ \\ ⋯, & (𝔬_{D}) , & ⋯, & ⋯, & ⋯ \\ (𝔬_{D}) , & ⋯, & ⋯, & (𝔭_{D}) , & ⋯ \\ (𝔬_{D}) , & ⋯, & ⋯, & (𝔬_{D}) , & (𝔭_{D}) \end{matrix} )^{\{n_{1},...,n_{r}\}} .

If 𝔄 is given in normal form 0.3, then, for the simple A-module V := ⨁_{i=1}^{m} D, we can choose the

numbering of the chain {X_{j}}_{j∈ℤ} such that ∀j ∈ ℤ, ^{t}X_{j} = 𝔭_{D}^{l}((⨁_{i=1}^{k}( ⨁_{μ=1}^{n_{i}} 𝔭_{D})) ⨁( ⨁_{i=k+1}^{r}( ⨁_{μ=1}^{n_{j}} 𝔬_{D}))), where

k, l ∈ ℤ such that j = lr + k, 0 ≤ k < r, and ^{t}(.) means the set of transposed vectors.

An order 𝔄 is called principal if its Jacobson radical 𝔓 is a principal ideal, especially by 0.1 such an 𝔄 is hereditary.

0.4 From [BF](1.3.2) we know that a herditary order 𝔄 with invariants n_{1}, …, n_{r} is a principal order if and only if n_{1} = … = n_{r} = s. In this situation we call s just the invariant of 𝔄.

0.5 Notation 𝔄 will always denote a hereditary 𝔬_{F} -order in A. It is not necessarily given in normal form 0.3, but will have period r and invariants n_{1}, …, n_{r}. Let t be the smallest period of the numbers n_{1}, …, n_{r}. Denote by {X_{j}}_{j∈ℤ} an associated lattice chain in V, by 𝔓 the Jacobson radical and by K_{𝔄} = {x ∈ A^{×} | x𝔄x^{−1} = 𝔄} the normalizer. For an extension E/F in A we consider a hereditary 𝔬_{E}-order 𝔅 in the centralizer B = C_{A}(E) := {x ∈ A | ∀y ∈ E, yx = xy}. We have B ≃ EndD̃(W) ≃ M_{m′}(D̃) for a right simple B-module and left D̃-vector space W, where D̃ = End_{B}(W) is a skew field of index d′. Denote by {Y_{j}}_{j∈ℤ} a lattice chain associated to 𝔅 in W, by 𝔔 the Jacobson radical of 𝔅, by r′ the period of 𝔅, by n′_{1}, …, n′_{r′} the invariants of 𝔅, by t′ the smallest period of the numbers n′_{1}, …, n′_{r′}, and by ⟨., …, .⟩ the greatest common divisior of integers. From the theory of Brauer groups it is well known that, in the latter situation, we have d′ = \frac{d}{⟨[E:F],d⟩} and m′ = \frac{⟨[E:F],d⟩m}{[E:F]} (see for example [Z](Proposition 1)), where m and d are as in the beginning of this section.

0.6 Definition Let E/F be a field extension of F in A such that E^{×} ⊂ K_{𝔄}. Let 𝔅 be a hereditary order in B.

(i) We say that 𝔄 is a continuation of B iff 𝔄 ∩ B = 𝔅 and K_{𝔅} ⊂ K_{𝔄}.

(ii) Such a continuation is called unramified iff there exists an h ∈ K_{𝔅} such that K_{𝔄} = ⟨h⟩𝔄^{×}

\underline{Remark:} 1) It is immediately ovious that K_{𝔄} ∩ B ⊂ K_{𝔄∩B} in general; hence, if 𝔄 is a continuation of 𝔅, then K_{𝔄} ∩ B = K_{𝔅}.

2) The notion of continuation has been chosen due to the notions of the valuation theory of algebras. Continuations of orders are the same as continuations of the associated pseudo valuations, and unramified continuations of orders are equivalent to unramified continuations of the associated pseudo valuations.

Our most important aim of this work stated in terms of orders (not valuations) is then to show that each hereditary order 𝔅 in B has a continuation 𝔄 to A and, secondly, to make this continuation unique by the consideration of pairs. We will also give some information about the associated embedding of E/F in the algebra A in this special case (see 1.9(ii)(iii)).

The general case of embeddings being good with respect to continuation will be investigated in a forthcoming paper, where we will give an explicit version of the criterion of Broussous [Br]. There we will also introduce a more general version of the notion of a pure element as in [Z]. The notion of [Z] could also be generalized by the results of this paper, but with less practicalness to the representation theory of simple algebras.

1 Hereditary orders from principal orders

It is straightforward that:

1.1 Lemma Let 𝔅, 𝔔, B, E/F and {Y_{j}}_{j∈ℤ} be as in 0.5 and g ∈ A^{×}. Then K_{g𝔅g^{−1}} = gK_{𝔅}g^{−1} and g𝔅g^{−1} is a hereditary 𝔬_{gEg^{−1}}-order in gBg^{−1} = C_{A}(gEg^{−1}) with the Jacobson radical g𝔔g^{−1} and the same invariants as 𝔅.

□

1.2 Proposition Let 𝔄 be as in 0.5 and use the associated notations, except for (at this moment) the notation for t. Let ν_{𝔄} : A → ℤ ∪ {∞} be defined by ν_{𝔄}(x) := max{j ∈ ℤ | x ∈ 𝔓^{j}} if x ∈ A\{0} and ν_{𝔄}(0) := ∞. From [BF](1.2.8)(i) we know that ν_{𝔄|K_{𝔄}} is a group homomorphism with kernel 𝔄^{×}. Consequently, letting t ∈ ℕ such that ν_{𝔄|K_{𝔄}}(K_{𝔄}) = tℤ, there exists an h ∈ K_{𝔄} such that ν_{𝔄}(h) = t, and, therefore, K_{𝔄} = ⟨h⟩𝔄^{×} = 𝔄^{×}⟨h⟩ (⟨h⟩ the infinite cyclic group generated by h). Let 𝔄_{i} := End_{𝔬_{D}}{X_{(i−1)+tj}}_{j∈ℤ} for i = 1, …, t. Then

(i) t is the smallest period of the sequence of natural numbers n_{1}, …, n_{r} as in 0.5, (ii) 𝔄_{1}, …, 𝔄_{t} are principal orders

(a) with invariant s := n_{1} + … + n_{t} and period \frac{r}{t}, hence pairwise conjugated,

(b) with Jacobson radicals 𝔓_{i} = h𝔄_{i} for i = 1, …, t, and

(c) such that for all j ∈ ℤ, 𝔓^{tj} = ⋂_{i=1}^{t} 𝔓_{i}^{j}. (iii) Let 𝔒 be a hereditary 𝔬_{F} -order in A such that 𝔄 ⊂ 𝔒 and K_{𝔄} ⊂ K_{𝔒}. Then there exist

l, j_{1}, …, j_{l} ∈ {1, …, t} such that 𝔒 = 𝔄_{j_{1}} ∩ … ∩ 𝔄_{j_{l}}.

(iv) 𝔄_{1}, …, 𝔄_{t} are the unique principal orders such that 𝔄 ⊂ 𝔄_{i} and ⟨h⟩𝔄^{×} = K_{𝔄} ⊂ K_{𝔄_{i}} = ⟨h⟩𝔄_{i}^{×} or, equivalently, the unique principal orders such that

(a) 𝔄_{1}, …, 𝔄_{t} are pairwise distinct but conjugate over A^{×},

(b) 𝔄 = 𝔄_{1} ∩ … ∩ 𝔄_{t}, and

(c) K_{𝔄} = K_{𝔄_{1}} ∩ … ∩ K_{𝔄_{t}} or, equivalently, the unique minimal set of principal orders satisfying (a),(b). \underline{Remark} If we look for the maximal set of hereditary orders satisfying (iv)(a),(b), we obtain the set

of maximal orders that contains 𝔄, and the order of this set is r (see [BF](1.2.8.(ii)) or [R](39.23)).

\underline{Proof} By 1.1 and replacing 𝔄 by g𝔄g^{−1}, h by ghg^{−1}, we can restrict ourselves to the case where 𝔄 is

given in normal form 0.3.

(ii)(a) Then π_{D}I_{m} ∈ K_{𝔄} and ν_{𝔄}(π_{D}I_{m}) = r, hence t | r and dim_{𝔨_{D}}(X_{j}/h^{\frac{r}{t}} X_{j}) = dim_{𝔨_{D}}(X_{j}/hX_{j})+ … + dim_{𝔨_{D}}(h^{\frac{r}{t} −1}X_{j}/h^{\frac{r}{t}} X_{j}) = m. Because of X_{i}/X_{j} ≃ hX_{i}/hX_{j} for i < j as 𝔨_{D} vector spaces, we obtain dim_{𝔨_{D}}(X_{(i−1)+tj}/X_{(i−1)+t(j+1)}) = dim_{𝔨_{D}}(X_{(i−1)+tj}/hX_{(i−1)+tj}) = \frac{mt}{r} =: s for all j ∈ ℤ.

(i) Since dim_{𝔨_{D}}(X_{j}/X_{j+t}) = dim_{𝔨_{D}}(X_{j}/X_{j+1})+…+dim_{𝔨_{D}}(X_{j+t−1}/X_{j+t}) = n_{j+1} +…+n_{j+t} for j = 0, …, r − 1, t is a period of the partition n_{1}, …, n_{r} of m. On the other hand, if t′ is an arbitrary period of n_{1}, …, n_{r}, it is easy to see from the explicit description of the chain {X_{j}}_{j∈ℤ} for V = D^{m} (column vectors) given after 0.3 that for, s′ := n_{1} + … + n_{t′},

h_{s′} := ( \begin{matrix}0 , & π_{D}I_{s′} \\ I_{m−s′} , & 0 \end{matrix} )  (2)

belongs to K_{𝔄}, hence t | t′.

(ii)(b) follows from [BF](1.2.9).

(ii)(c) follows from (ii)(b), and 𝔓^{tj} = h^{j}𝔄 = ⋂_{i=1}^{t} h^{j}𝔄_{i} = ⋂_{i=1}^{t} 𝔓_{i}^{j} for all j ∈ ℤ.

(iii) Let {Z_{j}}_{j∈ℤ} be a lattice chain for 𝔒. The Z_{j}′s span V over F, V is a simple A-module and 𝔄 spans A over F. Moreover, each isomorphism of 𝔄-modules that are complete 𝔬_{F} -lattices in V extends to an automorphism of V as a simple A-module. Since 𝔄 ⊂ 𝔒 and because the Z′_{j}s are direct indecomposable 𝔄-modules, we obtain by 0.3(1) and Schur’s Lemma that {Z_{j}}_{j∈ℤ} ⊂ {X_{j}}_{j∈ℤ}. Because of h ∈ K_{𝔄} ⊂ K_{𝔒} it follows that, if Z_{k} ∈ {X_{(i−1)+tj}}_{j∈ℤ} for an i ∈ {1, …, t}, then {X_{(i−1)+tj}}_{j∈ℤ} ⊂

{Z_{j}}_{j∈ℤ}; hence, our result follows from {X_{j}}_{j∈ℤ} = ⋃_{i=1}^{t}{X_{(i−1)+tj}}_{j∈ℤ}.

(iv) Clearly, the orders 𝔄_{1}, …, 𝔄_{t} fulfil all necessary conditions. The first assertion is clear from (iii) and the choice of h. From the arguments of the proof of (iii) it follows easily that a minimal set of hereditary orders fulfilling (iv)(a),(b) splits the chain {X_{j}}_{j∈ℤ} into disjoint subchains, each of them associated to a member of the set. As they all have the same invariants, it is easy to see that the order of this set must be a period of the numbers n_{1}, …, n_{r}, i. e., t, and the assertion follows from the definition of the 𝔄_{1}, …, 𝔄_{t} in terms of subchains. For the main assertion notice that each principal order contained in a set of principal orders satisfying (iv)(a)-(c) must contain h in its normalizer. From our last considerations and the choice of h it follows that the order of this set can be at most t and is a period of n_{1}, …, n_{r}. Hence, our result follows from (i).

□

1.3 Definition (i) In the following we often use the notation ν_{𝔄} for the ‘valuation map’ associated to a hereditary order 𝔄 as in 1.2.

(ii) We call t as in 1.2(i) the partition period.

(iii) Because of 1.2(iv), we call 𝔄_{1}, …, 𝔄_{t} (from 1.2) the canonical principal orders associated to 𝔄.

1.4 Corollary

(i) Let 𝔄 be a hereditary order in A with canonical principal orders 𝔄_{1}, …, 𝔄_{t}, and M/F a maximal extension in A (that is, [M : F] = md) such that M^{×} ⊂ K_{𝔄}. Then t = 1 and 𝔄 = 𝔄_{1} is a principal order.

(ii) Let 𝔅, E/F and B be as in 0.5, and assume that 𝔅 is principal. Then it has a unique hereditary continuation to A, which is principal.

\underline{Proof}

(i) Follows from the uniqueness assertion in [Be2](Chap. 2) or in [F](theorem 1.1) because K_{𝔄} = K_{𝔄_{1}} ∩ … ∩ K_{𝔄_{t}}.

(ii) Follows from (i) and [F]((2.7), Corollary 3).

□

1.5 Proposition

(i) Let 𝔄̃ be the hereditary order in normal form 0.3, where the order of the numbers in the partition n_{1}, …, n_{r} of m is fixed. Let t be the partition period as in 1.2 and 𝔄_{1} the principal order in normal form 0.3 with the invariant s := n_{1} + … + n_{t} and the period \frac{r}{t}. Then 𝔄̃_{1} is a canonical principal order of 𝔄̃.

(ii) Each pair (𝔄, 𝔄_{1}) of a hereditary order 𝔄 in A and an associated principal order 𝔄_{1} is conjugate under A^{×} to exactly one pair (𝔄̃, 𝔄̃_{1}) in simultanous normal form as in (i), i. e., there exists exactly one such pair (𝔄̃, 𝔄̃_{1}) such that (g𝔄g^{−1}, g𝔄_{1}g^{−1}) = (𝔄̃, 𝔄̃_{1}) for a suitable g ∈ A^{×}.

(iii) Let 𝔄, {X_{j}}_{j∈ℤ} and n_{1}, …, n_{r}, r be as in 0.5 and 𝔄_{1}, …, 𝔄_{t} be the canonical principal orders of 𝔄. Choose the index of the chain and the canonical principal orders in such a way that n_{i} = dim_{𝔨_{D}}(X_{(i−1)+tj}/X_{i+tj}) for i = 1, …, t (j ∈ ℤ is arbitrarily chosen and {X_{(i−1)+tj}}_{j∈ℤ} is the chain of 𝔄_{i}). Then the pair (𝔄, 𝔄_{i}) is conjugated to a pair (𝔄̃, 𝔄̃_{1}) in simultaneous normal form as in (i), where the invariants of 𝔄̃ appear in the normal form in the order n_{i}, …, n_{r}, n_{1}, …, n_{i−1}.

\underline{Proof}

(i) Clearly, 𝔄̃ ⊂ 𝔄̃_{1}. From 0.3 and the explicit descriptions of the chains in V = D^{m} (column vectors), we obtain that h_{s}, as in (2), is a common uniformizer for 𝔄̃ and 𝔄̃_{1} as required for the first assertion of 1.2(iv).

(ii) First we see that if (𝔄, 𝔄_{1}) is conjugate to (𝔄̃, 𝔄̃_{1}), where the invariants appear in the order of n_{1}, …, n_{r} in the normal form of 𝔄̃, and is also conjugate to another pair in simultanous normal form (𝔒̃, 𝔒̃_{1}), then 0.3 implies that, in the normal form of 𝔒̃, the invariants has to appear in a certain cyclic permuted order n_{i}, …, n_{r}, n_{1}, …, n_{i−1} for some i ∈ {1, …, t}, hence 𝔄̃_{1} = 𝔒̃_{1} in particular.

Therefore, 𝔄̃ and 𝔒̃ must be conjugated over an element of A^{×} that fixes 𝔄̃_{1}; hence, over K𝔄̃_{1}. If x, y ∈ A^{×} conjugate 𝔄̃ to 𝔒̃, then y^{−1}x ∈ K𝔄̃ ⊂ K𝔄̃_{1}; hence, if x ∈ K𝔄̃_{1}, then y ∈ K𝔄̃_{1}, too.

On the other hand, if i > 1, it is easy to see from the explicit description of the chain {X_{j}}_{j∈ℤ} in the case of V = D^{m} that h_{n_{1}+...+n_{i−1}}, as in (2), conjugates the chains, hence the orders 𝔄̃ and 𝔒̃, but it cannot be contained in K𝔄̃_{1}. Thus we must have i = 1 and (𝔄̃, 𝔄̃_{1}) = (𝔒̃, 𝔒̃_{1}). Therefore, (𝔄, 𝔄_{1}) can be conjugated to at most one such pair (𝔄̃, 𝔄̃_{1}).

For the next step let 𝔄 be as in 0.5, 𝔄_{1}, …, 𝔄_{t} its canonical principal orders, s = n_{1} + … + n_{t} and 𝔄^{(1)}, …, 𝔄^{(t)} the set of all hereditary orders in normal form 0.3 such that the invariants n_{1}, …, n_{r} appear one after another in the orders n_{1}, …, n_{r}; n_{2}, …, n_{r}, n_{1}; …; n_{t}, …, n_{r}, n_{1}, …, n_{t−1}. Clearly, each pair (𝔄^{(i)}, 𝔄̃_{1}) for i = 1, …, t is a pair in the sense of (i).

Now, by 0.3, each order 𝔄^{(i)} is conjugate to 𝔄 by an element g_{i} ∈ A^{×}, and, by 1.1 in the case of E = F and the first assertion of 1.2(iv), g_{i}𝔄̃_{1}g_{i}^{−1} is a canonical principal order of g_{i}𝔄^{(i)}g_{i}^{−1} = 𝔄 for i = 1, …, t, hence it is equal to one of the 𝔄_{1}, …, 𝔄_{t}.

From the above we know that none of the pairs (𝔄, g_{i}𝔄̃_{1}g_{i}^{−1}) can be conjugate; hence, all g_{i}𝔄̃_{1}g_{i}^{−1} must be different. Thus, there exists a k ∈ {1, …, t} such that (g_{k}^{−1}𝔄g_{k}, g_{k}^{−1}𝔄_{1}g_{k}) = (𝔄^{(k)}, 𝔄̃_{1}), as required.

(iii) By conjugation in A^{×}, (ii), with 1.1 in the case of E = F and the first assertion of 1.2(iv), we can restrict ourselves to the case where 𝔄 and 𝔄_{1} are given in normal form 0.3, and our assertion follows immediatly from the explicit description of the chain {X_{j}}_{j∈ℤ} in the simple A-module V = D^{m}.

□

1.6 Definition

(i) The pair (𝔄̃, 𝔄̃_{1}) as in 1.5(i) is obviously uniquely determined by the invariant tuple ((n_{1}, …, n_{t}), \frac{r}{t}) and our fixed isomorphism A ≃ M_{m}(D). Here we only consider tuples of the form ((k_{1}, …, k_{u}), κ), where k_{1}, …, k_{u} is a sequence of integers with minimal period u and m = (k_{1} + … + k_{u})κ; hence, κ is just the number of repetitions for which the sequence k_{1}, …, k_{u} appears in the invariants of the associated hereditary order, and k_{1} + … + k_{u} is the invariant of the associated canonical principal order. We call ((n_{1}, …, n_{t}), \frac{r}{t}) the invariants of the pair (of the conjugacy class of the pair, respectively).

(ii) A pair (𝔄, 𝔄_{1}) always means a pair in the sense of 1.5(ii).

\underline{Remark}: Note that the order in the invariant tuple ((n_{1}, …, n_{t}), \frac{r}{t}) is fixed but, in general, the invariants of a hereditary order 𝔄 are just unique up to cyclic permutation. Moreover, by the Skolem-Noether Theorem the invariants of a pair are independent of the fixed isomorphism A ≃ M_{m}(D).

1.7 Definition Let (𝔄, 𝔄_{1}) be a pair in A, E/F an extension in A, B := C_{A}(E) and (𝔅, 𝔅_{1}) a pair in B.

(i) We say that (𝔄, 𝔄_{1}) is a continuation of (𝔅, 𝔅_{1}) if 𝔄 continues 𝔅 and 𝔄_{1} continues 𝔅_{1}. (ii) Let ((n′_{1}, …, n′_{t′}), \frac{r′}{t′}) be the invariants of the pair (𝔅, 𝔅_{1}). We say that the pairs (𝔄, 𝔄_{1}) are

numerically proportional if there exists l_{1} ∈ ℕ and l_{2} ∈ ℚ_{>0} such that the pair (𝔄, 𝔄_{1}) has invariants ((n_{1}, …, n_{t}), \frac{r}{t}) = ((n′_{1}l_{1}, …, n′_{t′}l_{1}), \frac{r′}{t′} l_{2})

(iii) We call the pairs (𝔄, 𝔄_{1}), (𝔅, 𝔅_{1}) proportional if (𝔄, 𝔄_{1}) is a continuation of as well as numerically proportional to (𝔅, 𝔅_{1}).

1.8 Corollary (i) If the pairs (𝔄, 𝔄_{1}) and (𝔅, 𝔅_{1}) are numerically proportional and if we let s := n_{1} + … + n_{t} be the invariant of 𝔄_{1}, s′ := n′_{1} + … + n′_{t′} the invariant of 𝔅_{1}, where ((n′_{1}, …, n′_{t′}), \frac{r′}{t′}) (resp. ((n_{1}, …, n_{t}), \frac{r}{t}) = ((n′_{1}l_{1}, …, n′_{t′}l_{1}), \frac{r′}{t′} l_{2})) are the invariants of (𝔅, 𝔅_{1}) (resp. (𝔄, 𝔄_{1})) as in 1.7, then l_{1} = \frac{s}{s′}.

(ii) If (𝔄, 𝔄_{1}) continues (𝔅, 𝔅_{1}), then \frac{s}{s′} = ⟨f(E/F), \frac{r′}{t′} \frac{[E:F]}{⟨[E:F],d⟩}⟩. (iii) If the pairs (𝔄, 𝔄_{1}), (𝔅, 𝔅_{1}) are proportional, then l_{1} = ⟨f(E/F), \frac{r′}{t′} \frac{[E:F]}{⟨[E:F],d⟩}⟩ and l_{1}l_{2} = \frac{m}{m′} =

\frac{[E:F]}{⟨[E:F],d⟩}.

\underline{Proof}

(i) l_{1} = \frac{n′_{1}l_{1}+...+n′_{t′} l_{1}}{n′_{1}+...+n′_{t′}} = \frac{n_{1}+...+n_{t}}{n′_{1}+...+n′_{t′}} = \frac{s}{s′}. (ii) From [Z] (Corollary 3.(ii)) we conclude that \frac{s}{s′} = \frac{⟨s′f(E/F),m⟩}{s′} = ^{\frac{⟨s′f(E/F),m′ \frac{[E:F]}{⟨[E:F],d⟩} ⟩}{s′}} =

⟨f(E/F), \frac{r′}{t′} \frac{[E:F]}{⟨[E:F],d⟩}⟩.

(iii) Follows from (i),(ii) and 0.5. □

1.9 Theorem (main result) Let A, d, E/F and B be as in 0.5.

(i) Let (𝔅, 𝔅_{1}) be a pair in B. Then there exists a unique continuation (𝔄, 𝔄_{1}) to A. Furthermore, for this continuation we have

(a) (𝔄, 𝔄_{1}) and (𝔅, 𝔅_{1}) are numerically proportional and

(b) for the proportional factor l_{1} as in 1.8(iii) and all h ∈ K_{𝔅} we have ν_{𝔄}(h) = \frac{f(E/F)}{l_{1}}ν_{𝔅}(h). (ii) Let (𝔄, 𝔄_{1}) be a pair in A. Then the following assertions are equivalent:

(a) (𝔄 ∩ B, 𝔄_{1} ∩ B) is a pair in B continued by (𝔄, 𝔄_{1}).

(b) There exist field extensions M/K/E in A such that M^{×} ⊂ K_{𝔄_{1}}, K^{×} ⊂ K_{𝔄}, e(K/F) = e(M/F), [M : F] = md (M/F is of maximal degree in A) and d | e(M/F)f(E/F).

(iii) Let (𝔄, 𝔄_{1}) be a pair in A with invariants ((n_{1}, …, n_{t}), \frac{r}{t}). Then the following assertions are equivalent:

(a) There exists an embedding E ↪ A over F and field extensions M/K/E of the image of E in A such that M^{×} ⊂ K_{𝔄_{1}}, K^{×} ⊂ K_{𝔄}, e(K/F) = e(M/F), [M : F] = md (M/F is of maximal degree in A) and d | e(M/F)f(E/F).

(b) [E : F] | d\frac{r}{t} ⟨n_{1}, …, n_{t}⟩, e(E/F) | d\frac{r}{t} and f(E/F) | d̈⟨n_{1}, …, n_{t}⟩, where d̈ is the greatest divisor of d that is prime to \frac{r}{t}.

The proof of this theorem will be given in 4.

2 Continuation in the splittable case of ⟨f(E/F), d⟩ = 1

2.1 Proposition Let E/F, B, 𝔅, r′, t′ be as in 0.5, further let 𝔅_{1},…,𝔅_{t′} be the canonical principal orders of 𝔅 and 𝔄_{1},…,𝔄_{t′} the unique continuations as in 1.4(ii). Furthermore assume that the continuations of the 𝔅_{i}′s by the 𝔄_{i}′s are all unramified, then, if 𝔄 := 𝔄_{1} ∩ … ∩ 𝔄_{t′}, 𝔄 is a hereditary 𝔬_{F} -order in A and a continuation of 𝔅.

\underline{Remark}: One would like to show here that in this case the 𝔄_{1}, …, 𝔄_{t′} are the canonical principal orders of 𝔄. For that it remains to show that t′ is the partition period of 𝔄. This will be done later in the cases 2.2 and 3.2.

\underline{Proof}

Because of 1.2, we have 𝔄 ∩ B = ⋂_{i=1}^{t′} 𝔄_{i} ∩ B = ⋂_{i=1}^{t′} 𝔅_{i} = 𝔅. As ⋂_{i=1}^{t′} K_{𝔄_{i}} ⊂ K_{⋂_{i=1}^{t′} 𝔄_{i}} in general,

K_{𝔄} ∩ B ⊃ ⋂_{i=1}^{t′} K_{𝔄_{i}} ∩ B = ⋂_{i=1}^{t′} K_{𝔅_{i}} = K_{𝔅}. Thus, it is only necessary to show that 𝔄 is a hereditary order in A.

As a consequence of 1.2(ii)(b)(iv) and because the continuations of the 𝔅_{i}′s by the 𝔄_{i}′s are unramified by assumption,

there exists an h ∈ K_{𝔅} such that for i = 1, …, t′, K_{𝔄_{i}} = ⟨h⟩𝔄_{i}^{×}.  (3)

𝔅_{1}, …, 𝔅_{t′} have the same invariants; hence, there exist g_{1}, …, g_{t′} ∈ B^{×} such that

g_{1}𝔅_{1}g_{1}^{−1} = 𝔅_{2}, …, g_{t′−1}𝔅_{t′−1}g_{t′−1}^{−1} = 𝔅_{t′} and g_{t′}𝔅_{t′}g_{t′}^{−1} = 𝔅_{1}.  (4)

Furthermore, 1.1 and [F]((2.7), Corollary 3) imply that g_{i}𝔄_{i}g_{i}^{−1} is the unique continuation of g_{i}𝔅_{i}g_{i}^{−1} for i = 1, …, t′; hence,

g_{1}𝔄_{1}g_{1}^{−1} = 𝔄_{2}, …, g_{t′−1}𝔄_{t′−1}g_{t′−1}^{−1} = 𝔄_{t′} and g_{t′}𝔄_{t′}g_{t′}^{−1} = 𝔄_{1}.  (5)

If, for i = 1, …, t′, {Y_{j}^{i}}_{j∈ℤ} is the chain of 𝔅_{i} in a simple B-module W, then 0.2 (4) implies {g_{i}Y_{j}^{i}}_{j∈ℤ} = {Y_{j}^{i+1}}_{j∈ℤ} and {g_{t′}Y_{j}^{t′}}_{j∈ℤ} = {Y_{j}^{1}}_{j∈ℤ} for i = 1, …, t′ − 1. Now, let {Y_{j}}_{j∈ℤ} be the chain of 𝔅 in W and index the orders 𝔅_{1},…,𝔅_{t′} and the associated chains in such a way that Y_{j}^{i} = Y_{(i−1)+jt′} for i = 1, …, t′ and all j ∈ ℤ. So, by changing the g_{1}, …, g_{t′} via powers of h, one can obtain that g_{i}Y_{j}^{i} = Y_{j}^{i+1} and g_{t′}Y_{j}^{t′} = Y_{j+1}^{1} for i = 1, …, t′ − 1 and all j ∈ ℤ. In particular, we have g_{i} ∈ {y ∈ B | ∀j ∈ ℤ, yY_{j}^{i} ⊂ Y_{j}^{i}} = 𝔅_{i} for i = 1, …, t′. So, by changing g_{1} by an element of 𝔅_{1}^{×}, one can obtain g_{t′}…g_{1} = h. The conditions (5) are stable under all these changes.

Now let {X_{j}^{i}}_{j∈ℤ} be the chain of 𝔄_{i} in a simple A-module V for i = 1, …, t′. Then, again,

(5) means {g_{i}X_{j}^{i}}_{j∈ℤ} = {X_{j}^{i+1}}_{j∈ℤ} and {g_{t′}X_{j}^{t′}}_{j∈ℤ} = {X_{j}^{1}}_{j∈ℤ}. Hence, because g_{i} ∈ 𝔅_{i} ⊂ 𝔄_{i} for i = 1, …, t′, ∀j ∈ ℤ, X_{j}^{1} ⊃ g_{1}X_{j}^{1} = X_{j}^{2} ⊃ … ⊃ g_{t′}X_{j}^{t′} = g_{t′} … g_{1}X_{j}^{1} = hX_{j}^{1} = X_{j+1}^{1}.

Therefore ⋃_{i=1}^{t′}{X_{j}^{i}}_{j∈ℤ} is an 𝔬_{D}-lattice chain in V; hence, 𝔄 = 𝔄_{1} ∩…∩𝔄_{t′} = ⋂_{i=1}^{t′} End_{𝔬_{D}}{X_{j}^{i}}_{j∈ℤ} =

End_{𝔬_{D}} ⋃_{i=1}^{t′}{X_{j}^{i}}_{j∈ℤ} is a hereditary order.

□

2.2 Theorem Let E/F, A, d, B be as in 0.5 and ⟨f(E/F), d⟩ = 1. Then

(i) For 𝔅 being a hereditary 𝔬_{E}-order in B, there exists a unique hereditary continuation 𝔄 to A. For this continuation we have

(a) t = t′, r = r′ \frac{e(E/F)}{⟨e(E/F),d⟩} and n_{i} = n′_{i}f(E/F) for i = 1, …, t, where n′_{1}, …, n′_{r′}, r′, t′ are the invariants of 𝔅 and n_{1}, …, n_{r}, r, t are the invariants of 𝔄 as in 0.5,

(b) 𝔓^{j} ∩ B = 𝔔^{j} for all j ∈ ℤ, where 𝔔 is the Jacobson radical of 𝔅 and 𝔓 is the Jacobson radical of 𝔄.

(ii) Let 𝔄 be a hereditary 𝔬_{F} -order in A. Then the following statements are equivalent:

(a) 𝔄 ∩ B is a hereditary 𝔬_{E}-order in B continued by 𝔄.

(b) E^{×} ⊂ K_{𝔄}.

(iii) Let 𝔄 be a hereditary 𝔬_{F} -order in A and n_{1}, …, n_{r}, r, t all invariants of 𝔄 as in 0.5. Then the following statements are equivalent:

(a) There exists an embedding j of E over F into A such that j(E^{×}) ⊂ K_{𝔄}.

(b) f(E/F) | ⟨n_{1}, …, n_{t}⟩ and e(E/F) | d\frac{r}{t}.

\underline{Remark}: 1) As we use 2.1 for the existence part of (i), it follows from the remark to 2.1 and (i)(a) that under the assumptions of (i) all canonical principal orders of 𝔅 are continued by canonical principal orders of 𝔄.

2) (iii) is a direct generalization of [BK2](1.2), and (ii)(iii) is a stronger form of [F](Theorem 3(ii)) in the case of a principal order.

\underline{Proof}

(i) Let 𝔅_{1}, …, 𝔅_{t′} be the canonical principal orders of 𝔅 in B and 𝔄_{1}, …, 𝔄_{t′} their unique continuations to A. Choose an unramified extension L/F of degree d in D, then because ⟨f(E/F), d⟩ = 1, E ⊗_{F} L/F ⊗_{F} L is a field extension in the split algebra A ⊗_{F} L and B ⊗_{F} L = C_{A⊗_{F} L}(E ⊗_{F} L). Clearly, the 𝔄_{i} ⊗_{𝔬_{F}} 𝔬′_{L}s are continuations of the 𝔅_{i} ⊗_{𝔬_{F}} 𝔬_{L} and by [BK3](1.2.4) these continuations are unramified. By [BT](4.4) 𝔅 ⊗_{𝔬_{F}} 𝔬_{L} has the Jacobson radical 𝔔 ⊗_{𝔬_{F}} 𝔬_{L} and, therefore, a common uniformizer with 𝔅 if we consider B to be canonically embedded in B ⊗_{F} L. The same holds true for the relation of the 𝔅_{i}′s and 𝔅_{i} ⊗_{𝔬_{F}} 𝔬_{L}′s and, finally, for the relation of the 𝔄_{i}′s and 𝔄_{i} ⊗_{𝔬_{F}} 𝔬_{L}′s. We deduce that the continuation of the 𝔅_{i}′s by the 𝔄_{i}′s must be unramified and the existence of 𝔄 now follows from 2.1.

To see that it is unique let V, {X_{j}}_{j∈ℤ} be as in 0.5. Then, again by [CR](26.29) or [BT](4.4), 𝔄⊗_{𝔬_{F}} 𝔬_{L} is a hereditary 𝔬_{L}-order in A⊗_{F} L ≃ End_{L}(V) ≃ M_{md}(L) and 𝔄⊗_{F} L∩B ⊗_{F} L = 𝔅⊗_{𝔬_{F}} 𝔬_{L}. Moreover, 𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L} ⊂ 𝔄 ⊗_{𝔬_{F}} 𝔬_{L} and the prime element of E ⊗_{F} L is π_{E⊗_{F} L} = π_{E} ⊗ 1 ∈ K_{𝔄} ⊗ 1 ⊂ K_{𝔄⊗_{𝔬_{F}} 𝔬_{L}}; hence, (E ⊗_{F} L)^{×} =< π_{E⊗_{F} L} > (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L})^{×} ⊂ K_{𝔄⊗_{𝔬_{F}} 𝔬_{L}}. Consequently [BK3](1.2.1) implies that 𝔄 ⊗_{𝔬_{F}} 𝔬_{L} and 𝔅 ⊗_{𝔬_{F}} 𝔬_{L} have the same chain {X_{j}}_{j∈ℤ} in the simple A ⊗_{F} L- resp. B ⊗_{F} L-module V. Hence, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, and therefore 𝔄, is unique.

(i)(b) Again from [BT](4.4) we know that Jac(𝔄 ⊗_{𝔬_{F}} 𝔬_{L}) = 𝔓 ⊗_{𝔬_{F}} 𝔬_{L} and Jac(𝔅 ⊗_{𝔬_{F}} 𝔬_{L}) = 𝔔⊗_{𝔬_{F}} 𝔬_{L}; hence, we see from [BK3](1.2.4) that, for all j ∈ ℤ, 𝔓^{j} ∩ B ⊗_{𝔬_{F}} 𝔬_{L} = 𝔓^{j} ⊗_{𝔬_{F}} 𝔬_{L} ∩ B ⊗_{F} L = Jac(𝔄 ⊗_{𝔬_{F}} 𝔬_{L})^{j} ∩ B ⊗_{F} L = Jac(𝔅 ⊗_{𝔬_{F}} 𝔬_{L})^{j} = 𝔔^{j} ⊗_{𝔬_{F}} 𝔬_{L}. Thus, for all j ∈ ℤ, 𝔓^{j} ∩ B = 𝔔^{j}, as required.

(i)(a) Because of 0.2 and 0.3(1), it is obvious that 𝔄 and 𝔄 ⊗_{𝔬_{F}} 𝔬_{L} have the same chain {X_{j}}_{j∈ℤ} in the simple A- resp. A ⊗_{F} L-module V. Therefore, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L} has the same partition period as 𝔄, period rd, because the right action of π_{D}^{d} has the same period on the chain as the left or right action of π_{F} and, moreover, the invarints are n_{1}, …, n_{r}, …, n_{1}, …, n_{r} (d-times), because dim_{𝔨_{L}}(X_{i−1}/X_{i}) = dim_{𝔨_{D}}(X_{i−1}/X_{i}) for i = 1, …, rd. Now 𝔅 ⊗_{𝔬_{F}} 𝔬_{L}, as in the proof of the first assertion of (i), has also the same partition period, period \frac{rd}{e(E/F)}, because the left action of π_{E}^{e(E/F)} has the same period on the chain as the left or right action of π_{F} and, moreover, the invariants are \frac{n_{1}}{f(E/F)}, …, \frac{n_{t}}{f(E/F)}, …, \frac{n_{1}}{f(E/F)}, …, \frac{n_{t}}{f(E/F)} (\frac{rd}{e(E/F)t}-times), because dim_{𝔨_{E⊗_{F} L}}(X_{i−1}/X_{i}) = \frac{dim_{𝔨_{L}} (X_{i−1}/X_{i})}{f(E/F)} for i = 1, …, \frac{rd}{e(E/F)}.

Next we consider the transformation of the invariants from 𝔅 to 𝔅 ⊗_{𝔬_{F}} 𝔬_{L}. First note that 𝔅 ⊗_{𝔬_{F}} 𝔬_{L} ≃ 𝔅 ⊗_{𝔬_{E}} (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L}) ≃ (𝔅 ⊗_{𝔬_{E}} (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L′})) ⊗_{𝔬_{E}⊗_{𝔬_{F}} 𝔬_{L′}} 𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L}, where L/L′/F is an intermediate extension of degree d′ = \frac{d}{⟨[E:F],d⟩} = \frac{d}{⟨e(E/F),d⟩} over F. Hence, because B ⊗_{F} L′ ≃ M_{m′d′}(L′) (where m′, d′ are as in 0.5), one sees from 0.3 that 𝔅 ⊗_{𝔬_{E}} (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L}) and 𝔅 ⊗_{𝔬_{E}} (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L′}) have the same invariants. Again, we see that 𝔅 and 𝔅 ⊗_{𝔬_{E}} (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L′}) have the same chain {Y_{j}}_{j∈ℤ} in a simple B resp. B ⊗_{E} (E ⊗_{F} L′)-module W; hence, again 𝔅 ⊗_{𝔬_{E}} (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L′}) has the same partition period as 𝔅, the period being r′d′, because the right action of πd′ ; D̃ has the same period on the chain as the left or right action of π_{E} and, moreover, the invariants are n′_{1}, …, n′_{r′}, …, n′_{1}, …, n′_{r′} (d′-times), because dim_{𝔨_{E⊗_{F} L′}} (Y_{i−1}/Y_{i}) = dim_{𝔨D̃}(Y_{i−1}/Y_{i}) for i = 1, …, r′d′.

Summarizing we obtain the required relations for 𝔄 and 𝔅.

(ii) (a)⇒(b) E^{×} ⊂ K_{𝔄∩B} ⊂ K_{𝔄}.

(b)⇒(a) As in the proof of (i) we see that (E ⊗_{F} L)^{×} ⊂ K_{𝔄⊗_{𝔬_{F}} 𝔬_{L}} and that 𝔄 ⊗_{𝔬_{F}} 𝔬_{L} ∩ B ⊗_{F} L ≃ 𝔄 ∩ B ⊗_{𝔬_{E}} (𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L}) is hereditary; therefore, by [CR](26.29), 𝔄 ∩ B is hereditary. 𝔄 ∩ B has a unique continuation 𝔄′ to A and, clearly, again E^{×} ⊂ K_{𝔄′}. However, in the same situation the latter condition was sufficient to prove in (i) that 𝔄 is unique; hence, 𝔄 = 𝔄′, as required.

(iii) (a)⇒(b) Follows from (ii) and the relation of the invariants in (i)(a). Here we must note that the partition period is always a divisor of the period of an order.

(b)⇒(a) We embed E first arbitrarily over F into A. If the relations of the invariants hold, there exists a hereditary order ℭ in C := C_{A}(E) with the same invariants as 𝔅 in (i)(a). ℭ has a continuation 𝔄′ to A with the same invariants as 𝔄; hence, by 0.3, there exist g ∈ A^{×} such that g𝔄′g^{−1} = 𝔄. By 1.1 we have (gEg^{−1})^{×} ⊂ K_{𝔄}; hence, if we change our first embedding according to the adjoint action of g, we obtain an embedding j, as required.

□

3 Continuation of pairs in the case of an unramified extension

in the division algebra D

3.1 Lemma Let E ⊂ L ⊂ D, L/E/F unramified extensions and L/F maximal in D (i.e., [L : F] = d). Let A = M_{m}(D), B = M_{m}(D̃) = C_{A}(E); hence, D̃ = C_{D}(E). Let 𝔄 be a hereditary 𝔬_{F} -order in A and {X_{j}}_{j∈ℤ} its chain, as in 0.5, in V = D^{m} (column vectors) and 𝔅 := 𝔄 ∩ B.

(i) 𝔅 is a hereditary 𝔬_{E}-order in B.

(ii) Let f:=[E:F], {Y_{j}}_{j∈ℤ} be the chain of 𝔅 in W = D̃^{m} (column vectors) and π be a prime element of D which normalizes L under the adjoint action. Then we have a decomposition V = ⨁_{i=1}^{f} W π^{i−1} of V as a (D̃,B)-bimodule and, under this decomposition, X_{j} decomposes for all j ∈ ℤ into a direct sum of indecomposable (𝔅,𝔬D̃)-bimodules to X_{j} = Y_{j}^{(1)} ⊕ Y_{j}^{(2)}π ⊕ … ⊕ Y_{j}^{(f)}π^{f−1}, where {Y_{j}^{(i)} | j ∈ ℤ} = {Y_{j} | j ∈ ℤ} for i = 1, …, f.

\underline{Proof}

We give here a quick proof of these assertions, which are just special versions of the more general results of [Br].

E ⊗_{F} L is a commutative semisimple algebra over L isomorphic to the direct sum of f copies of L. This gives us a decomposition 1_{A} = e_{1} + … + e_{f} of the unit element of the algebra A (resp. A ⊗_{F} L) into a direct sum of central primitive idempotents due to the decomposition E ⊗_{F} L ≃ ⨁_{i=1}^{f} E_{i} (E_{i} ≃ L for i = 1, …, f) into simple components. On each E ⊗_{F} L-module the e_{i} act as projectors to the isotypic components, which are the sum of all simple submodules of the same isomorphy class of simple E ⊗_{F} L-modules. It is easy to see that L is an E ⊗_{F} L-module in f non-isomorphic ways via (e ⊗ l)l′ := el′σ^{i}(l) (i ∈ {1, …, f}), where σ is the generating element of Gal(L/F) induced by the adjoint action of π on L. Therefore, L, Lπ, …, Lπ^{f−1} comprise a full set of simple E ⊗_{F} L-modules and the isotypic components of D (as an E ⊗_{F} L-module via (e ⊗ l)v := evl) are D̃, D̃π, …, D̃π^{f−1}; hence,

V = ⨁_{i=1}^{f} e_{i}V = ⨁_{i=1}^{f} W π^{i−1},  (6)

as required.

Clearly, X_{j} = ⨁_{i=1}^{f} e_{i}X_{j} = ⨁_{i=1}^{f} Y_{j}^{(i)}π^{i−1} for all j ∈ ℤ, where the Y_{j}^{(i)} are indecomposable 𝔅-modules. Clearly 𝔅 is an o_{E}-order in B. Since π^{f} is a prime element of D̃, its right action maps the inclusion ordered sets {Y_{j}^{(i)} | j ∈ ℤ} bijectively to themselves (with inverse π^{−f}), and the action of π permutes the components of (6). We conclude that the inclusion ordered sets {Y_{j}^{(i)} | j ∈ ℤ} are all pairwise equal and form 𝔬D̃-lattice chains in W.

Finally, by 0.2, 𝔅 is hereditary.

□

3.2 Lemma Let E/F, B, 𝔅, r′, t′ be as in 0.5. Assume further that E/F is unramified and [E : F] | ⟨\frac{r′}{t′} , d⟩. Let 𝔅_{1}, …, 𝔅_{t′} be the canonical principal orders of 𝔅 and 𝔄_{1}, …, 𝔄_{t′} their unique continuations, then 𝔄 := 𝔄_{1} ∩…∩𝔄_{t′} is a hereditary order with canonical principal orders 𝔄_{1}, …, 𝔄_{t′}. Moreover, each pair (𝔄, 𝔄_{i}) is numerically proportional to the pair (𝔅, 𝔅_{i}) for i = 1, …, t′.

\underline{Proof}

Note that we fixed an isomorphism A ≃ M_{m}(D) and consider D to be canonically embedded into M_{m}(D). By 1.1 and conjugation in A^{×}, we can restrict ourselves to the case of E ⊂ D. Let f := [E : F], V := D^{m}, D̃ := C_{D}(E), W := D̃^{m} and {Y_{j}}_{j∈ℤ} be a lattice chain of 𝔅 in W. Set for all j ∈ ℤ and i = 1, …, t′: Y_{j}^{i} := Y_{(i−1)+jt′}. Let n′_{1}, …, n′_{r′} be the invariants of 𝔅 and s′ := n′_{1} + … + n′_{t′}. Assume that the index of the chain {Y_{j}}_{j∈ℤ} was chosen in such a way that dim_{𝔨D̃}(Y_{0}/Y_{1}) = n′_{1}.

We construct the chains of the 𝔄′_{i}s explicitly and will see that 𝔄 has the same partition period as 𝔅. The result will then follow from 1.2(iv) (see also the remark to 2.1). Recall the canonical decomposition (6) in 3.1 and let

X_{j}^{i} := ⨁_{k=0}^{f−1} Y_{j− \frac{r′}{t′f} k}^{i}π^{k}.  (7)

We want to show that the sequences {X_{j}^{i}}_{j∈ℤ} are the lattice chains of the 𝔄′_{i}s for i = 1, …, t′.

As π^{f} is a prime element of 𝔬D̃, one sees that the \{X_{j}^{i}\}_{j∈ℤ} are lattice chains with period \frac{r′}{t′f} in V, and if we set 𝔄′_{i} := End_{𝔬_{D}}{X_{j}^{i}}_{j∈ℤ}, clearly 𝔄′_{i} ∩ B = 𝔅_{i}. From [BF](1.2.8 (i)) and because K_{𝔅} acts componentwise with respect to the decomposition (7), one obtains that the 𝔄′′_{i}s are continuations of the 𝔅′_{i}s; hence, by 1.4(ii) 𝔄′_{i} = 𝔄_{i} for i = 1, …, t′, as required.

Moreover, again by [BF](1.2.8 (i)), a uniformizer of 𝔅_{i} acts by definition of the chains as a uniformizer of 𝔄_{i}, too; hence, the 𝔄_{i}′s are unramified continuations of the 𝔅_{i}′s.

We know from the proof of 2.1 that the inclusion ordered set ⋃_{i=1}^{f}{X_{j}^{i} | j ∈ ℤ} can be indexed as a lattice chain for 𝔄 in V, and it is easy to see that this is possible in such a way that X_{i−1} := X_{0}^{i} for i = 1, …, t′ and X_{t′} := X_{1}^{1}; hence, we obtain for the first numbers of the invariants of 𝔄 that n_{i} =

dim_{𝔨_{D}}(X_{i−1}/X_{i}) = dim_{𝔨_{D}}(X_{0}^{i}/X_{0}^{i+1}) = (resp. dim_{𝔨_{D}}(X_{0}^{t′}/X_{1}^{1}) =) ∑_{k=0}^{f−1} dim_{𝔨D̃}(Y_{0− \frac{r′}{t′f} k}^{i}/Y_{0− \frac{r′}{t′f} k}^{i+1}) =

(resp. ∑_{k=0}^{f−1} dim_{𝔨D̃}(Y_{0− \frac{r′}{t′f} k}^{t′}/Y_{1− \frac{r′}{t′f} k}^{1}) =) n′_{i}f for i = 1, … t′ − 1 (resp. for i = t′). Therefore, by 1.2(i) the partition period of 𝔄 must be at least t′. In the situation of 2.1 with the condition of the last assertion of 1.2(iv), we obtain that this can be at most t′, which finishes the proof by the remark following 2.1.

□

3.3 Lemma Let A, d, E/F, B be as in 0.5 and 𝔅 be a hereditary 𝔬_{E}-order in B with canonical principal orders 𝔅_{1}, …, 𝔅_{t′} and period r′. Assume that E/F is unramified, [E : F] | d and ⟨[E : F], \frac{r′}{t′} ⟩ = 1. Then, to a fixed canonical principal order 𝔅_{1}, there exists a pair (𝔄, 𝔄_{1}) in A which is proportional to (𝔅, 𝔅_{1}).

\underline{Proof}

Because of 1.1, we can restrict ourselves to the case where E ⊂ D and (𝔅, 𝔅_{1}) is given in simultaneous normal form 1.5. Here we identify A with M_{m}(D) and B with M_{m}(D̃), where D̃ = C_{D}(E). Let ((n′_{1}, …, n′_{t′}), \frac{r′}{t′}) be the invariants of (𝔅, 𝔅_{1}) and 𝔄_{1} be the unique continuation of 𝔅_{1} as in 1.4(ii). Let s′ := n′_{1} + … + n′_{t′} and s be the invariant of 𝔄_{1}. Then, by the assumptions on our field extension E/F and the calculations in 1.8(ii), we see that \frac{s}{s′} = 1; hence,

𝔄_{1} and 𝔅_{1} have the same invariant.  (8)

Let f := [E : F], {X_{j}^{1}}_{j∈ℤ} be the chain of 𝔄_{1} in V = D^{m} and {Y_{j}^{1}}_{j∈ℤ} be the chain of 𝔅_{1} in W = D̃^{m}. Then, by 3.1(ii) and with its notations, there exist Y_{j}^{(1)}, …, Y_{j}^{(f)} such that X_{j}^{1} = ⨁_{i=1}^{f} Y_{j}^{(i)}π^{i−1} for all j ∈ ℤ, where this is a decomposition into a direct sum of indecomposable (𝔅_{1}, 𝔬D̃)-bimodules due to (6).

Let

h := ( \begin{matrix}0 , & π^{f}I_{s′} \\ I_{m−s′} , & 0 \end{matrix} )

be the usual uniformizer for 𝔅_{1}, i. e., h ∈ K_{𝔅_{1}} and ν_{𝔅_{1}}(h) = 1. Then there exist functions [.]_{i} : ℤ → ℤ for i = 1, …, f such that X_{j}^{1} = ⨁_{i=1}^{f} h^{[j]_{i}}Y_{0}^{(i)}π^{i−1}. Because of (8), we obtain s′ = dim_{𝔨_{D}}(X_{j}/X_{j+1}) =

∑_{i=1}^{f} dim_{𝔨D̃}(h^{[j]_{i}}Y_{0}^{(i)}/h^{[j+1]_{i}}Y_{0}^{(i)}) for all j ∈ ℤ; hence, because of the uniqueness of the decomposition (6) and since the chain {X_{j}^{1}}_{j∈ℤ} is ordered by inclusion, we conclude that

for j ∈ ℤ, there exists an i_{j} ∈ {1, …, f} such that [j + 1]_{i_{j}} = [j]_{i_{j}} + 1 and [j + 1]_{i} = [j]_{i} for i ≠ i_{j}.

(9)

Since both 𝔅_{1} and 𝔄_{1} have period \frac{r′}{t′} and π^{f} is a prime element of 𝔬D̃, X_{j}^{1}π = X_{j+ \frac{r′}{t′}}^{1} =

⨁_{i=1}^{f} h^{[j+ \frac{r′}{t′}]_{i}}Y_{0}^{(i)}π^{i−1}, Y_{j}^{1}π^{f} = Y_{j+ \frac{r′}{t′}}^{1} and, thus, X_{j}^{1}π = ⨁_{i=1}^{f} h^{[j]_{i}}Y_{0}^{(i)}π^{i} = Y_{0}^{(f)}π^{f} ⨁_{i=1}^{f−1} h^{[j]_{i}}Y_{0}^{(i)}π^{i}. Com-

paring the coefficients one sees from the action of π on X_{j}^{1} that

[j + \frac{r′}{t′}]_{1} = [j]_{f} + \frac{r′}{t′} , [j + \frac{r′}{t′}]_{2} = [j]_{1}, …, [j + \frac{r′}{t′}]_{f} = [j]_{f−1}.  (10)

Let (𝔄′, 𝔄′_{1}) be the pair in A which is given in normal form 1.5, and let {X_{j}^{′1}}_{j∈ℤ} be the chain of 𝔄′_{1} in V = D^{m}. We can index the chain such that ^{t}X_{j}^{′1} = 𝔭_{D}^{k}(( ⨁_{ν=1}^{μs′} 𝔭_{D}) ⊕ ( ⨁_{ν=1}^{m−μs′} 𝔬_{D})), where j ∈ ℤ and where k ∈ ℤ, 0 ≤ μ < \frac{r′}{t′} are such that j = k \frac{r′}{t′} + μ.

Next, we show the existence of g ∈ A^{×} such that g𝔄_{1}g^{−1} = 𝔄′_{1}, where g = P M is an \frac{r′}{t′} × \frac{r′}{t′} -(s′×s′)- block monomial matrix, which means that M = diag(π^{k_{1}}I_{s′}, …, π^{k_{\frac{r′}{t′}}} I_{s′}), where k_{1}, …, k_{\frac{r′}{t′}} ∈ ℤ and P = (p_{i,j}I_{s′})_{i,j=1,..., \frac{r′}{t′}}, where (p_{i,j})_{i,j=1,..., \frac{r′}{t′}} is an \frac{r′}{t′} × \frac{r′}{t′} -permutation matrix. By (10) one sees that h^{\frac{r′}{t′}} Y_{0}^{(f)} = h^{[ \frac{r′}{t′}]_{1}}Y_{0}^{(1)}, Y_{0}^{(1)} = h^{[ \frac{r′}{t′}]_{2}}Y_{0}^{(2)}, …, Y_{0}^{(f−1)} = h^{[ \frac{r′}{t′}]_{f}} Y_{0}^{(f)}; hence, for all k ∈ ℤ,

Y_{0}^{(2)} = h^{k}h^{−[ \frac{r′}{t′}]_{2}}Y_{0}^{(1)}, …, Y_{0}^{(f)} = h^{k}h^{− ∑_{i=2}^{f}[ \frac{r′}{t′}]_{i}}Y_{0}^{(1)}. For k ∈ ℤ such that h^{k}Y_{0}^{(1)} = ⨁_{i=1}^{m} 𝔭D̃ and noting

that 𝔬_{D} = 𝔬D̃+𝔬D̃π+…+𝔬D̃π^{f−1}, 𝔭_{D} = 𝔭D̃+𝔬D̃π+…+𝔬D̃π^{f−1}, …, 𝔭_{D}^{f} = 𝔭D̃+𝔭D̃π+…+𝔭D̃π^{f−1}, we

obtain ^{t}(h^{k}X_{0}^{1}) = ⨁_{i=1}^{s′[ \frac{r′}{t′}]_{1}} 𝔭_{D}^{f} ⨁_{i=1}^{s′[ \frac{r′}{t′}]_{f}} 𝔭_{D}^{f−1} … ⨁_{i=1}^{s′[ \frac{r′}{t′}]_{2}} 𝔭_{D}^{1}. As the index in the chain \{X_{j}^{1}\}_{j∈ℤ} can be shifted

and X_{0}^{1} was arbitrarily chosen as a finitely generated direct indecomposable 𝔄_{1}-submodule of V, we

obtain for all j ∈ ℤ the existence of k_{1}^{j}, …, kj_{\frac{r′}{t′}} ∈ ℤ such that ^{t}X_{j}^{1} = (⨁_{i=1}^{s′} 𝔭_{D}^{k_{1}^{j}}) ⊕ … ⊕ (⨁_{i=1}^{s′} 𝔭_{D}^{kj ; \frac{r′}{t′}} ).

Let M := diag(π^{−k_{1}^{0}}I_{s′}, …, π^{−k0 ; \frac{r′}{t′}} I_{s′}). Then MX_{0}^{1} = ⨁_{i=1}^{m} 𝔬_{D} = X_{0}^{′1} and MX_{0}^{1}π = ⨁_{i=1}^{m} 𝔭_{D} = X′1_{\frac{r′}{t′}}.

Since dim_{𝔨_{D}}(X_{j}^{1}/X_{j+1}^{1}) = s′ for all j ∈ ℤ (see (8)), there exists an \frac{r′}{t′} × \frac{r′}{t′} -(s′ × s′)-block permutation matrix P such that P MX_{j}^{1} = X_{j}^{′1} for j = 0, …, \frac{r′}{t′}. Therefore, as X_{j}^{1}π = X_{j+ \frac{r′}{t′}}^{1} for all j ∈ ℤ, {P MX_{j}^{1}}_{j∈ℤ} = {X_{j}^{′1}}_{j∈ℤ}. Consequently, for g := P M we obtain g𝔄_{1}g^{−1} = 𝔄′_{1}, as required.

Clearly, (g^{−1}𝔄′g, g^{−1}𝔄′_{1}g) = (g^{−1}𝔄′g, 𝔄_{1}) is a pair, and it remains to show that 𝔄 := g^{−1}𝔄′g is a continuation of 𝔅. To see this we consider the order 𝔄_{|s′}, which denotes the hereditary order given in normal form 0.3 in A_{s′} = M_{s′}(D) with invariants n′_{1}, …, n′_{t′} (in this order), as well as 𝔅_{|s′}, which denotes the hereditary order given in normal form 0.3 in B_{s′} = M_{s′}(D̃) = C_{A_{s′}} (E) with invariants n′_{1}, …, n′_{t′} (in this order). Because of 𝔬_{D} ∩ D̃ = 𝔬D̃, 𝔭_{D} ∩ D̃ = 𝔭D̃, K_{𝔄_{|s′}} =< π > 𝔄_{|s′}^{×} and K_{𝔅_{|s′}} =< π^{f} > 𝔅_{|s′}^{×}, it is clear that 𝔄_{|s′} is a continuation of 𝔅_{|s′}. As conjugation by \frac{r′}{t′} × \frac{r′}{t′} - (s′ × s′)-block permutation matrices just permutes the diagonal s′ × s′-blocks and conjugates them by certain powers of π, 𝔄 has the same diagonal blocks as 𝔄′ and 𝔄_{1} has the same diagonal blocks as 𝔄′_{1}. Therefore, 𝔄 can be obtained from 𝔄_{1} just by replacing the diagonal (s′ ×s′)-blocks by 𝔄_{|s′}. As also 𝔅 can be obtained from 𝔅_{1} just by replacing the diagonal s′ ×s′-blocks by 𝔅_{|s′}, we see immediately that 𝔄 ∩ B = 𝔅. We also know that conjugation of 𝔄 by an \frac{r′}{t′} × \frac{r′}{t′} -(s′ × s′)-block monomial matrix leads to a permutation action on the diagonal blocks combined with the adjoint action of certain powers of π on each diagonal block. Since h is an \frac{r′}{t′} × \frac{r′}{t′} -(s′ × s′)-block monomial matrix that normalizes 𝔄_{1}, we obtain that h normalizes 𝔄, too. It follows that K_{𝔅} ⊂ K_{𝔄}; hence, 𝔄 is a continuation of 𝔅, as required.

□

3.4 Lemma Let A, d, E/F, B be as in 0.5, and assume that E/F is unramified with [E : F] | d. Let 𝔅_{1} be a principal order with the Jacobson radical 𝔔_{1} and 𝔅 a hereditary order in B such that 𝔅_{1} contains 𝔅. Then there exists at most one pair (𝔄, 𝔄_{1}) in A such that

(a) E^{×} ⊂ K_{𝔄},

(b) 𝔄_{1} is a continuation of 𝔅_{1},

(c) 𝔅 = 𝔄 ∩ B,

(d) there exists an h ∈ K_{𝔄} ∩ K_{𝔅_{1}} such that h𝔅_{1} = 𝔔_{1},

and if that pair exists, then (𝔅, 𝔅_{1}) is a pair, too, and is continued by (𝔄, 𝔄_{1}).

\underline{Proof}

Again taking in account our fixed isomorphism A ≃ M_{m}(D), by 1.1 we can restrict ourselves to the case of E ⊂ D; hence, in particular, we can identify B with M_{m}(D̃), where D̃ = C_{D}(E). Assume that the pair (𝔄, 𝔄_{1}) exists, and denote by ((n_{1}, …, n_{t}), \frac{r}{t}) its invariants. Let n′_{1}, …, n′_{r′}, r′ be the invariants of 𝔅, t′ its partition period, r̃ the period of 𝔅_{1}, and E/K/F an intermediate extension such that [E : K] = ⟨[E : F], r̃⟩. Let L/E be a maximal, unramified extension in D, π be a prime element of 𝔬_{D} that normalizes L, f := [K : F], f′ := [E : K] and C := C_{A}(K) = M_{m}(D̄), where D̄ = C_{D}(K). Then π^{f} is a prime element of 𝔬D̄ and π^{ff′} a prime element of 𝔬D̃. Let V := D^{m}, T := D̄^{m}, W := D̃^{m}, {X_{k}}_{k∈ℤ} be a lattice chain of 𝔄 in V, {X_{k}^{1}}_{k∈ℤ} be a lattice chain of 𝔄_{1} in V, {Y_{k}^{1}}_{k∈ℤ} be a lattice chain of 𝔅_{1} in W and {Y_{k}}_{k∈ℤ} be a lattice chain of 𝔅 in W.

By [Z](Theorem 2), ℭ_{1} := 𝔄_{1} ∩ C is a principal order in C which continues 𝔅_{1} and is continued by 𝔄_{1}. Clearly, h ∈ K_{𝔅_{1}} ⊂ K_{ℭ_{1}} ⊂ K_{𝔄_{1}}, and from [Z](Theorem 2(iii) and Corollary 2(ii)(and its proof)) we obtain ν_{ℭ_{1}}(h) = 1 and ν_{𝔄_{1}}(h) = f; hence, because of ν_{𝔄} = tν_{𝔄_{1}},

ν_{𝔄}(h) | fr.  (11)

We have decompositions

V = T ⊕ T π ⊕ … ⊕ T π^{f−1},  (12)
T = W ⊕ W π^{f} ⊕ … ⊕ W π^{f(f′−1)}  (13)

and associated projections ρ_{i} : V → T π^{i−1} → T for i = 1, …, f, and σ_{i} : T → W π^{f(i−1)} → W for i = 1, …, f′. Let {Z_{k}^{1}}_{k∈ℤ} be a lattice chain of ℭ_{1} in T. Because of 3.1(i), ℭ := 𝔄 ∩ C is a hereditary order, and we denote by {Z_{k}}_{k∈ℤ} its lattice chain in T. From 3.1(ii), we obtain ρ_{i}(X_{j}^{1}) ∈ {Z_{k}^{1}}_{k∈ℤ} and ρ_{i}(X_{j}) ∈ \{Z_{k}\}_{k∈ℤ} for all j ∈ ℤ and i = 1, …, f. As ⟨f, \frac{r̃}{f′} ⟩ = 1 and ℭ_{1} has period \frac{r̃}{f′} (by [F], [Z], or the proof of 3.2), we deduce, as in the proof of 3.3, the existence of (ℭ_{1}, 𝔬D̄)-bimodules Z^{(1)}, …, Z^{(f)} ∈ {Z_{k}^{1}}_{k∈ℤ} and functions [.]_{1}, …, [.]_{f} : ℤ → ℤ such that for all j ∈ ℤ,

X_{j}^{1} = h^{[j]_{1}}Z^{(1)} ⊕ h^{[j]_{2}}Z^{(2)}π ⊕ … ⊕ h^{[j]_{f}} Z^{(f)}π^{f−1},  (14)

where, as in (9), there exists exactly one i_{j} ∈ {1, …, f} for each j ∈ ℤ such that [j + 1]_{i_{j}} = [j]_{i_{j}} + 1 and [j + 1]_{i} = [j]_{i} for all the other i ∈ {1, …, f}\{i_{j}}.

Let M_{j}^{1} := h^{[j]_{i_{j}}} Z^{(i_{j})}. For all j ∈ ℤ there exists exactly one l_{j} such that X_{j}^{1} ⊋ X_{l_{j}} ⊋ X_{l_{j}+1} ⊋ … ⊋ X_{l_{j}+t−2} ⊋ X_{j+1}^{1}. These (𝔄, 𝔬_{D})-bimodules all have the same summands Z^{(i)}π^{i−1} in the component T π^{i−1} of (12) for i ≠ i_{j}, but if we set M_{j}^{u} := ρ_{i_{j}}(X_{l_{j}+u−2}) (u = 2, …, t), then

M_{j}^{1} ⊋ M_{j}^{2} ⊋ … ⊋ M_{j}^{t} ⊋ hM_{j}^{1}.  (15)

Let X_{j}^{u} := X_{l_{j}+u−2} for u = 2, …, t. We have h ∈ K_{𝔄}∩K_{𝔅_{1}} ⊂ K_{𝔄}∩C ⊂ K_{ℭ}. Since {Z_{k}^{1}}_{k∈ℤ} ⊂ {Z_{k}}_{k∈ℤ} and hZ_{j}^{1} = Z_{j+1}^{1} for all j ∈ ℤ, we obtain ν_{ℭ}(h) > 0. Hence, there exist ν̄ ∈ ℕ such that ν_{ℭ}(h) = ν̄t̄, where t̄ is the partition period of ℭ. For the residual fields we have canonical isomorphisms

𝔨_{D} ≃ 𝔨D̄ ≃ 𝔨D̃;  (16)

hence, for i = 1, …, t − 1,

n_{i} = dim_{𝔨_{D}}(X_{j}^{i}/X_{j}^{i+1}) = dim_{𝔨D̄}(M_{j}^{i}/M_{j}^{i+1}) and n_{t} = dim_{𝔨_{D}}(X_{j}^{t}/X_{j+1}^{1}) = dim_{𝔨D̄}(M_{j}^{t}/hM_{j}^{1}) (17)

for all j ∈ ℤ. Since all M_{j}^{1}, hM_{j}^{1} ∈ {Z_{k}^{1}}_{k∈ℤ}, which is a subchain of {Z_{k}}_{k∈ℤ} of period ν̄t̄, and for all α < β we have an isomorphism of 𝔨D̄ vector spaces Z_{α}/Z_{β} ≃ hZ_{α}/hZ_{β}, the dimension (17) determines the subchains of period ν̄t̄ where the modules M_{j}^{i} lie in. Therefore, there exists a number η(i) ∈ {1, …, ν̄t̄} depending only on i for i = 1, …, t such that M_{j}^{i} ∈ {Z_{(η(i)−1)+ν̄t̄k}}_{k∈ℤ} for all j ∈ ℤ.

As the right action of π^{f} on the subchain {X_{k}^{1}}_{k∈ℤ} of {X_{k}}_{k∈ℤ} is componentwise with respect to the decomposition (14) (resp. componentwise on the decomposition of the X_{j} with respect to (12)) and has period fr on {X_{k}}_{k∈ℤ} from (11) and the componentwise left action of h we obtain

ν_{ℭ}(h) | r̄,  (18)

where r̄ is the period of ℭ.

On the other hand, we have

𝔄 ∩ C = {x ∈ C | ∀j ∈ ℤ, i = 1, …, f, xρ_{i}(X_{j}) ⊂ ρ_{i}(X_{j})} = ℭ.  (19)

However, the ρ_{i}(X_{j}) are either contained in the chain {Z_{k}^{1}}_{k∈ℤ} or equal to some M_{j}^{i}. Because of 0.3(1), all isomorphy classes of direct indecomposable ℭ-modules must appear under the ρ_{i}(X_{j}) to enable equation (19). It follows that all modules of the chain {Z_{k}}_{k∈ℤ} between M_{j}^{1} and hM_{j}^{1} must occur in the list (15). With the equations (17), it follows that for an appropriate choice of the invariants n̄_{1}, … n̄r̄ of ℭ, which are only unique up to cyclic permutation, we have n̄_{1} = n_{1}, …, n̄_{t} = n_{t}. By componentwise action of h ∈ K_{𝔄} with respect to (12), all modules of the chain {Z_{k}^{1}}_{k∈ℤ} occur as summands of the modules of {X_{k}}_{k∈ℤ}, and {Z_{k}^{1}}_{k∈ℤ} is a subchain of period ν̄t̄ in {Z_{k}}_{k∈ℤ}. We deduce that ν̄ = 1, t̄ = t and r̄ = r. In particular, h is a uniformizer for ℭ, and consequently K_{ℭ} = ⟨h⟩ℭ^{×} ⊂ ⟨h⟩ℭ_{1}^{×} = K_{ℭ_{1}} and K_{ℭ} = ⟨h⟩ℭ^{×} ⊂ ⟨h⟩𝔄^{×} ⊂ K_{𝔄}; hence

(ℭ, ℭ_{1}) is a pair continued by (𝔄, 𝔄_{1}).  (20)

Since ℭ_{1} ∩ B = 𝔅_{1} and ℭ ∩ B = 𝔅, 3.1 applies, and thus σ_{i}(Z_{j}^{1}) ∈ {Y_{k}^{1}}_{k∈ℤ} and σ_{i}(Z_{j}) ∈ {Y_{k}}_{k∈ℤ} for all j ∈ ℤ and i = 1, …, f′. For all j ∈ ℤ there exists exactly one l_{j} ∈ ℤ such that Z_{j}^{1}⊋Z_{l_{j}}⊋Z_{l_{j}+1}⊋ … ⊋Z_{l_{j}+t−2}⊋Z_{j+1}^{1}. Let Y_{j}^{(i,1)} := σ_{i}(Z_{j}^{1}) and Y_{j}^{(i,u)} := σ_{i}(Z_{j}^{u}) for i = 1, …, f′ and all j ∈ ℤ, where Z_{j}^{u} := Z_{l_{j}+u−2} for u = 2, …, t. Then Y_{j}^{(i,u)} ∈ {Y_{k}}_{k∈ℤ} for i = 1, …, f′, u = 1, …, t and j ∈ ℤ. For u = 1, …, t and all j ∈ ℤ we obtain a decomposition

Z_{j}^{u} = Y_{j}^{(1,u)} ⊕ Y_{j}^{(2,u)}π^{f} ⊕ … ⊕ Y_{j}^{(f′,u)}π^{f(f′−1)}  (21)

into a direct sum of indecomposable (𝔅, 𝔬D̃)-bimodules due to (13).

By assumption (d) we have h ∈ K_{𝔄} ∩ K_{𝔅_{1}} ⊂ K_{𝔅}, ν_{𝔅_{1}}(h) = 1 > 0, and by (b)(c), 𝔅 ⊂ 𝔅_{1}; hence, there exist ν′ ∈ ℕ such that ν_{𝔅}(h) = ν′t′ > 0, and using arguments as in the proof of 1.2(iii) we see that {Y_{k}^{1}}_{k∈ℤ} is a subchain of period ν′t′ of {Y_{k}}_{k∈ℤ}. As π^{ff′} (right action) and h (left action) act componentwise on the Y_{j}^{u} with respect to the decomposition (21), we obtain from (18) that

ν_{𝔅}(h) | r′.  (22)

The decomposition (21) shows that Z_{j}^{u}π^{f} = Y_{j}^{(f′,u)}π^{ff′} ⊕ Y_{j}^{(1,u)}π^{f} ⊕ … ⊕ Y_{j}^{(f′−1,u)}π^{f(f′−1)}; hence, by (18),(22) and the fact that h acts componentwise with respect to (21), we obtain from the iterated action of π^{f} that, for fixed j ∈ ℤ and u ∈ {1, …, t}, all the Y_{j}^{(i,u)} lie in the same subchain {Y_{υ+ν′t′k}}_{k∈ℤ} (υ ∈ {0, …, ν′t′ − 1}) of {Y_{k}}_{k∈ℤ}. Together with (16) for all j ∈ ℤ and an arbitrary i ∈ {1, …, f′} this leads to

f′dim_{𝔨D̃}(Y_{j}^{(i,u)}/Y_{j}^{(i,u+1)}) = dim_{𝔨D̄}(Z_{j}^{u}/Z_{j}^{u+1}) = n_{u} for u = 1, …, t − 1 and

f′dim_{𝔨D̃}(Y_{j}^{(i,t)}/Y_{j+1}^{(i,t)}) = dim_{𝔨D̄}(Z_{j}^{t}/Z_{j+1}^{1}) = n_{t}.

(23)

The last equation implies that, for u = 1, …, t, there exists a number μ(u) ∈ {1, …, ν′t′} only depending on u such that for all j ∈ ℤ and i = 1, …, f′, Y_{j}^{(i,u)} ∈ {Z_{(μ(u)−1)+ν′tk}}_{k∈ℤ}. For all j ∈ ℤ and i = 1, …, f′ we have, moreover,

Y_{j}^{(i,1)} ⊋ Y_{j}^{(i,2)} ⊋ … ⊋ Y_{j}^{(i,t)} ⊋ Y_{j+1}^{(i,1)}.  (24)

On the other hand, we have

ℭ ∩ B = {x ∈ B | ∀j ∈ ℤ, ∀i = 1, …, f′, xσ_{i}(Z_{j}) ⊂ σ_{i}(Z_{j})} = 𝔅.  (25)

Due to 0.3, (1), (25) implies that all isomorphy classes of direct indecomposable 𝔅-modules have to appear under the σ_{i}(Z_{j}); hence, by (22) all modules of the chain {Y_{k}}_{k∈ℤ} that lie between Y_{j}^{(i,1)} and Y_{j+1}^{(i,1)} must occur in the list (24). The equations (23) now provide that ν′ = 1. We obtain f′n′_{1} = n_{1}, …, f′n′_{t} = n_{t}, for an appropriate numbering of the invariants n′_{1}, …, n′_{r′} of 𝔅, which are unique up to cyclic permutation only, and using (22) also t′ = t. Finally, by n_{1} + … + n_{r} = m, we obtain \frac{r′}{f′} = r. In particular, h is a uniformizer for 𝔅; hence, K_{𝔅} = ⟨h⟩𝔅^{×} ⊂ ⟨h⟩𝔅_{1}^{×} = K_{𝔅_{1}} and K_{𝔅} = ⟨h⟩𝔅^{×} ⊂ ⟨h⟩ℭ^{×} = K_{ℭ}. Therefore, (𝔅, 𝔅_{1}) is a pair continued by (ℭ, ℭ_{1}); hence, by (20) and the transitivity of continuation, the pair (𝔅, 𝔅_{1}) is continued by (𝔄, 𝔄_{1}). This proves the last assertion of our proposition.

To prove the uniqueness of (𝔄, 𝔄_{1}), we consider the way in which the chain of 𝔄 is built up from the chain of 𝔄_{1} and 𝔅. We have 𝔄 ∩ B = {x ∈ B | ∀j ∈ ℤ, ∀i = 1, …, f,∀u = 1, …, f′,xσ_{u} ∘ ρ_{i}(X_{j}) ⊂ σ_{u} ∘ ρ_{i}(X_{j})} = 𝔅. By (14), (15) and (21) the σ_{u} ∘ ρ_{i}(X_{j}) are either contained in the chain {Y_{k}^{1}}_{k∈ℤ} of 𝔅_{1} or direct summands of an M_{j}^{u} (for a certain other choice of u ∈ {2, …, t} and j ∈ ℤ). To consider the latter case, choose for each j ∈ ℤ and i ∈ {1, …, f′} an l_{j} such that σ_{i}(M_{j}^{u}) = Y_{l_{j}}^{(i,u)}. One can see from the structure of the chain {Y_{k}}_{k∈ℤ} that the Y_{l_{j}}^{(i,u)} are uniquely determined by the relations (24); hence, the modules M_{j}^{u} for u = 2, …, t are uniquely determined by knowing the modules M_{j}^{1} and hM_{j}^{1}. Finally, the modules X_{j}^{u} for u = 2, …, t are uniquely determined by knowing the modules X_{j}^{1} and X_{j+1}^{1}, because they conside in all components except for one and are determined there by the M_{j}^{u} for u = 2, …, t. Thus, because {X_{k}}_{k∈ℤ} = ⋃_{u=1}^{t}{X_{k}^{u}}_{k∈ℤ}, the chain {X_{k}}_{k∈ℤ} is (up to a shift of the index) uniquely determined by the chains {X_{k}^{1}}_{k∈ℤ} and {Y_{k}}_{k∈ℤ}. Therefore, 𝔄 is uniquely determined by 𝔄_{1} and 𝔅. Because of [F](Corollary 3), 𝔄_{1} is uniquely determined by 𝔅_{1}. Consequently, (𝔄, 𝔄_{1}) is uniquely determined by (𝔅, 𝔅_{1}), as required.

□

4 Continuation of pairs in the general case

Proof of Theorem 1.9

(i),(i)(a) \underline{Existence:} Let E/L/F be an unramified intermediate extension over F of degree [L : F] = ⟨f(E/F), d⟩. The algebra C := C_{A}(L) has index d′ := \frac{d}{[L:F]} and

⟨f(E/L), d′⟩ = 1;  (26)

hence, 2.2 applies and there exists a numerically proportional continuation (ℭ, ℭ_{1}) of (𝔅, 𝔅_{1}) to B. If ℭ_{1} has period r̃, then let L/K/F be the unique intermediate extension of degree [L : K] = ⟨[L : F], r̃⟩. By 0.5, H := C_{A}(K) is a central simple algebra of index \frac{d}{[K:F]}; hence, 3.2 applies and there exists a numercially proportional continuation (𝔒, 𝔒_{1}) of (ℭ, ℭ_{1}) to H, where 𝔒_{1} has period \frac{r̃}{[L:K]}. Clearly, ⟨[K : F], \frac{r̃}{[L:K]}⟩ = 1; hence, 3.3 applies and there exists a numerically proportional continuation (𝔄, 𝔄_{1}) of (𝔒, 𝔒_{1}) to A. By transitivity (𝔄, 𝔄_{1}) is a numerically proportional continuation of (𝔅, 𝔅_{1}) to A.

\underline{Uniqueness:} Again let E/L/F and C as in the existence part of our proof and, moreover, h ∈ K_{𝔅} such that ν_{𝔅}(h) = t′; hence, ν_{𝔅_{1}}(h) = 1. By [Z](Theorem 2), ℭ_{1} := 𝔄_{1} ∩ C is a principal order continued by 𝔄_{1} and continuing 𝔅_{1}. By [Br](Theorem 1) (or using 1.1 and 3.1), ℭ := 𝔄 ∩ C is a hereditary order, and, by (26) and 2.2(i)(b), ν_{ℭ_{1}}(h) = 1, where h ∈ K_{𝔄} ∩ K_{ℭ_{1}}; hence, 3.4 applies and (ℭ, ℭ_{1}) is a pair continued uniquely by (𝔄, 𝔄_{1}). Because of (26), 2.2 applies, and (ℭ, ℭ_{1}) is the unique continuation of (𝔅, 𝔅_{1}). This proves uniqueness.

(i)(b) From (a) we know that 𝔄 and 𝔅 have the same partition period t. Furthermore, ν_{𝔄} = tν_{𝔄_{1}} and ν_{𝔅} = tν_{𝔅_{1}}; hence, our assertion follows from [Z](Theorem 2(iii), proof of Corollary 3(ii)).

(ii) (a)⇒(b) Let K/E be a totally ramified extension of E of degree \frac{r′}{t′} d′, where d′ = \frac{d}{⟨[E:F],d⟩} is the index of B. Because of 2.2(iii), we can embed K^{×} over E^{×} in K_{𝔅}, and, because of 2.2(ii), ℭ_{1} := 𝔅_{1}∩C is a principal order in C := C_{B}(K) continued by 𝔅_{1}. By (i)(a), ℭ_{1} has period r̄ = \frac{r′}{t′} \frac{⟨e(K/E),d′⟩}{e(K/E)} = 1. Because of [F](Corollary 1), there exists an unramified maximal extension M/K in C such that E^{×} ⊂ K_{ℭ_{1}}.

Now we have E^{×}⊂K^{×}⊂M^{×} ⊂ K_{𝔄_{1}}, K^{×}⊂E^{×} ⊂ K_{𝔄} and e(K/F) = e(M/F). As e(M/F)f(E/F) = d′ \frac{r′}{t′} e(E/F)f(E/F) = d\frac{r′}{t′} \frac{[E:F]}{⟨[E:F],d⟩}, we obtain d | e(M/F)f(E/F). Clearly, M/F is also of maximal degree in A.

(b)⇒(a) Let E/L/F be an unramified intermediate extension over F of degree [L : F] = ⟨f(K/F), d⟩ and C = C_{A}(L), then, because of [Z](Theorem 2), ℭ_{1} := 𝔄_{1}∩C is a principal order in C continued by 𝔄_{1}, and, because of Lemma 1.1 and Lemma 3.1, ℭ := 𝔄 ∩ C is a hereditary order. Let π be a prime element of o_{K}, whence (e(M/F) = e(K/F)) of o_{M}. Let r̄ be the period of ℭ_{1}. Clearly, π ∈ K_{𝔄} ∩ K_{ℭ_{1}}. By [F](Theorem 1(iii), Remark 1) we obtain for d′ := \frac{d}{[L:F]} (the index of C) that ν_{ℭ_{1}}(π) = \frac{r̄d′}{e(M/F)} = ^{\frac{\frac{e(M/F)}{⟨e(M/F),d′⟩} d′}{e(M/F)}} = \frac{d′}{⟨e(M/F),d′⟩} = \frac{d}{⟨e(M/F)f(K/F),d⟩} = 1, because of d | e(M/F)f(E/F).

Therefore, 3.4 applies, and (ℭ, ℭ_{1}) is a pair in C continued by (𝔄, 𝔄_{1}). As ⟨f(K/L), d′⟩ = 1, 2.2 also applies, and (𝔄 ∩ B, 𝔄_{1} ∩ B) is a pair in B continued by (ℭ, ℭ_{1}); hence, our result follows by the transitivity of continuation.

(iii) Let ((n_{1}, …, n_{t}), \frac{r}{t}) be the invariants of (𝔄, 𝔄_{1}).

(a)⇒(b) By assumption and (ii) (𝔄 ∩ B, 𝔄_{1} ∩ B) is a pair in B continued by and numerically proportional to (𝔄, 𝔄_{1}). We denote its invariants by ((n′_{1}, …, n′_{t′}), \frac{r′}{t′}). We know from [F](Corollary 1) that there exists a divisor d̃ of d̈ such that f(M/F) = (n_{1} + … + n_{t})d̃ and, clearly,

e(E/F) | e(M/F) = \frac{md}{f(M/F)} = \frac{r}{t} \frac{d}{d̃}.  (27)

As in 1.8 we calculate l_{1} = \frac{n_{1}+...+n_{t}}{n′_{1}+...+n′_{t}} = _{\frac{n_{1}+...+n_{t}}{⟨f(M/E), \frac{m⟨[E:F],d⟩}{[E:F]} )}} = _{\frac{(n_{1}+...+n_{t})f(E/F)}{⟨f(M/F), \frac{⟨[E:F],d⟩m}{e(E/F)} ⟩}} = _{\frac{f(E/F)}{⟨d̃, \frac{\frac{r}{t} ⟨[E:F],d⟩}{e(E/F)} ⟩}} = \frac{f(E/F)}{⟨d̃, \frac{⟨[E:F],d⟩}{⟨e(E/F),d⟩} ⟩} = \frac{f(E/F)}{⟨d̃,⟨f(E/F), \frac{d}{⟨e(E/F),d⟩} ⟩⟩}; hence, l_{1} | f(E/F) | l_{1}d̃ and, according to (27), [E : F] | \frac{r}{t} dl_{1}. Now everything follows from l_{1} | ⟨n_{1}, …, n_{t}⟩ (see 1.8).

(b)⇒(a) From our assumptions, the existence of a divisor l_{1} of ⟨n_{1}, …, n_{t}⟩ such that l_{1} | f(E/F) | l_{1}d̈ immediately follows and if we choose l_{1} to be maximal with this property, we have ⟨\frac{f(E/F)}{l_{1}}, \frac{⟨n_{1},...,n_{r}⟩}{l_{1}}⟩ = 1. From [E : F] | d\frac{r}{t} ⟨n_{1}, …, n_{r}⟩ and e(E/F) | d\frac{r}{t} it follows that \frac{f(E/F)}{l_{1}} | \frac{d \frac{r}{t}}{e(E/F)}; hence, [E : F] | d\frac{r}{t} l_{1}. Let E ↪ A be an arbitrary embedding of E over F into A (which exists because of [E : F]|md) and B = C_{A}(E), where we identify E with its image in A. There exists a pair (𝔅, 𝔅_{1}) in B with invariants ((\frac{n_{1}}{l_{1}} , …, \frac{n_{t}}{l_{1}} ), \frac{rl_{1}⟨[E:F],d⟩}{t[E:F]}).

Choose a totally ramified extension K/E such that e(K/E) = \frac{rl_{1}⟨[E:F],d⟩}{t[E:F]}d′, where d′ = \frac{d}{⟨[E:F],d⟩}. Then, by 2.2(ii), there exists an embedding K^{×} ↪ K_{𝔅} over E^{×}. Because of 2.2(ii), for C = C_{B}(K), (ℭ, ℭ_{1}) = (𝔅 ∩ C, 𝔅_{1} ∩ C) is a pair in C with invariants ((\frac{n_{1}}{l_{1}} , …, \frac{n_{t}}{l_{1}} ), 1). By [F](Corollary 1), we can find an unramified maximal extension M/K in C such that M^{×} ⊂ K_{ℭ_{1}}. Part (i) shows that there exists a proportional continuation (𝔄′, 𝔄′_{1}) of (𝔅, 𝔅_{1}).

Because of 1.8(iii), the choice of the invariants of (𝔅, 𝔅_{1}) and the definition of l_{1}, we deduce that (𝔄′, 𝔄′_{1}) has the same invariants as (𝔄, 𝔄_{1}); hence, both pairs are conjugated over A^{×}. So, by conjugating the first embedding and the fields M, K, we can assume without loss of generality that (𝔄′, 𝔄′_{1}) = (𝔄, 𝔄_{1}), M^{×} ⊂ K_{𝔄_{1}}, and K^{×} ⊂ K_{𝔄}. Finally, we obtain d|\frac{r}{t} l_{1}d = \frac{rl_{1}⟨[E:F],d⟩}{t[E:F]}d′e(E/F)f(E/F) = e(M/F)f(E/F) from our assumptions.

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5 References

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[BF] C. J. Bushnell and A. Fröhlich: Non-abelian congruence Gauss sums and p-adic simple algebras, Proc. London Math. Soc. (3) 50, 1985, 207-264

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[BK2] C. J. Bushnell and P. C. Kutzko: Simple types in Gl(N), Computing conjugacy classes, Contemporary Mathematics Vol. 177, 1994

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[BT] F. Bruhat and J. Tits, Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France, 112, (1984), 259-301

[CR] C. W. Curtis and I. Reiner: Methods of representation theory - with applications to finite groups and orders, Volume I, John Wiley & Sons Inc., New York, 1981

[F] A. Fröhlich: Principal orders and embeddings of local fields in algebras, Proc. London Math. Soc. (3) 54, 1987, 247-266

[G] M. Grabitz: Erbliche Ordnungen in lokalen zentralen einfachen Algebren, Diplomarbeit unter Anleitung von E.-W. Zink, Humboldt Universität zu Berlin, Januar 1997 (Nachfassung: Mai 1997)

[K] I. Kersten: Brauergruppen von Körpern, Vieweg, Braunschweig, 1990

[R] I. Reiner: Maximal Orders, Academic Press, New York, 1975

[SZ] A. Silberger and E.-W. Zink: An explicit matching theorem for level zero discrete series of unit groups of p-adic simple algebras, preprint, 1997

[Z] E.-W. Zink: More on embeddings of local fields in simple algebras, Max-Planck-Institut für Mathematik (Bonn, Germany) (MPI 96-155), 1996