Krull dimension of Generalized Weyl Algebras with non-commutative coefficients ^{∗}

V V Bavula  T H Lenagan

Department of Mathematics

Kiev University

Volodymyrs’ka Str, 64

Kiev 252617

Ukraine Department of Mathematics University of Edinburgh James Clerk Maxwell Building King’s Buildings Mayfield Road Edinburgh EH9 3JZ

sveta@kinr.kiev.ua  tom@maths.ed.ac.uk

Abstract We obtain formulae for the Krull dimension of the Generalized Weyl Algebra

T = R(σ, a), where R is a left noetherian ring.

1991 Mathematics subject classification: 16P60, 16P40, 16W50, 16S34, 16S36.

Introduction

Krull dimension, in the sense of Rentschler and Gabriel, is one of the most useful invariants in the study of noetherian rings. However, it is notoriously difficult to calculate its exact value in general. For example, until recently, the Krull dimension of the enveloping algebra of a finite dimensional simple Lie algebra was known only for U(sl(2)), [16], and U(sl(3)), [13]. In this paper, we consider the problem of calculating the Krull dimension of the Generalised Weyl Algebras, introduced by the first author in [1]. A start has already been made on this problem in [7] where the Krull dimension of Generalised Weyl Algebras over a commutative noetherian base ring is calculated. However, in order to make progress on iterated Generalised Weyl Algebras it is necessary first to consider the case of a Generalised Weyl Algebra over a noncommutative base ring, and this is the problem we answer in this paper.

∗This research was done while the first author held a Royal Society/NATO Fellowship at the University of Edinburgh

Generalised Weyl Algebras are defined in the following way. Let R be a ring, and let σ be an automorphism of the ring R. Let a be a central element of R. Then the Generalised Weyl Algebra T = R(σ, a) is defined to be the ring generated over R by two indeterminates X, Y subject to the relations

Xα = σ(α)X Y α = σ^{−1}(α)Y Y X = a XY = σ(a),

for all α ∈ R. The terminology, Generalised Weyl Algebra, is appropriate, since the Weyl algebras can be presented as iterated Generalised Weyl Algebras. The first author has used this fact to place the classification of the simple modules over the first Weyl algebra in a more general framework, see, for example, [2], [3] and [6].

In the case that a = 1, or more generally, a is a unit, then the Generalised Weyl Algebra T is a skew Laurent polynomial ring (with Y = aX^{−1}). In this case, Goodearl and the second author, [8], have obtained formulae for the Krull dimension of T. The methods of clean modules and height developed there are adapted here for use in the case of Generalised Weyl Algebras. However, when a is not a unit new phenomena appear which are not present in the skew Laurent case, mainly concerned with the problem that aM = 0 is possible for non-trivial modules. In order to deal with this problem, the methods of stars and holes introduced in [7] are adapted to the non-commutative setting.

Generalised Weyl Algebras are ℤ-graded algebras: T = ⊕v_{i} ⊗R, where v_{i} = X^{i}, for i ≥ 0, and v_{i} = Y^{−i} for i ≤ 0. Wherever possible, we exploit this graded structure of T.

A reference for most of the basic notions concerning Krull dimension that we need is [15, Chapter 6]. We denote the Krull dimension of a module M by K(M). If R is a left noetherian ring, and T = R(σ, a) is a Generalised Weyl Algebra then it is known that

K(R) ≤ K(T) ≤ K(R) + 1

see [7, Proposition 2.2], and the hard problem is to decide which of the two possibilities occur.

The strategy we adopt is as follows. The ring T, considered as a left module, is an induced module T = T ⊗_{R} R. This suggests studying the structure of induced modules. This is a strategy that has proved successful in earlier results on Krull dimension, see, for example, [9], [8] and [7], for the cases of differential operator rings, skew Laurent extensions, and Generalized Weyl Algebras over commutative base rings, respectively. The aim is to reach T = T ⊗_{R} M, via a chain of induced modules, starting from the induced modules T (M) = T ⊗_{R} M, where M runs through the simple modules of R. At each step, the induced modules are ℤ-graded modules, and this leads us to suspect that the structure of ℤ-graded modules should play a large rôle in the proof. Indeed, in the end, the result we obtain shows that the graded Krull dimension and the ordinary Krull dimension coincide.

However, we are not able to prove this directly, and then use this fact; it just emerges as a bonus at the end. It would be interesting to investigate which classes of graded rings can be shown to have the same graded and ordinary Krull dimensions.

The graded submodules that occur as subfactors of induced modules fall into two distinct classes: the graded modules with infinite support, and those with finite support. The former class can be dealt with by developing existing methods. There are two cases to consider: the graded modules which are infinite in both directions, and those which are either left or right restricted. It is the class of graded modules with finite support which causes most problems in the analysis; these modules are both left and right restricted. Specifically, we need to be able to deal with the extension theory for graded T -modules with finite support. In some of the earlier results on calculating Krull dimension progress was possible precisely because this subcategory of the module category was known to be semisimple, see, for example [16] and [11]. In the latter part of the proof of Theorem 3.5, we employ graded techniques to reduce the study of the extensions of the finite support modules that occur within certain factors of induced modules to the study of the modules over the ring T/[m], where [m] is the ideal generated by the powers X^{m} and Y^{m}, for a suitable m.

Here is a brief outline of the contents of the sections. In the first section, we develop the methods of leading coefficients which are needed to deal with the modules with infinite support. We find it necessary to introduce three types of modules to deal with this case: T (M) = T ⊗_{R} M, and V_{±}(M), the latter being needed to deal with the one-sided restricted modules. In the second section, we investigate the structure of the R-module subfactors of induced modules, and introduce the ideas of stars and holes which are not needed in the skew Laurent case, but which are basic tools for modules over Generalized Weyl Algebras. One of the problems that arises is that the induced module T (M) need not be a critical module, even if M is a critical R-module, see [8], for example. In the earlier studies, this problem is overcome by introducing clean modules, and then proving that there are enough clean modules in the sense that every module contains a clean submodule. Unfortunately, this is not the case here, because of the existence of restricted induced modules, and we find it necessary to have three kinds of cleanliness in order to have enough clean submodules.

Section three is the heart of the paper. Here we establish the basic inductive step, Theorem 3.5, from which all the formulae follow. It is here that we have to come to grips with the problems caused by the modules with finite support, and graded techniques are used to overcome these problems. After this, in section four, the required formulae follow in a (relatively) routine way. In section five, we specialize to the case of fully bounded noetherian base rings, and obtain the same theorem as that obtained for commutative base rings in [7]. Finally, in section six, we provide examples of each of the two possible kinds of behaviour for the Krull dimension of T. In a subsequent paper, [5], we exploit the results obtained here and in [7] to calculate the Krull dimension for many classes of algebras which can be presented as Generalised Weyl Algebras.

Throughout the paper, we use “module” to stand for “left module”, and abbreviate “left Krull dimension” to “Krull dimension”.

1 Induced modules

Let T = R(σ, a) be a Generalised Weyl Algebra. The ring T = ⊕_{n∈ℤ} T_{n} is ℤ-graded, where T_{n} = Rv_{n} and

v_{n} = X^{n} (n > 0), v_{n} = Y^{−n} (n < 0), v_{0} = 1.

Moreover,

v_{n}v_{m} = (n, m)v_{n+m}

for some (n, m) ∈ R. If n > 0 and m > 0, then

n ≥ m : (n, −m) = σ^{n}(a) … σ^{n−m+1}(a), (−n, m) = σ^{−n+1}(a) … σ^{−n+m}(a),
n ≤ m : (n, −m) = σ^{n}(a) … σ(a), (−n, m) = σ^{−n+1}(a) … a,

and (n, m) = 1 in all other cases.

The ring T contains the skew polynomial rings T^{+} = ⊕_{i≥0} T_{i} = R[X; σ] and T^{−} = ⊕_{i≤0} T_{i} = R[Y; σ^{−1}].

The ring isomorphism

R(σ, a) ≃ R(σ^{−1}, σ(a)), X ↔ Y, Y ↔ X, r ↔ r, r ∈ R,

is called the ±-symmetry ( or the left-right symmetry).

If the element a ∈ Z(R) is not nilpotent, then the (multiplicative) submonoid S of R\0 generated by all σ^{i}(a), i ∈ ℤ, satisfies the (left and right) Ore condition in T. In other words, there exists the (left and right) localization S^{−1}T = T_{S} of the ring T at S. Moreover, σ extends to R_{S}, and T_{S} ≃ R_{S}[X, X^{−1}; σ] is the skew Laurent polynomial ring. A T -module M contains the S-torsion (or the a-torsion, for short) submodule tor(M) := {m ∈ M | sm = 0 for some s ∈ S}. A T -module M is called a-torsion, if M = tor(M); and a-torsionfree, if tor(M) = 0. If a is a nilpotent element, then, by definition, any T -module is a-torsion.

Let M be an R-module and τ ∈ Aut(R). The twisted module ^{τ}M as an abelian group coincides with M and the action of R on M is given as follows: rm := τ(r)m. We often write_{τ^{i}}M ≡ ^{τ^{−i}}M, in order to avoid sign changes. Note that v_{i}⊗M ≃ _{σ^{i}}M, as R-modules.

Let M be an R-module. The induced T -module

T (M) := T ⊗_{R}M = ⊕_{i∈ℤ} v_{i} ⊗ M

is the direct sum of R-submodules _{R}v_{i} ⊗ M ≃ _{σ^{i}}M. The T -module T (M) is ℤ-graded:

T (M) = ⊕_{i∈ℤ} T (M)_{i}, where T (M)_{i} = v_{i} ⊗ M.

Alternatively, we can write

T ⊗_{R}M = … (Y^{2} ⊗ M) ⊕ (Y ⊗ M) ⊕ (1 ⊗ M) ⊕ (X ⊗ M) ⊕ (X^{2} ⊗ M) ….

It is also useful to keep in mind the following kind of pictures to help see what is happening.

⋯ •^{v_{−2}⊗M} − •^{v_{−1}⊗M} − •^{v_{0}⊗M} − •^{v_{1}⊗M} − •^{v_{2}⊗M} − •^{v_{3}⊗M} ⋯

⋯ •^{Y^{2}⊗M} − •^{Y^{1}⊗M} − •^{1⊗M} − •^{X^{1}⊗M} − •^{X^{2}⊗M} − •^{X^{3}⊗M} ⋯

Looking at the pictures, we see that multiplying T ⊗ M by elements from R preserves the grading, while multiplying by X shifts to the right, and multiplying by Y shifts to the left. If we are in a position to the right of 1 ⊗ M then X· acts as a monomorphism. However, if we are to the left of 1 ⊗ M then the action of X· depends on the action of some σ^{i}(a)·. For example, consider X · (Y ⊗ m), for some m ∈ M. We have

X · (Y ⊗ m) = XY ⊗ m = σ(a) ⊗ m = 1 ⊗ σ(a)m.

In a similar manner, Y · acts as a monomorphism to the left of 1 ⊗ M, but depends on σ^{i}(a)· to the right. Thus, points where σ^{i}(a)M = 0 are especially important: for example, if aM = 0, then Y · (X ⊗ M) = a ⊗ M = 1 ⊗ aM = 0, and so

L(M)_{+} := ∑_{i≥1} v_{i} ⊗ M = ∑_{i≥1} X^{i} ⊗ M

which at first sight is only a module over T^{+} := R[X; σ], is in fact a T -module.

Integers i such that a ·_{σ^{i}} M = 0, or, equivalently a · (v_{i} ⊗ M) = 0, are called the stars of M, while the set of integers i + 1, where i is a star, are called the holes of M. The structure of the induced module T ⊗ M depends to a large extent on whether the set of stars of M is empty, finite or infinite.

Let M be an R-module with aM = 0. The induced module T ⊗ _{R}M contains the T - submodule

L(M)_{+} := ∑_{i≥1} v_{i} ⊗ M = ∑_{i≥1} X^{i} ⊗ M,

and we set

V_{−}(M) := T ⊗_{R}M/L(M)_{+} = ⊕_{i≤0} v_{i} ⊗ M.

and draw the picture

\underbrace{… •^{Y^{3}⊗M} − •^{Y^{2}⊗M} − •^{Y ⊗M} − ⋆^{1⊗M}}_{V_{−}(M)} ] ⋯ ∘^{X^{1}⊗M} …

to illustrate this T -module. Similarly, let N be an R-module with σ(a)N = 0. The induced module T ⊗_{R}N contains the T -submodule

L(N)_{−} := ∑_{i≤−1} v_{i} ⊗ N = ∑_{i≥1}Y^{i} ⊗ M,

and we have the factor module

V_{+}(N) := T ⊗_{R}N/L(N)_{−} = ⊕_{i≥0} v_{i} ⊗ N.

which is represented by the picture

… ⋆^{Y ⊗M} ⋯ [ \underbrace{∘^{1⊗M} − •^{X⊗M} − •^{X^{2}⊗M} − •^{X^{3}⊗M} …}_{V_{+}(M)}

The T -module V = V_{+}(N) is uniquely characterized by the following properties:

• it is a ℤ_{0}^{+}-graded T -module; that is, V = ⊕_{i≥0}V_{i} is a ℤ-graded T -module,

• V_{0} = N, and T V_{0} = V,

• the map X_{V} : V → V, given by v ↦ Xv, is injective. Similarly, the T -module V = V_{−}(N) is uniquely characterized by the following properties:

• it is a ℤ_{0}^{−}-graded T -module; that is, V = ⊕_{i≤0}V_{i} is a ℤ-graded T -module,

• V_{0} = N, and T V_{0} = V,

• the map Y_{V} : V → V, given by v ↦ Y v, is injective.

Given an R-module M, the nonzero elements of the induced module T ⊗_{R}M can be written uniquely in the form

u = v_{m} ⊗ u_{m} + v_{m+1} ⊗ u_{m+1} + ⋯ + v_{n} ⊗ u_{n},

where all u_{i} ∈ M and both of u_{m} and u_{n} are nonzero. The u_{i} are called the coefficients of u. The (+)-leading coefficient of u is u_{n}, and the (−)-leading coefficient of u is u_{m}. The integer n is the (+)-degree of u, denoted by deg_{+}(u), and m is the (−)-degree, denoted by deg_{−}(u). The non-negative integer n − m is the length of u, denoted by l(u). The element 0 is defined to have deg_{±}(0) = ∓∞, leading coefficients 0 and length −∞.

There are doubly infinite filtrations

… ⊆ U_{−1} ⊆ U_{0} ⊆ U_{1} ⊆ …

and

… ⊇ V_{−1} ⊇ V_{0} ⊇ V_{1} ⊇ …

on T ⊗_{R}M given by the R-submodules

U_{n} = {u ∈ T ⊗_{R}M | deg_{+}(u) ≤ n},

and

V_{n} = {v ∈ T ⊗_{R}M | deg_{−}(v) ≥ n},

for n ∈ ℤ.

Also,

U_{n}/U_{n−1} ≃ v_{n} ⊗ M ≃ ^{σ^{−n}}M = _{σ^{n}}M and V_{n}/V_{n+1} ≃ v_{n} ⊗ M ≃ ^{σ^{−n}}M = _{σ^{n}}M,

for all n.

For a T -submodule N of T ⊗_{R}M denote by λ_{−}(N) the set of (−)-leading coefficients of elements of N which have non-positive (−)-degree. Similarly, λ_{+}(N) denotes the set of (+)-leading coefficients of elements of N which have non-negative (+)-degree. The sets λ_{−}(N), λ_{+}(N) are R-submodules of M.

For i > 0, let N_{i} be the set of all (+)-leading coefficients of U_{i} together with 0. Similarly, let N_{−i} be the set of all (−)-leading coefficients together with 0. Then

… ≥ N_{−i} ≥ … ≥ N_{−1}, N_{1} ≤ … ≤ N_{i} ≤ …,

are chains of R-modules and

λ_{±}(N) = ∪_{i≥1} N_{±i}.

The most important case of an induced module is the ring T, where we have

T T = ⊕_{i∈ℤ} v_{i}R ≃ T ⊗_{R}R = ⊕_{i∈ℤ} v_{i} ⊗ R, v_{i}r ↔ v_{i} ⊗ r, r ∈ R.

Lemma 1.1 1. If I is a left ideal of T, then λ_{±}(I) = I_{±n} for some n > 0.

2. If I, J are left ideals of T with J ⊆ I and λ_{+}(J) = λ_{+}(I), λ_{−}(J) = λ_{−}(I), then R(I/J) is a finitely generated R-module.

Proof. Straightforward.

Lemma 1.2 1. Let M an R-module that is a-torsionfree and let N be a nonzero T - submodule of T ⊗_{R}M. Then there exists a nonzero R-submodule L of M such that T ⊗_{R}L embeds in N.

2. Let M an R-module such that σ(a)M = 0 and let N be a nonzero T -submodule of V_{+}(M). Then there exists a nonzero R-submodule L of _{σ^{i}}M, for some i ≥ 0, such that V_{+}(L) embeds in N. Similarly, if, instead, aM = 0 and N is a nonzero T -submodule of V_{−}(M), then there exists a nonzero R-submodule L of _{σ^{i}}M for some i ≤ 0 such that V_{−}(L) embeds in N.

Proof. The first statement is exactly [7, Lemma 2.6]. In the second case, the T^{±}- module V_{±}(M) is isomorphic to the induced module T^{±} ⊗_{R} M. Now we may repeat the arguments of [8, Lemma 1.1].

A module N is a subfactor of a module M if there exist submodules V ≤ U in M such that N = U/V. Also, if 0 ≠ V then N is a minor subfactor of M.

Proposition 1.3 ([7, 2.7]) Let M be a Noetherian R-module, and let I, J be T -submodules of T ⊗_{R}M such that I < J. Suppose that N is a nonzero Noetherian R-module such that

1. T ⊗ _{R}N is isomorphic to a T -submodule subfactor of J/I. Then there exists a nonzero subfactor L of λ_{±}(J)/λ_{±}(I) such that L ≤ _{σ^{±i}}N, for some i ≥ 0.

2. V_{+}(N) is isomorphic to a T -submodule subfactor of J/I. Then there exists a nonzero subfactor L of λ_{+}(J)/λ_{+}(I) such that L ≤ _{σ^{i}}N, for some i ≥ 0.

3. V_{−}(N) is isomorphic to a T -submodule subfactor of J/I. Then there exists a nonzero subfactor L of λ_{−}(J)/λ_{−}(I) such that L ≤ _{σ^{−i}}N, for some i ≥ 0.

2 Submodules of induced modules

The results in the next Proposition are easy to establish, by using the graded structure of Generalised Weyl Algebras.

Proposition 2.1 Let T = R(σ, a) be a generalized Weyl Algebra.

1. ([3]) If R is left (right) Noetherian, then so is T.

2. ( [7, 2.2]) Let R be a left Noetherian ring. Then K(R) ≤ K(T) ≤ K(R) + 1.

In order to study the Krull dimension of an induced module T (M) = T ⊗_{R} M, we find it necessary to deal with the R-module structure of certain T -module factors T (M)/C. The next result gives the details necessary to do this.

Proposition 2.2 Let R be a ring and

A ≥ B_{1} ≥ B_{2} ≥ … ≥ B > 0

be a descending chain of R-submodules with B ≠ 0. Suppose that α is an ordinal, and let … ≤ A_{n} ≤ A_{n+1} ≤ … ≤ A be an ascending chain of R-submodules, with K(A_{n}) ≤ α, for each n, and such that A = ∪A_{n}. Suppose that, for sufficiently large n, each factor A_{n+1}/A_{n} is an α-critical R-module and B ∩ A_{n−1} ≠ B ∩ A_{n}. Then there exists an integer p such that for any integer j ≥ p, all finitely generated R-module subfactors of B_{j}/B_{j+1} have Krull dimension less than α.

Proof. Choose an integer t such that, for all n ≥ t, each factor A_{n}/A_{n+1} is α-critical and B ∩ A_{n−1} ≠ B ∩ A_{n}. Note that (A_{n} ∩ B + A_{n−1})/A_{n−1} ≃ (A_{n} ∩ B)/(A_{n−1} ∩ B) ≠ 0, for all n ≥ t.

Since

(B + A_{n})/(B + A_{n−1}) ≃ A_{n}/(A_{n} ∩ B + A_{n−1}) ≃ (A_{n}/A_{n−1})/((A_{n} ∩ B + A_{n−1})/A_{n−1})

for n ≥ t, the R-module (B + A_{n})/(B + A_{n−1}) is a proper epimorphic image of A_{n}/A_{n−1}, so K((B + A_{n})/(B + A_{n−1})) < α, for each n ≥ t, by the choice of t. As A/(B + A_{t}) is the union of the submodules

(B + A_{t+1})/(B + A_{t}) ≤ (B + A_{t+2})/(B + A_{t}) ≤ …

it follows that all finitely generated R-module subfactors of A/(B +A_{t}) have Krull dimension less than α. In the R-module A/B, consider the following two chains of submodules

A/B ≥ (B + A_{t})/B ≥ 0, A/B ≥ B_{1}/B ≥ B_{2}/B ≥ … ≥ 0.

On applying the Schreier Refinement Theorem to the two chains above we obtain a common refinement of them, cf. [8, 3.2(b)]. Namely, there exist R-submodules

A ≥ V_{01} ≥ V_{02} ≥ … ≥ B + A_{t} ≥ V_{11} ≥ V_{12} ≥ … ≥ B,

B_{j} = W_{0j} ≥ W_{1j} ≥ W_{2j} = B_{j+1}, j = 1, 2, …

such that V_{ij}/V_{i,j+1} ≃ W_{ij}/W_{i+1,j} for all i, j. Now

K((B + A_{t})/B) ≤ K(A_{t}) = α.

Hence, there exists an integer p such that K(V_{1j}/V_{1,j+1}) < α, for all j ≥ p. Consequently, for j ≥ p: K(W_{1j}/W_{2,j}) < α. Since W_{0j}/W_{1,j} is isomorphic to V_{0j}/V_{0,j+1} which is a subfactor of A/(B+A_{t}), all finitely generated subfactors of W_{0j}/W_{1,j} have Krull dimension less than α. Therefore for j ≥ p, every finitely generated subfactor of B_{j}/B_{j+1} has Krull dimension less than α.

Let M be a nonzero R-module and let N be a nonzero T -submodule of T ⊗_{R} M. We say that N has type (−) if λ_{−}(N) ≠ 0, but λ_{+}(N) = 0. Similarly, N has type (+) if λ_{+}(N) ≠ 0, while λ_{−}(N) = 0; and type (−, +) if λ_{−}(N) ≠ 0 and λ_{+}(N) ≠ 0. Finally, N has type (0, 0) if both λ_{±}(N) = 0. In this latter case, N is contained in a finite sum ⊕ v_{i} ⊗ M; so N is finitely generated as an R-module, provided that M is a noetherian R-module, by Lemma 1.1.(2). Let N_{1} ≥ N_{2} ≥ … be a chain of nonzero submodules of T ⊗_{R} M; then, for sufficiently large i, each of the submodules N_{i} has the same type, say t, and we say that the chain above has type t.

Corollary 2.3 Let M be an α-critical noetherian R-module, for some ordinal α. Let

T ⊗_{R} M ≥ N_{1} ≥ N_{2} ≥ … ≥ N > 0

be a chain of nonzero T -submodules of T ⊗_{R} M which is of the the same type t as N, with t ≠ (0, 0). Then there is a positive integer n such that, for any i ≥ n, all finitely generated R-module subfactors of N_{i}/N_{i+1} have Krull dimension less than α.

Proof. Without loss of generality, we may assume that n = 1. Suppose that the T -module N has one of the types (i) (−, +), (ii) (+), or (iii) (−). In cases (ii) and (iii), there exists k ∈ ℤ such that N_{1} is contained in A^{+} := ⊕_{i≥k} T (M)_{i}, or A^{−} := ⊕_{i≤k} T (M)_{i}, respectively. The T -module T ⊗_{R} M = ⊕_{i∈ℤ} T (M)_{i} is ℤ-graded where T (M)_{i} = v_{i} ⊗ M ≃ _{σ^{i}}M, as an R-module. Consider an ascending chain of R-modules: … ≤ A_{i} ≤ A_{i+1} ≤ …, where, in case (i) A_{−1} = 0, A_{2n} = T (M)_{−n} ⊕ ⋯ ⊕ T (M)_{n} and A_{2n+1} = T (M)_{−n} ⊕ ⋯ ⊕ T (M)_{n+1}, for n ≥ 1; while in cases (ii) or (iii), set A_{i} = ⊕_{k≤j≤i} T (M)_{j}, or A_{i} = ⊕_{k≥j≥−i} T (M)_{j}, respectively. Observe that the conditions of Proposition 2.2 hold and the result follows. In more detail, we set A = T ⊗_{R} M, A^{+} and A^{−}, in cases (i), (ii) and (iii), respectively.

For an R-module M define the sets

St(M) = {i ∈ ℤ | a_{σ^{i}}M = σ^{−i}(a)M = 0}, Ho(M) = {i + 1 | i ∈ St(M)}.

The elements of St(M) and Ho(M) are the stars and holes respectively. Set St_{−}(M) = {i ∈ St(M) | i < 0} and Ho_{+}(M) = {j ∈ Ho(M) | j > 0}. Denote by s_{−}(M) the largest element of St_{−}(M), and by h_{+}(M) the smallest element of Ho_{+}(M) (if they exist). The importance of stars and holes is as follows: if i is a positive hole, then Y · (X^{i} ⊗ M) = 0; while if i is a negative star then X · (Y^{−i} ⊗ M) = 0.

An R-module M is called a-monic if for every i ∈ ℤ the map σ^{i}(a)_{M} : M → M, given by m → σ^{i}(a)m is either injective or zero. It is obvious that a submodule of an a-monic module is a-monic, and easy to check that any critical R-module is a-monic. Let M be an a-monic R-module. Then

ker X_{T (M)} = ⊕_{i∈St_{−}(M)} T (M)_{i},  (1)

ker Y_{T (M)} = ⊕_{j∈Ho_{+}(M)} T (M)_{j}.  (2)

If M is a-monic then either M is a-torsionfree, or St(M) ≠ ∅. In the latter case, there is either a negative star, or a positive hole (or both), and so, either s_{−}(M) or h_{+}(M) exists (they may both exist).

The T -submodule of T (M)

L(M) := ∑_{i≤s_{−}(M)} T (M)_{i} ⊕ ∑_{j≥h_{+}(M)} T (M)_{j}

is the largest homogeneous submodule of T (M) which has zero intersection with 1 ⊗ M. Denote by L(M)_{−} and L(M)_{+} the first and the second summand, respectively. If St_{−}(M) = ∅ or Ho_{+}(M) = ∅, we set L(M)_{−} = 0 and L(M)_{+} = 0 respectively. The L(M)_{±} are T -submodules of T (M). If L(M)_{−} ≠ 0, then set

V_{+}(M) = T (M)/L(M)_{−} = ⊕_{i>s_{−}(M)} T (M)_{i},

pictured as

… ⋆^{s_{−}(M)} ⋯ [ \underbrace{∘ − … − •^{1⊗M} − ⋯}_{V_{+}(M)} ,

and if L(M)_{+} ≠ 0, then set

V_{−}(M) = T (M)/L(M)_{+} = ⊕_{i<h_{+}(M)} T (M)_{i},

pictured as

\underbrace{… − •^{1⊗M} − ⋯ − ⋆}_{V_{−}(M)} ] ⋯ ∘^{h_{+}(M)} ….

Finally, if both L(M)_{−} ≠ 0 and L(M)_{+} ≠ 0 then set

L(M) := T (M)/L(M) = ⊕_{s_{−}(M)<i<h_{+}(M)} v_{i} ⊗ M,

… ⋆^{s_{−}(M)} ⋯ [ \underbrace{∘ − … − ⋯ − ⋆}_{L(M)} ] ⋯ ∘^{h_{+}(M)} ….

If s_{−}(M) = −1 then V_{+}(M) = V_{+}(M), while if h_{+}(M) = 1 then V_{−}(M) = V_{−}(M). Let

St_{−}(M) = {… < i_{2} < i_{1} = s_{−}(M)},  Ho_{+}(M) = {h_{+}(M) = j_{1} < j_{2}, …}. (3)

Then there are two strictly descending chains of T -submodules of T (M):

… < V_{−}(i_{n}) < … < V_{−}(i_{1}) < T (M),  (4)
T (M) > V_{+}(j_{1}) > … > V_{+}(j_{m}) > …,  (5)

where

V_{−}(i_{n}) := ⊕_{i≤i_{n}} T (M)_{i} ≃ V_{−}(_{σ^{in}}M),

and

V_{+}(j_{m}) = ⊕_{j≥j_{m}} T (M)_{j} ≃ V_{+}(_{σ^{jm}}M).

Define

L_{−}(i_{n+1}, i_{n}] ≡ L_{−}(_{σ^{i_{n+1}}}M,_{σ^{in}}M] := V_{−}(i_{n})/V_{−}(i_{n+1}) ≃ L(_{σ^{in}}M),  (6)

and

L_{+}(j_{m} − 1, j_{m+1} − 1] ≡ L_{+}(_{σ^{jm−1}}M,_{σ^{j_{m+1}−1}}M] := V_{+}(j_{m})/V_{+}(j_{m+1}) ≃ L(_{σ^{jm}}M). (7)

The T -module V = V_{+}(M) is uniquely characterized by the following properties:

• the T -module V = ⊕_{i∈ℤ} V_{i} is ℤ-graded and left restricted (that is, V_{i} = 0, for all i ≪ 0);

• V = T V_{j}, for some j; • the map X_{V} : V → V, given by v ↦ Xv, is injective.

Clearly, the T -module V_{+}(M) has the properties listed above, with V_{j} = M.

Conversely, suppose that the T -module V satisfies the conditions above. It follows from the first two conditions that V is isomorphic to a T -module T (V_{j})/N, where N is a homogeneous submodule of T (V_{j}). Further, N contains L_{−}(V_{j}) follows from the third condition. Therefore, V is isomorphic to the factor module V_{+}(V_{j})/L, where L is homogeneous submodule of V_{+}(V_{j}). Now, L ⊆ ⊕_{i≥1} X^{i} ⊗ V_{j}, since the map X_{V} is injective and (1 ⊗ V_{j}) ∩ L = 0. If L ≠ 0 then there exists a nonzero element X^{i} ⊗ l ∈ L. Thus, X^{i}(1 ⊗ l) ∈ L, and so 0 = 1 ⊗ l, by the third condition. However, 0 ≠ 1 ⊗ l ∈ L, a contradiction. Thus L = 0, and V ≃ V_{+}(V_{j}).

Similarly, T -module V = V_{−}(M) is uniquely characterized by the following properties:

• the T -module V = ⊕_{i∈ℤ} V_{i} is ℤ-graded and right restricted (that is, V_{i} = 0, for all i ≫ 0);

• V = T V_{j}, for some j;

• the map Y_{V} : V → V, given by v ↦ Y v, is injective.

Lemma 2.4 Let M be an a-monic R-module.

1. If s_{−}(M) = h − 1 exists, then the T -submodule T v_{h} ⊗ M of V_{+}(M) is isomorphic to V_{+}(_{σ^{h}}M); and the T -submodule T v_{−h} ⊗ _{σ^{h}}M of V_{+}(_{σ^{h}}M) is isomorphic to V_{+}(M). So, we have

V_{+}(M) ≥ V_{+}(_{σ^{h}}M) ≥ V_{+}(M) and K(V_{+}(M)) = K(V_{+}(_{σ^{h}}M)).

2. If h_{+}(M) = s + 1 exists, then the T -submodule T v_{s} ⊗ M of V_{−}(M) is isomorphic to V_{−}(_{σ^{s}}M); and the T -submodule T v_{−s} ⊗_{σ^{s}}M of V_{−}(_{σ^{s}}M) is isomorphic to V_{−}(M). So, we have

V_{−}(M) ≥ V_{−}(_{σ^{s}}M) ≥ V_{−}(M) and K(V_{−}(M)) = K(V_{−}(_{σ^{s}}M)).

Proof. This follows immediately from the characterization of the modules V_{±}(M) and V_{±}(M).

Definition A critical R-module M is called T -clean if M is a-torsionfree and T (M) is a critical T -module. In a similar manner, a critical R-module M is called (T, +)-clean if s_{−}(M) exists and V_{+}(M) is a critical T -module, and M is (T, −)-clean if h_{+}(M) exists and V_{−}(M) is a critical T -module.

In view of Lemma 2.4, if M is a critical R-module, then the following hold:

• M is (T, +)-clean if and only if the T -module V_{+}(_{σ^{h}}M) is critical, where h = s_{−}(M)+1;

• M is (T, −)-clean if and only if the T -module V_{−}(_{σ^{s}}M) is critical, where s = h_{+}(M)−1.

It follows from Lemma 2.4, and (4), (5) that if M is (T, +)-clean then so is _{σ^{i}}M, for all i ≥ s_{−}(M), and if M is (T, −)-clean, then so is _{σ^{i}}M, for all i ≤ h_{+}(M).

The next result shows that there are enough (T, •)-clean modules.

Lemma 2.5 Let M be a noetherian critical R-module.

1. If M is a-torsionfree, then M contains a T -clean submodule.

2. If s_{−}(M) exists, then for some i ≥ 0 the R-module _{σ^{i}}M contains a (T, +)-clean submodule.

3. Similarly, if h_{+}(M) exists, then for some i ≤ 0 the R-module _{σ^{i}}M contains a (T, −)- clean submodule.

Proof. The T -module T (M) is noetherian, and so are V_{±}(M), whenever they exist, since the R-module M is noetherian.

1. Assume that M is a-torsionfree. Fix a critical T -submodule, N say, of T (M). By Lemma 1.2.(1), there exists a nonzero R-submodule L of M such that T (L) embeds in N. Let U be a critical R-submodule of L. Since T is a flat right R-module, T (U) embeds in T (L), and so embeds in the critical module N. Hence, the T -module T (U) is critical; and so U is T -clean.

2. Assume that s_{−}(M) exists. By Lemma 2.4, the module V_{+}(M) contains the submodule V_{+}(M′ := _{σ^{h}}M), where h = s_{−}(M) + 1. Fix any critical T -submodule, N say, of V_{+}(M′). By Proposition 1.3.(2), there exists a nonzero R-submodule L of σ^{i}M′, for some i ≥ 0 such that V_{+}(L) embeds in N. Let U be a critical R-submodule of L. Since T is a flat right R-module, V_{+}(U) embeds in V_{+}(L), and so embeds in the critical module N. Hence, the T -module V_{+}(U) is critical; and so U is (T, +)-clean. Unfortunately, if i < |h|, then σ^{i}M′ is still to the left of M, so we need to shift to the right, and get the submodule σ^{|h|}U of σ^{i+|h|}M′ = σ^{i}M, which is also (T, +)-clean by the comment immediately before the lemma.

3. The case where h_{+}(M) exists is similar to the previous case.

Let M = ⊕_{i∈ℤ} M_{i} be a ℤ -graded module. Given a submodule N of M, then

N^{hom} := ⊕_{i∈ℤ} N ∩ M_{i}

is the largest homogeneous submodule of N.

The next lemma establishes various easy properties of the Krull dimension of the various kinds of T -modules that have just been introduced.

Lemma 2.6 Let M be an α-critical noetherian R-module.

1. α ≤ K(T (M)) ≤ α + 1 and if the set St(M) is infinite, then K(T (M)) = α + 1.

2. K(L_{±}(_{σ^{i}}M,_{σ^{j}}M]) = α.

3. If M is a-torsion, then α ≤ K(V_{±}(M)) ≤ α + 1. If the set Ho_{+}(M) is infinite, then K(V_{−}(M)) equals α + 1. Similarly, if St_{−}(M) is infinite, then K(V_{+}(M)) equals α + 1.

4. Let V be one of the following T -modules: (i) V_{−}(M) and St_{−}(M) = ∅; (ii) V_{+}(M) and Ho_{+}(M) = ∅; (iii) L(M) and both s_{−}(M) and h_{+}(M) exists. If N is a non-zero T -submodule of V, then N^{hom} ≠ 0.

5. If V_{+}(M) exists, then K(V_{+}(M)) = K(V_{+}(_{σ^{i}}M)), for i > s_{−}(M), while if V_{−}(M) exists, then K(V_{−}(M)) = K(V_{−}(_{σ^{i}}M)), for i < h_{+}(M).

Proof. 2. As a left R-module L_{±}(_{σ^{i}}M,_{σ^{j}}M] = ⊕_{i<k≤j} _{σ^{k}}M and the result follows.

1 and 3. Follow from (4) and (5) and statement 2.

4. Evident.

5. By Lemma 2.4.(1), K(V_{+}(M)) = K(V_{+}(_{σ^{h}}M)), h = s_{−}(M) + 1 and K(V_{+}(_{σ^{i}}M)) = K(V_{+}(_{σ^{n}}M)) for some n ∈ Ho_{+}(_{σ^{i}}M) ⊆ Ho_{+}(M). Since V_{+}(_{σ^{h}}M) ≥ V_{+}(_{σ^{n}}M) and, by 2, K(V_{+}(_{σ^{h}}M)/V_{+}(_{σ^{n}}M)) = K(L_{+}(_{σ^{h−1}}M,_{σ^{n−1}}M]) = K(M), the result follows from 3. The case (−) is symmetrical to (+).

In the following lemma, we establish the existence of clean subfactors in induced modules, which have the same Krull dimension as the modules, and see the need for the various types of clean modules that we have introduced.

Lemma 2.7 Let M be a nonzero noetherian R-module.

1. If σ(a)M = 0 then, for some i ≥ 0, there exists a (T, +)-clean subfactor N of _{σ^{i}}M such that σ(a)N = 0 and K(V_{+}(M)) = K(V_{+}(N)), while if aM = 0, then, for some i ≤ 0, there exists a (T, −)-clean subfactor N of _{σ^{i}}M, such that aN = 0 and K(V_{−}(M)) = K(V_{−}(N)).

2. There exists either (i) a T -clean subfactor L of M such that K(T ⊗_{R} M) = K(T ⊗_{R} L), or (ii) a (T, +)-clean subfactor N of _{σ^{i}}M, for some i ≥ 0, such that K(T ⊗_{R} M) = K(V_{+}(N)), or (iii) a (T, −)-clean subfactor N of _{σ^{i}}M, for some i ≤ 0, such that K(T ⊗_{R} M) = K(V_{−}(N)).

Proof. 1. Let us consider the first case. Without loss of generality we may assume that M is critical. In fact, the R-module M is noetherian, thus we can choose a chain of submodules M = M_{1} > M_{2} > ⋯ > M_{n} = 0 with critical factors M_{i}/M_{i+1}. Observe V_{+}(M) = T^{+}⊗_{R} M where T^{+} = R[X; σ] is the skew polynomial ring. The right R-module T^{+} is faithfully flat, so we have the chain of T -submodules:

V_{+}(M) = V_{+}(M_{1}) > V_{+}(M_{2}) > ⋯ > V_{+}(M_{n}) = 0

with factors V_{+}(M_{i})/V_{+}(M_{i+1}) ≃ V_{+}(M_{i}/M_{i+1}). So

K(V_{+}(M)) = max{K(V_{+}(M_{i}/M_{i+1}), i = 1, …, n − 1}.

By replacing M by a suitable factor M_{i}/M_{i+1}, we may assume that M is critical. Recall that, in this case, M is also a-monic. By Lemma 2.5.(2), there exists 0 ≤ j = j_{1} ∈ Ho_{+}(M) such that _{σ^{j}}M contains a (T, +)-clean submodule L_{1}. By Lemma 2.6.(5), n := K(V_{+}(M)) = K(V_{+}(_{σ^{j}}M)). If K(V_{+}(L_{1})) = n, then L = L_{1} and we are done. If not, that is, if K(V_{+}(L_{1})) < n, then n = K(V_{+}(_{σ^{j}}M)/V_{+}(L_{1})) = K(V_{+}(_{σ^{j}}M/L_{1})). Using the same sort of arguments, we find an integer j_{2} ≥ j_{1} and R-submodules L_{2}, N_{2} such that _{σ^{j_{2}}}M ≥ L_{2} > N_{2} > _{σ^{j_{2}−j_{1}}}L_{1}, L_{2}/N_{2} is (T, +)-clean and V_{+}(L_{2}/N_{2}) exists. If n = K(V_{+}(L_{2}/N_{2})), then set L = L_{2}/N_{2}. If not, we find L_{3} and so on. Since _{R}M is noetherian and σ is automorphism this process must stop.

2. The module T_{R} is faithfully flat. Applying T ⊗_{R} − to the above chain M = M_{1} > M_{2} > ⋯ > M_{n} = 0 we obtain the chain of T -submodules

T (M) = T (M_{1}) > T (M_{2}) > ⋯ > T (M_{n}) = 0

with i’th factor isomorphic to T (M_{i}/M_{i+1}), hence

K(T (M)) = max{K(T (M_{i}/M_{i+1}), i = 1, …, n − 1}.

Thus, without loss of generality, we may assume that M is critical.

If St(M) ≠ ∅, then either i_{1} = s_{−}(M) or j_{1} = h_{+}(M) exists (or both). By (4), (5), we have at least one of the following exact sequence of T -modules:

0 → V_{−}(i_{1}) → T (M) → V_{+}(M) → 0 or 0 → V_{+}(j_{1}) → T (M) → V_{−}(M) → 0.

Observe that K(V_{+}(M)) = K(V_{+}(_{σ^{i_{1}+1}}M)) and K(V_{−}(M)) = K(V_{−}(_{σ^{j_{1}−1}}M)), by Lemma 2.4, so that K(T (M)) is either equal to max{K(V_{−}(i_{1})), K(V_{+}(_{σ^{i_{1}+1}}M))}, or to max{K(V_{+}(j_{1})), K(V_{−}(_{σ^{j_{1}−1}}M))} and the result follows from statement 1.

If St(M) = ∅; then M is a critical a-torsionfree R-module. By Lemma 2.5, M contains a T -clean submodule L_{1} ≤ M. If n := K(T (M)) = K(T (L_{1})), there is nothing to prove. Otherwise, if n > K(T (L_{1})), then the Krull dimensions of the modules T (M) and T (M/L_{1}) coincide. Applying the same argument to the factor module M/L_{1} we either reduce the problem to the case considered in statement 1 or find a nonzero subfactor \overline{L}_{2} of M/L_{1} (i.e. submodules L_{2} > N_{2} ≥ L_{1}, \overline{L}_{2} = L_{2}/N_{2}) which has the same property as L_{1}. The module M is noetherian, so this process must stop.

The next result is part of the inductive step in proving a formula, Theorem 4.1, for the Krull dimension of an induced module T ⊗M, in the case that M is is an a-torsion, critical noetherian R-module. This is the easy part of the inductive step. The most difficult part is the case where M is a-torsionfree, see Theorem 3.5, which requires a refined study of the submodule structure of induced modules. This is done in the remainder of the present section, see Lemma 2.13.

Proposition 2.8 Let M be a noetherian non-simple R-module of finite Krull dimension.

1. If M is (T, +)-clean, then

K(V_{+}(M)) = max{K(V_{+}(N)) | N ∈ a_{+}} + 1,

where a_{+} is the family of (T, +)-clean minor subfactors of _{σ^{i}}M, i > s_{−}(M).

2. If M is (T, −)-clean, then

K(V_{−}(M)) = max{K(V_{−}(N)) | N ∈ a_{−}} + 1,

where a_{−} is the family of (T, −)-clean minor subfactors of _{σ^{i}}M, i < h_{+}(M).

Proof. 1. First, suppose that Ho_{+}(M) is a finite set (it could be empty). Note that M has a negative star, and so, M has a non-positive hole. Let i be the maximal hole of M. By Lemma 2.4, K(V_{+}(M)) = K(V_{+}(_{σ^{i}}M)). So, without loss of generality, we may replace M by _{σ^{i}}M, which is also (T, +)-clean, and assume that i = 0; that is, V_{+}(M) = V_{+}(M) and Ho_{+}(M) = ∅.

The R-module M is noetherian, so V_{+}(M) is a noetherian T -module. Set n := K(V_{+}(M)), so that 1 ≤ K(M) ≤ n = K(V_{+}(M)) ≤ K(M) + 1. Let N be a nonzero submodule of V_{+}(M) such that K(V_{+}(M)/N) = n − 1. The homogeneous component N_{0} := N ∩ M ≠ 0, by Lemma 2.6.(4)(ii), since Ho_{+}(M) = ∅, and so 0 ≠ V_{+}(N_{0}) ≤ N ≤ V_{+}(M).

The T -module V_{+}(M) is critical, since M is assumed to be (T, +)-clean; so n − 1 ≥ K(V_{+}(M)/V_{+}(N_{0})) = K(V_{+}(M/N_{0})). On the other hand, n − 1 = K(V_{+}(M)/N) ≤ K(V_{+}(M)/V_{+}(N_{0})), so that K(V_{+}(M/N_{0})) = n − 1. By Lemma 2.7.(1), for some i ≥ 0 there exists a (T, +)-clean subfactor L of _{σ^{i}}(M/N_{0}) such that K(V_{+}(L)) = n − 1. The module N_{0} is nonzero, so L ∈ a_{+}.

Next, suppose that Ho_{+}(M) is infinite. As above, we may assume that σ(a)M = 0; that is, V_{+}(M) = V_{+}(M). The R-module M is noetherian of finite Krull dimension, say m, and m ≥ 1, since M is not simple. We can choose a minor subfactor C of M such that C is critical of Krull dimension m − 1. For some i ≥ 0, the R-module _{σ^{i}}C contains a (T, +)-clean submodule, say C′, by Lemma 2.5.(2). By going to the next hole bigger than i, we may assume that σ(a)C′ = 0. The R-module _{σ^{i}}C is critical, since C is critical, and K(C′) = K(_{σ^{i}}C) = K(C) = m − 1. The sets Ho_{+}(M) and Ho_{+}(C′) are infinite, so K(V_{+}(C′)) = K(C′) + 1 = m and K(V_{+}(M)) = K(M) + 1 = m + 1, by Lemma 2.6.(3). Finally, observe that C′ ∈ a_{+}.

2. The proof of the second part is similar to 1, or one can invoke the ±-symmetry. .

Definition. Let M and N be T -clean noetherian R-modules. Define h(N : M) = 1 if N is isomorphic to a minor subfactor of M, but no nonzero submodule of N is isomorphic to a minor subfactor of a T -clean minor subfactor of M.

Note that if h(N : M) = 1 then h(_{σ^{i}}N′ : _{σ^{i}}M) = 1, for all nonzero submodules N′ of N and for all i ∈ ℤ.

Definition. Let M be a noetherian R-module that is either T -clean or (T, +)-clean, and let N be a (T, +)-clean noetherian R-module. Define h_{+}(N : M) = 1 if N is isomorphic to a minor subfactor of M, but, for any i ≥ 0, no nonzero submodule of _{σ^{i}}N is isomorphic to a minor subfactor of a T -clean or (T, +)-clean minor subfactor of _{σ^{i}}M. Similarly, if M is either T -clean or (T, −)-clean, and N is a (T, −)-clean noetherian R-module then h_{−}(N : M) = 1 if N is isomorphic to a minor subfactor of M, but, for any i ≤ 0, no nonzero submodule of _{σ^{i}}N is isomorphic to a minor subfactor of a T -clean or (T, −)-clean minor subfactor of _{σ^{i}}M.

Again, note the following.

• If h_{±}(N : M) = 1 then h_{±}(_{σ^{±i}}N′ : _{σ^{±i}}M) = 1, for all nonzero (T, ±)-clean submodules N′ of N and for all i ≥ 0.

• Let M and _{σ^{i}}M be noetherian R-modules which are either T -clean or (T, ±)-clean, and let N and _{σ^{i}}N be (T, ±)-clean noetherian R-modules, for some i ∈ ℤ. Then h_{±}(N : M) = 1 if and only if h_{±}(_{σ^{i}}N : _{σ^{i}}M) = 1.

Lemma 2.9 Let M be a noetherian R-module and let

M ≥ C_{1} ≥ C_{2} ≥ … ≥ C > 0

be R-submodules of M. If M is (T, +)-clean then there exists an integer m ≥ 1 such that each module _{σ^{j}}(C_{i}/C_{i+1}), for i ≥ m and j ≥ 0, has no (T, +)-clean submodules N for which h_{+}(N : _{σ^{j}}M) = 1. Similarly, if M is (T, −)-clean there exists an integer m ≥ 1 such that each module _{σ^{j}}(C_{i}/C_{i+1}), for i ≥ m and j ≤ 0, has no (T, −)-clean submodules N for which h_{−}(N : _{σ^{j}}M) = 1.

Proof. Suppose that M is (T, +)-clean. The module M is noetherian, so we can choose C to be maximal among those submodules of M contained in a descending chain for which the conclusion fails. Then, for each submodule _{σ^{i}}C of _{σ^{i}}M, for i ≥ 0, the conclusion fails as well. Thus, C can be chosen in such a way that all _{σ^{i}}C, for i ≥ 0, are maximal with failure.

Refine the chain, so that each of the modules C_{i}/C_{i+1} is a critical module. By Lemma 2.5.(2), there exists an integer n ≥ 0 and a submodule V of _{σ^{n}}M such that _{σ^{n}}C ⊆ V and V/_{σ^{n}}C is (T, +)-clean.

By the choice of C, there exist two sequences of non-negative integers, 1 < i_{1} < i_{2} < … and j_{1}, j_{2}, … such that each of the factor modules _{σ^{jν}} (C_{i_{ν}}/C_{i_{ν}+1}), for ν ≥ 1, contains a (T, +)-clean submodule, N_{ν}, say, for which h_{+}(N_{ν} : _{σ^{jν}} M) = 1.

Without loss of generality, we may assume that N_{ν} = _{σ^{jν}} (C_{i_{ν}}/C_{i_{ν}+1}), and that j_{1} ≤ j_{2} ≤ …, since if h_{+}(N : M) = 1 then h_{+}(_{σ^{i}}N : _{σ^{i}}M) = 1, for all i ≥ 0. Fix j = j_{ν}, for any ν, and consider the two chains of modules

σ^{n+j}M ⊇ _{σ^{j}}V ⊇ _{σ^{n+j}}C

and

σ^{n+j}M ≥ _{σ^{n+j}}C_{1} ≥ _{σ^{n+j}}C_{2} ≥ … ≥ _{σ^{n+j}}C.

Comparing these chains, as in Proposition 2.2, we obtain chains of modules

σ^{n+j}M ≥ V_{01} ≥ V_{02} ≥ … ≥ _{σ^{j}}V ≥ V_{11} ≥ V_{12} ≥ … ≥ _{σ^{n+j}}C

and

σ^{n+j}(C_{l}) = W_{0l} ≥ W_{1l} ≥ W_{2l} = _{σ^{n+j}}(C_{l+1}), l = 1, 2, …

such that V_{kl}/V_{k,l+1} ≃ W_{kl}/W_{k+1,l} for all k, l.

Of course, the modules V_{0l}, V_{1l} and W_{1l} depend on j; that is, V_{kl} = V_{kl}(j), etc. For two integers j_{ν} ≤ j_{μ}, it follows from the Schrier Refinement Theorem, that

V_{kl}(j_{μ}) = _{σ^{p}}V_{kl}(j_{ν}),  W_{kl}(j_{μ}) = _{σ^{p}}W_{kl}(j_{ν}),  (8)

where p = j_{μ} − j_{ν}.

Now, h_{+}(_{σ^{n}}N_{ν} : _{σ^{n+j}}M) = 1, since h_{+}(N_{ν} : _{σ^{j}}M) = 1.

The module _{σ^{n+j}}C is nonzero, and the module

\frac{_{σ^{j}}V}{_{σ^{n+j}}C} ≃ _{σ^{j}}(\frac{V}{_{σ^{n}}C} )

is a (T, +)-clean minor subfactor of _{σ^{n+j}}M.

It follows that W_{1jν}(j_{ν}) = W_{2jν}(j_{ν}), for all j_{ν}, since h_{+}(_{σ^{n}}N_{ν} : _{σ^{n+j}}M) = 1. By (8), W_{1j_{μ}}(j_{ν}) = W_{2j_{μ}}(j_{ν}), for all μ, ν. Now,

\frac{V_{0l}(j)}{V_{0,l+1}(j)} ≃ \frac{_{σ^{n+j}}C_{l}}{_{σ^{n+j}}C_{l+1}} ,

for all l = j_{1}, j_{2}, . . .. Thus, we obtain the chain

σ^{n+j}M ≥ V_{0,1} ≥ V_{0,2} ≥ … ≥ _{σ^{j}}V,

for which the conclusion of the lemma fails. However, _{σ^{j}}V > _{σ^{n+j}}C, a contradiction.

Lemma 2.10 Let M be a T -clean noetherian R-module and let

M ≥ C_{1} ≥ C_{2} ≥ … ≥ C > 0

be R-submodules of M. Then, there exists an integer n ≥ 1 such that the following conditions hold.

1. Each module C_{i}/C_{i+1}, for i ≥ n, has no T -clean submodule N for which h(N : M) = 1.

2. Each module _{σ^{±j}}(C_{i}/C_{i+1}), for i ≥ n and j ≥ 0, has no (T, ±)-clean submodule N such that h_{±}(N : _{σ^{±j}}M) = 1.

Proof. Assume that the result is not true, and suppose that C is maximal among submodules of M contained in a chain for which the result fails. Thus, the result also fails above _{σ^{i}}C, for each i ∈ ℤ, and each of these submodules is maximal for failure also.

Choose a submodule M′ of M, containing C, such that M′/C is critical; this is possible since M is noetherian. If M′/C is a-torsionfree then there exists a submodule V with C ⊆ V ⊆ M′ such that V/C is T -clean. We now use the argument of [9, Lemma 4.3] and Lemma 2.9 to obtain a contradiction.

If M′/C is a-torsion then, for some i ∈ ℤ, there exists a submodule V with _{σ^{i}}C ⊆ V ⊆ σ^{i}M′ such that V/_{σ^{i}}C is (T, +)-clean, or (T, −)-clean, by Lemma 2.5.(2, 3). In this case, we finish the proof as in Lemma 2.9 Lemma 2.11 Let R be any ring, and let M be an R-module which contains an ascending chain of submodules 0 = M_{1} ≤ M_{2} ≤ … ≤ M such that ∪_{i=1}^{∞}M_{i} = M. Let N be a nonzero R-module subfactor of M. Then there exists a nonzero submodule N′ of N such that N′ is isomorphic to a subfactor of some M_{k}/M_{k−1}.

Proof. Let N = E/F, for some R-submodules F < E ≤ M. We use [9, Proposition 3.2(a)] to obtain a common refinement of the two chains

0 ≤ F < E ≤ M, 0 = M_{1} ≤ M_{2} ≤ … ≤ M.

Namely, we can choose R-submodules

0 = V_{01} ≤ V_{02} ≤ … ≤ F = V_{11} ≤ V_{12} ≤ … ≤ E = V_{21} ≤ V_{22} ≤ … ≤ M
M_{i} = W_{0i} ≤ W_{1i} ≤ W_{2i} ≤ W_{3i} = M_{i+1}, for i = 1, 2, …

such that

V_{i,j+1}/V_{ij} ≃ W_{i+1,j}/W_{ij},

for all i, j, and E = ∪_{i=1}^{∞}V_{1i}. Let k be the least positive integer such that V_{1k} > F. Set N′ := V_{1k}/F. Then N′ is a nonzero submodule of N and N′ ≃ V_{1k}/V_{1,k−1} ≃ W_{2,k−1}/W_{1,k−1}. Hence, N′ is isomorphic to a subfactor of M_{k}/M_{k−1}.

Corollary 2.12 Let R be a ring and let M be an R-module. Suppose that M is one of the following T -modules: (i) T (M); (ii) V_{+}(M); (iii) V_{−}(M). Let N be a nonzero R-module subfactor of M. Then there exists a nonzero submodule N′ of N such that N′ is isomorphic to a subfactor of _{σ^{i}}M, where, in the first case, i ∈ ℤ, in the second case i > s_{−}(M), and in the third case i < h_{+}(M).

Proof. Observe that M is isomorphic to ⊕_{σ^{i}}M, as an R-module, and apply the previous lemma.

Proposition 2.13 Let M be a noetherian R-module, and let M be a nonzero subfactor of the T -module T (M). Then, there exists a chain of T -submodules

M = M_{0} > M_{1} > … > M_{n−1} > M_{n} = 0

such that to each i there corresponds a cyclic critical R-module V = V_{i}, which is a subfactor of some _{σ^{n(i)}}M, and each factor module M_{i}/M_{i+1} is critical and is of one of the following seven types.

1. T (V), where V is T -clean.

2. V_{−}(V), where V is (T, −)-clean.

3. L_{−}(_{σ^{s}}V, V], where s = s_{−}(V).

4. V_{+}(V), where V is (T, +)-clean.

5. T (V)/N, where V is T -clean and N is a nonzero T -submodule of T (V), and N^{hom} = 0.

6. V_{−}(V)/N, where V is (T, −)-clean with |St_{−}(V)| = ∞ and N is a nonzero T - submodule of V_{−}(V) such that N^{hom} = 0.

7. V_{+}(V)/N, where V is (T, +)-clean with |Ho_{+}(V)| = ∞ and N is a nonzero T - submodule of V_{+}(V) such that N^{hom} = 0.

Remark. If V is a critical R-module then L = L_{−}(_{σ^{s}}V, V], where s = s_{−}(V), is a critical T -module.

Proof. The R-module M is noetherian, hence the T -modules T (M) and M are also noetherian. It is enough to show that M has a submodule in one of the seven categories. Without loss of generality, we may assume that M is a critical T -module, in which case, any nonzero submodule of M is also critical. If follows from the decomposition T (M) = ⊕_{σ^{i}}M and the noetherianity of _{R}M that any finitely generated R-submodule of T (M) or M is noetherian.

Suppose that the proposition is true without the following assumption on the module V = V_{i}

(subfac): “which is a subfactor of some _{σ^{n(i)}}M.”

Then V is a subfactor of an R-module N := _{σ^{i_{1}}}M ⊕ … ⊕_{σ^{is}}M. The module N has the ascending chain of submodules

0 < _{σ^{i_{1}}}M < _{σ^{i_{1}}}M ⊕_{σ^{i_{2}}}M < … < N

with subfactors {_{σ^{i_{j}}} M, j = 1, …, s}. By Lemma 2.11, there exists a nonzero cyclic submodule V′ of V such that V′ is isomophic to subfactor of some _{σ^{i_{j}}} M. Let A(V) be one of the modules from categories 1-4. Then the T -module A(V) naturally contains the submodule A(V′) and the inclusion A(V′) ⊆ A(V) respects the grading. Moreover, the transition from A(V) to A(V′) preserves all of the properties of A(V). Let A(V) be one of the modules from the remaining three cases. Repeating the same argument, we conclude that the transition from A(V) to A(V′) preserves all of the properties of A(V) except possibly the property N ≠ 0. However, in this case A(V′) is in one of the categories 1,2 or 4, respectively.

Thus, we need only prove the proposition without the assumption (subfac).

Choose a cyclic, critical, noetherian R-submodule V of M, with least possible Krull dimension among R-submodules of M. There is no nonzero R-submodule of M that is a proper epimorphic image of the R-module _{σ^{i}}V, for any i ∈ ℤ.

The R-module inclusion V → M induces a T -module map f : T (V) → M. Suppose that f is injective. If St(V) = ∅, then Im(f) belongs to category 1. If St(V) ≠ ∅, then, by considering the chains (4) and (5) we see that the T -module T (V) ≃ Im(f) contains either V_{−}(U) or V_{+}(W), hence the submodules f(V_{−}(U)) and f(V_{+}(W)) belong to the categories 2 and 4, respectively.

Suppose that ker(f) ≠ 0. If St(V) = ∅ then by the choice of V, we have (ker(f))^{hom} = 0, for otherwise, if (ker(f))^{hom} ≠ 0 then V ∩ ker(f) ≠ 0, a contradiction. So, Im(f) belongs to category 5, provided that V is T -clean. If not, the module V contains a T -clean submodule V′, by Lemma 2.5.(1). Hence, the image of the homogeneous T -submodule T (V′) of T (V) under the map f belongs either to category 1 or 5.

It remains to consider the case when St(V) ≠ ∅. In this case, either (i) h = h_{+}(V) exists, or (ii) s = s_{−}(V) exists, or both exist.

If h exists, then there is an exact sequence of T -submodules

0 → V_{+} → T (V) → V_{−}(V) → 0,

where V_{+} = ⊕_{i≥h}T (V)_{i} = V_{+}(_{σ^{h}}V). If s exists, then there is an exact sequence of T - submodules

0 → V_{−} → T (V) → V_{+}(V) → 0,

where V_{−} = ⊕_{i≤s}T (V)_{i} = V_{+}(_{σ^{s}}V).

Consider the first case. We subdivide this into two. Either (I) V_{+} ≰ ker(f), or (II) V_{+} ≤ ker(f). In case (I), denote by g the restriction of f to V_{+}. This is a nonzero map, let N = ker(g). The nonzero T -module V_{+}/N is isomorphic to Im(g), a submodule of M. If N = 0, then V_{+} ≃ Im(g) belongs to category 4. Suppose that N ≠ 0. Then Ho_{+}(_{σ^{h}}V) ≠ ∅. For, if Ho_{+}(_{σ^{h}}V) = ∅, then it follows from Lemma 2.6.(4) that N^{hom} ≠ 0, since N ≠ 0. Hence, _{σ^{h}}V ∩ N^{hom} ≠ 0, which contradicts the choice of V.

Suppose next that Ho_{+}(_{σ^{h}}V) = {h_{1} < h_{2} < … < h_{k}} is a finite nonempty set. We will show that

N = N^{hom} = ⊕_{i≥h_{j}}(X^{i} ⊗_{σ^{h}}V),

for some j. Observe that the map X_{V_{+}} : V_{+} → V_{+}, given by v ↦ Xv is injective and the map Y_{V_{+}} : V_{+} → V_{+}, given by v ↦ Y v, has ker(Y_{V_{+}}) = ⊕(X^{j} ⊗_{σ^{h}}V), where j runs through Ho_{+}(_{σ^{h}}V) ∪ {h}. Using this observation, we conclude that N ∩ (X^{h_{k}} ⊗_{σ^{h}}V) ≠ 0, since N ≠ 0. Thus N ⊇ X^{h_{k}} ⊗ _{σ^{h}}V by the choice of V, and so N ⊇ T X^{h_{k}} ⊗ _{σ^{h}}V = ⊕_{i≥h_{k}}(X^{i} ⊗_{σ^{h}}V).

Set h_{0} = h. Let h_{j} be the minimal positive hole such that N ≥ N′ = T X^{h_{j}} ⊗ _{σ^{h}}V = ⊕_{i≥h_{j}}(X^{i} ⊗_{σ^{h}}V). Then j > 0, since V_{+}/N ≠ 0. We aim to show that N = N′, so that N = N′ = N^{hom}. The factor module V_{+}/N′ contains the homogeneous T -submodule L = ⊕_{h_{j−1}≤i<h_{j}}X^{i} ⊗_{σ^{h}}V. If N ≠ N′ then we see that (X^{h_{j}−1} ⊗_{σ^{h}}V) ∩ N ≠ 0, by the observation above. By the choice of h_{j}, we see that L ≰ N, so the restriction of the map g to X^{h_{j}−1} ⊗_{σ^{h}}V is nonzero, and is not injective, which contradicts the choice of V, since R(X^{h_{j}−1} ⊗_{σ^{h}}V) ≃ _{σ^{h+h_{j} −1}}V.

Hence,

N = N^{hom} = ⊕_{i≥h_{j}}(X^{i} ⊗_{σ^{h}}V),

for some j. Thus V_{+}/N′ ≃ Im(g) is a T -submodule of M which contains the submodule L. The T -submodule L generated by X^{h_{j}−1} ⊗_{σ^{h}}V is isomorphic to L_{−}(_{σ^{h+h_{j−1}−1}}V,_{σ^{h+h_{j} −1}}V], which belongs to category 3.

Suppose now that Ho_{+}(_{σ^{h}}V) is an infinite set, so that |Ho_{+}(V)| = ∞.

If N^{hom} = 0, then Im(g) ≃ V_{+}/N belongs to category 7, if _{σ^{h}}V is (T, +)-clean. If σ^{h}V is not (T, +)-clean then, by Lemma 2.5.(2), there exists a (T, +)-clean submodule U of _{σ^{i}}V, for some i ∈ Ho_{+}(V). Then V_{+}(U) is a homogeneous T -submodule of V_{+}. If N_{1} := N ∩V_{+}(U) = 0 then V_{+}(U)/N belongs to category 4, while if N_{1} ≠ 0 then V_{+}(U)/N belongs to category 7, since N_{1}^{hom} ≤ N^{hom} = 0.

If N^{hom} ≠ 0, then, repeating the same argument as in the case when Ho_{+}(_{σ^{h}}V) is finite, we have that N = N^{hom} = ⊕_{i≥h_{j}}(X^{i} ⊗_{σ^{h}}V) holds. Applying the same arguments as in the case above to the module V_{+}/N^{hom} we find a submodule of M from category 3.

Consider the case (II) where V_{+} ≤ ker(f). The T -submodule of V_{−}(V) generated by X^{h−1} ⊗ V is V_{−}(_{σ^{h−1}}V). By the choice of V, we conclude that V_{−}(_{σ^{h−1}}V) ≰ ker(f). This situation is dual to case (I).

The remaining case, when s exists is dual to the case when h exists. Observe that the dual cases are dealt with immediately, by applying the ± symmetry.

3 Krull dimension of induced modules: the inductive step

Definition. Let U, W be critical noetherian R-modules. We say that U and W are subisomorphic if there exist submodules U′ of U and W′ of W such that U′ ≃ W′. The relation, ∼, of subisomorphism is an equivalence relation on the set C of noetherian critical R-modules. For U ∈ C, we denote by {U} the ∼ equivalence class of U.

Let M be a noetherian module with K(M) = d. Following [10] and [12], we construct a chain of submodules in the following way. Let M_{1} be a submodule maximal such that K(M/M_{1}) = d. Then M/M_{1} is a d-critical module. Continue in this way to construct a chain

M = M_{0} > M_{1} > M_{2} > … > M_{k}

such that each factor is d-critical and K(M_{k}) < d. Note the procedure stops, or we contradict K(M) = d. Let V be any d-critical subfactor of M. Then V is subisomorphic to one of the factors M_{i}/M_{i+1} in the above chain.

Proposition 3.1 Let M be an a-torsionfree noetherian R-module and suppose that the ideal Ra of R has the Artin-Rees property. Suppose that K(M) = β, and that K(M/aM) = γ < β. Let V be a γ-critical subfactor of M such that aV = 0. Then V is subisomorphic to a subfactor of M/aM. In particular, if V is γ-critical then V is subisomorphic to one of the (finitely many up to subisomorphism) γ-critical subfactors of M/aM.

Proof. Suppose that V = E/F, for some submodules F < E of M. The ideal I = Ra has the Artin-Rees property. Apply the AR property to the module M/F, to find an n such that a^{n}M ∩ E ≤ F. It follows that E/F ≃ (E + a^{n}M)/(F + a^{n}M), so without loss of generality, we may assume that a^{n}M ≤ F. By comparing the chain a^{n}M ≤ F < E ≤ M with the chain a^{n}M ≤ a^{n−1}M ≤ … ≤ aM ≤ M, we see that some submodule V′ of V is isomorphic to a subfactor of some a^{i}M/a^{i+1}M. However, M is a-torsionfree, so a^{i}M/a^{i+1}M ≃ M/aM. Thus V′ is isomorphic to a subfactor of M/aM. Now, construct a chain for M/aM as above and V′ is isomorphic to a subfactor of one of the finitely many γ-critical factors of this chain. In fact, since each of these factors is γ-critical and K(V′) = γ, the submodule V′ is isomorphic to a submodule of one of these factors.

If V ∼ U, then St(V) = St(U) and ^{σ^{i}}V ∼ ^{σ^{i}}U, for all i ∈ ℤ. The subgroup G of Aut(R) generated by the automorphism σ acts on the set C/ ∼ of subisomorphism classes via σ^{i}{V } := {^{σ^{i}}V }. Denote by O{V } the orbit {^{σ^{i}}V | i ∈ ℤ} of {V }. The elements in the set St(O) of O = O{V } annihilated by a are called stars. Clearly,

St(O) = {{_{σ^{i}}V } | i ∈ St(V)}.

An orbit containing a star is called degenerate, otherwise it is non-degenerate. Note that O{V } is degenerate if and only if St(V) ≠ ∅.

Any non-degenerate orbit O{V } is equal to O{V′} for some T -clean submodule of V; similarly, any degenerate orbit O{V } is equal to O{V′_{±}} for some (T, ±)-clean submodule of _{σ^{±i}}V, for some i ≥ 0, by Lemma 2.5.

Lemma 3.2 Let M be a nonzero R-module and let M = T (M), V_{+}(V) or V_{−}(V).

1. If there exists a nonzero T -submodule N f M with N^{hom} = 0, then there exists nonzero elements a and b of M such that ann_{R} b = σ^{l}(ann_{R} a) for some natural l ≥ 1.

2. Hence, the map

Ra ≃ R/ann_{R} a → ^{σ^{l}}(R/ann_{R} b) ≃ ^{σ^{l}}(Rb), r + ann_{R} a → σ^{l}(r) + ann_{R} b,

is an isomorphism of R-modules. If, in addition, M is a critical R-module, then the orbit O{M} is finite.

Proof. 1. Let u = v_{m} ⊗u_{m} +⋯+v_{n} ⊗u_{n} be a nonzero element in N of minimal length l = n−m where u_{i} ∈ M, u_{n} ≠ 0, u_{m} ≠ 0, n < ⋯ < m. Observe that l ≥ 1 since N^{hom} = 0. Denote by I and J the annihilators in R of the elements u_{m} and u_{n} respectively. Then σ^{m}(I) and σ^{n}(J) are the annihilators of v_{m} ⊗ u_{m} and v_{n} ⊗ u_{n}. By the minimality of l we conclude that σ^{m}(I) = σ^{n}(J) (otherwise, for any nonzero x ∈ σ^{m}(I)\σ^{n}(J)∪σ^{m}(J)\σ^{n}(I) the element xu is nonzero and has length less than l). So, I = σ^{l}(J).

2. Evident.

Lemma 3.3 Let 0 < C ≤ B < A be a chain of essential T -submodules such that _{R}C = ⊕_{j∈J} C_{j} is a direct sum of noetherian critical R-modules C_{j} and A/B is an epimorphic image of a T -module D which, as R-module, is a direct sum ⊕_{k∈K} D_{k} of noetherian critical R-modules D_{k} such that O{D_{k}} ≠ O{C_{j}} and K(D_{k}) ≤ K(C_{j}) for all k, j. Then every homogeneous ideal in T which annihilates B also annihilates A.

Proof. Let a = ⊕_{i∈ℤ} v_{i} ⊗ a_{i} be a nonzero homogeneous ideal in T with aB = 0. Let f : D → A/B be the epimorphism as above. Since aB = 0, we have an R-module epimorphism

g : a ⊗_{R} D → aA, a ⊗ d → af(d).

As an R-module a ⊗_{R} D is the direct sum ⊕_{i,k} D_{i,k} of R-submodules D_{i,k} = v_{i} ⊗ a_{i}D_{k} ≃ σ^{i}(a_{i}D_{k}). The D_{i,k} are critical provided they are nonzero and then O(D_{i,k}) = O(D_{k}). Suppose aA ≠ 0, then aA ∩ C ≠ 0, since C is an essential submodule of A. Choose 0 ≠ u ∈ aA ∩ C, then Ru is a submodule of C_{j_{1}} ⊕ ⋯ ⊕ C_{j_{s}} where s is chosen as small as possible. Clearly, K(Ru) = n := max{K(C_{j_{α}}) | α = 1, …, s} = K(C_{j_{β}}) for some β. The projection W of Ru on the summand C_{j_{β}} is a nonzero (by the choise of s) n-critical subfactor of Ru with O(W) = O(C_{j_{β}}). Let ũ ∈ a ⊗_{R} D satisfy g(ũ) = u. Thus there is a natural R-module epimorphism Rũ → Ru, given by rũ → ru. The R-submodule Rũ of a ⊗_{R} D is a submodule of some direct sum D_{i_{1}k_{1}} ⊕ ⋯ ⊕ D_{i_{t}k_{t}} with k as small as possible. Since K(D_{i_{α}k_{α}}) ≤ n, for all α, we conclude that K(Rũ) = n. Hence O(W) = O(D_{i_{γ}k_{γ}}) for some γ; that is, O(C_{j_{β}}) = O(D_{k_{γ}}), a contradiction.

For any natural number i denote by [i] the ideal of T generated by X^{i} and Y^{i}. Then [i] = ⊕_{j∈ℤ} v_{j} ⊗ [i]_{j} is the homogeneous ideal with the ideals [i]_{j} of R satisfying

R = [i]_{−i} ⊇ ⋯ ⊇ [i]_{−1} ⊇ [i]_{0}, [i]_{0} ⊆ [i]_{1} ⊆ ⋯ ⊆ [i]_{i} = R,

and [i]_{j} = R, for all |j| ≥ i. The left and right R-module T/[i] is finitely generated. Clearly,

T ≥ [1] ≥ ⋯ ≥ [i] ≥ ⋯, [i][j] ⊇ [i + j].

Note that any module of type L_{−}(_{σ_{s}}V, V] is annihilated by [|s|].

Lemma 3.4 Let M be a noetherian R-module and let M be a nonzero subfactor of the T -module T (M). Let M = M_{0} > M_{1} > ⋯ > M_{n−1} > M_{n} = 0 be the chain of sumbodules from Proposition 2.13. Then

1. [m]M = 0 for some m ≥ 1 if and only if all subfactors M_{i}/M_{i+1} belong to category 3.

2. X^{m}M = 0 for some m ≥ 1 if and only if all subfactors M_{i}/M_{i+1} belong to categories 2, 3, 6 and the module V in case of categories 2 or 6 has finite orbit O{V }.

3. Y^{m}M = 0 for some m ≥ 1 if and only if all subfactors M_{i}/M_{i+1} belong to categories 3, 4, 7 and the module V in case of categories 4 or 7 has finite orbit O{V }.

Proof. 1. (⇐) Suppose that all subfactors M_{i}/M_{i+1} belong to category 3. Let j be the maximum of the corresponding s = s(i) as in Proposition 2.13. Then the ideal [j] annihilates all subfactors M_{i}/M_{i+1}, hence [j]^{n} annihilates M. Observe that [jn] ⊆ [j]^{n}.

(⇒) Suppose [m]M = 0, for some m ≥ 1. Then [m]M_{i}/M_{i+1} = 0 for all i. Let V be one of the T -modules T (V), V_{±}(V). The T -submodule [m]V of V is nonzero and homogeneous, hence each M_{i}/M_{i+1} does not belong to classes 1, 2, 4-7.

2. (⇐) Let j be the maximum of the corresponding s = s(i) and |O(V)|. Then evidently X^{j} annihilates all subfactors, hence X^{jn} annihilates M.

(⇒) Suppose X^{m}M = 0 for some m ≥ 1. Then X^{m}(M_{i}/M_{i+1}) = 0 for all i. Let V be either T (V) or V_{+}(V). The T -submodule T X^{m}V of V is nonzero and homogeneous, hence each M_{i}/M_{i+1} does not belong to classes 1, 4, 5, 7.

3. The proof is similar to case 2.

Let S be any ring and let B be a submodule of an S-module A. Take an additive complement, B^{•}, of B in A; that is, a submodule that is maximal subject to having zero intersection with B^{•}. Then B ⊕ B^{•} is an essential submodule of A and B ≡ (B ⊕ B^{•})/B^{•} is an essential submodule of A/B^{•}. Whenever we pass from the chain of submodules 0 ≤ B ≤ A to the chain 0 ≤ B^{•} ≤ B ⊕ B^{•} ≤ A, we will say that we make a perestroika with the first chain.

Theorem 3.5 Let M be a nonzero, non-simple T -clean noetherian R-module of finite Krull dimension and suppose that the ideal Ra of R have the left Artin-Rees property (for example, if R is left noetherian). Then

K(T ⊗_{R} M) = max{α_{−}, α_{0}, α_{+}} + 1

where α_{0} = max{K(T ⊗_{R} N) | N ∈ A_{0}}, and A_{0} is the set of T -clean minor subfactors of M, while α_{±} = max{K(V_{±}(N)) | N ∈ A_{±}}, and A_{±} is the set of (T, ±)-clean minor subfactors of _{σ^{±i}}M, for i ≥ 0.

Remark. It follows, from Lemma 2.6.(5) and Lemma 2.4, that α_{±} = max{K(V_{±}(N)) | N ∈ A′_{±}}, where A′_{±} is the set of (T, ±)-clean minor subfactors of _{σ^{i}}M, for i ∈ ℤ.

Proof. At least one of the sets A_{0}, A_{±} is nonempty, by Lemma 2.5, since the R-module M is nonzero and non-simple. Set β = K(M) < ∞, and note that K(T (M)) ≤ β + 1 < ∞, and that α := max{α_{−}, α_{0}, α_{+}} < ∞. Clearly, K(T (M)) ≥ α + 1. Suppose that K(T (M)) = K(M) = β. There exists either a T -clean, or a (T, +)-clean, or a (T, −)-clean minor subfactor, V say, of M with Krull dimension β−1. Now, β−1 = K(V) ≤ K(W(V)), where W(V) = T (V), V_{+}(V), V_{−}(V), respectively; so α ≥ β − 1. Thus the claim of the theorem holds. Hence, we need only concentrate on the case where K(T (M)) = β + 1. Choose N ∈ A := A_{−} ∪ A_{0} ∪ A_{+} with K(N) = α, where N = T (N), V_{+}(N) or V_{−}(N), respectively, according to whether N ∈ A_{0}, A_{+}, or A_{−}. Then N is isomorphic to a minor subfactor of T (M), and so K(T (M)) > α.

If K(T (M)) > α+1 then α = β −1, and K(T (M)/C) > α, for some nonzero T -submodule C of T (M) such that T (M)/C is critical. Hence, there exists a chain of T -submodules

T (M) ≥ C_{1} ≥ C_{2} ≥ … > C > 0

such that K(C_{j}/C_{j+1}) = α, for infinitely many j. After refining the chain, we may assume that each C_{j}/C_{j+1} is one of the modules from Proposition 2.13. By Corollary 2.3, there exists a positive integer p such that, for all j ≥ p, any finitely generated R-module subfactor of C_{j}/C_{j+1} has Krull dimension less than β. Hence, for each j ≥ p, every C_{j}/C_{j+1} from the categories 5-7 in Proposition 2.13 has Krull dimension less than α.

Thus, if, for some j ≥ p, we have K(C_{j}/C_{j+1}) = α, then C_{j}/C_{j+1} belongs to one of the sets 1-4 in Proposition 2.13.

Set λ = λ_{+} or λ_{−}. Inside M there is a chain of R-submodules

M ≥ λ(C_{1}) ≥ λ(C_{2}) ≥ … ≥ λ(C) > 0

(note that λ(C) ≠ 0, since M is a-torsionfree and C ≠ 0).

By Lemma 2.10, there exists a positive integer q such that, for all j ≥ q,

(i) each module λ(C_{j})/λ(C_{j+1}) has no T -clean subfactors N for which h(N : M) = 1; (ii) each module _{σ^{±i}}(λ(C_{j})/λ(C_{j+1})), for i ≥ 0, has no (T, ±)-clean submodule N such that h_{±}(N : _{σ^{±i}}M) = 1.

Choose m ≥ max{p, q} such that K(C_{m}/C_{m+1}) = α. We aim to show that C_{m}/C_{m+1} belongs to category 3 of Proposition 2.13. It is enough to prove that C_{m}/C_{m+1} is not in categories 1 and 4 (since the category 2 is dual to category 4). Suppose that the module C_{m}/C_{m+1} = T (V), or C_{m}/C_{m+1} = V_{+}(V), belongs to category 1 or 4, respectively. Then K(V) < β, since m ≥ p, and hence V is a minor subfactor of some _{σ^{k}}M, since M is a critical R-module. Set λ = λ_{+}. By Proposition 1.3, there exists a nonzero subfactor N of λ(C_{m})/λ(C_{m+1}) such that N ≤ _{σ^{i}}V, for some i ∈ ℤ, or i ≥ 0 respectively. The R-module V is T -clean in case 1, or (T, +)-clean in case 4, and hence so is _{σ^{i}}V, for i ∈ ℤ, or i ≥ 0, respectively. In any case, the R-module N is either T -clean (in case 1) or (T, +)-clean (in case 4); and

α = K(T (V)) = K(T (_{σ^{i}}V)) = K(T (N))

in case 1, while

α = K(V_{+}(V)) = K(V_{+}(_{σ^{i}}V)) = K(V_{+}(N))

in case 4.

The R-module N is T -clean (or (T, +)-clean) and m ≥ q. Thus there exists either

(a) a nonzero R-submodule L of N and a T -clean minor subfactor G of M such that L is isomorphic to a minor subfactor of G (case 1), or

(b) a nonzero R-submodule L of _{σ^{n}}N, for some n ≥ 0, and a (T, +)-clean minor subfactor G of _{σ^{n}}M such that L is isomorphic to a minor subfactor of G (case 1 or 4).

Suppose that case (a) holds. Then

α = K(T (N)) = K(T (L)) < K(T (G)) ≤ α,

a contradiction.

Similarly, if case 1 and case (b) hold, then

α = K(V_{+}(N)) = K(V_{+}(L)) < K(T (G)) ≤ α,

while if case 4 and case (b) hold, then

α = K(V_{+}(N)) = K(V_{+}(L)) < K(V_{+}(G)) ≤ α.

In each case, we have a contradiction.

So, for every m ≥ r = max{p, q}, a factor C_{m}/C_{m+1} that has Krull dimension β − 1 must belong to category 3 of Proposition 2.13. The set {C_{m(i)}/C_{m(i)+1} | i = 1, 2, …} of all such factors is infinite.

Consider the chain of submodules

A_{1} ≥ B_{1} ≥ A_{2} ≥ B_{2} ≥ A_{3} ≥ … > C  (9)

where A_{1} = C_{r}, B_{i} = C_{m(i)} and A_{i+1} = C_{m(i)+1}. Observe that K(B_{i}/A_{i+1}) = β − 1 and K(A_{i}/B_{i}) < β − 1, for all i. Making a perestroika with the chain A_{1} ≥ B_{1} ≥ A_{2}, we obtain a chain A_{1} ≥ B′_{1} ≥ A′_{2} ≥ A_{2}, where the module B′_{1}/A′_{2} is essential in A_{1}/A′_{2}, and is isomorphic to B_{1}/A_{2}, and the first and third factors in the chain have Krull dimension less than β − 1. Continuing, by making a perestroika with the chain A′_{2} ≥ B_{2} ≥ A_{3}, we get a chain A′_{2} ≥ B′_{2} ≥ A′_{3} ≥ A_{3} with the same properties as the new chain above. After making all perestroikas, we have a chain

A′_{1} = A_{1} ≥ B′_{1} ≥ A′_{2} ≥ B′_{2} ≥ A′_{3} ≥ … > C  (10)

where every subfactor B′_{i}/A′_{i+1} is essential in A′_{i}/A′_{i+1}, belongs to category 3 of Proposition 2.13 and has Krull dimension β − 1. Further, all other subfactors in the chain have Krull dimension less than β − 1.

The module T (M)/C is critical with the same Krull dimension as A′_{1}/C′, where C′ := ∩_{i=1}^{∞}A_{i}, hence, C′ = C. Without loss of generality, we may assume that chains 9 and 10 coincide. Now, note that the T -module B_{i}/A_{i+1} is isomorphic to L_{−}(_{σ^{s(i)}}V_{i}, V_{i}], by Proposition 2.13.(3) and T (M) = ⊕_{j∈ℤ} (v_{j} ⊗ M), where _{R}(v_{j} ⊗ M) ≃ _{σ^{j}}M. Thus, we conclude that the critical R-module V_{i} is isomorphic to a subfactor of some _{σ^{j}}M, for j = j(i).

The R-module M is a-torsionfree, thus

K(_{σ^{j}}M/a_{σ^{j}}M) < K(_{σ^{j}}M) = β.

By Lemma 2.6.(2), K(V_{i}) = β − 1 and, by Proposition 3.1, V_{i} is isomorphic to a subfactor of _{σ^{j}}M/a_{σ^{j}}M. It follows that

β − 1 = K(V_{i}) = K(_{σ^{j}}M/a_{σ^{j}}M) = K(M/aM),

that V_{i} is a (β − 1)-critical subfactor of _{σ^{j}}M/a_{σ^{j}}M , and that the orbit O{V_{i}} belongs to the finite set of orbits, Λ, of the (β − 1)-critical subfactors of the R-module M/aM. Note that the orbit O{V_{i}} is infinite, and St(V_{i}) is finite, since α = β −1, using Lemma 2.6.(3). Denote by Ω the subset of Λ which consists of the infinite orbits O_{γ} = O_{γ}{U_{γ}}, where γ = 1, …, s, which have finitely many, but at least two, stars; that is, subisomorphism classes annihilated by a. Note that {W } is a star of O_{γ} if and only if {W } = {_{σ^{j}}U_{γ}}, for j ∈ St(U_{γ}). Observe that O{V_{i}} ∈ Ω and V_{i} is a star of O{V_{i}}.

Denote by m the maximum value of m_{γ} := S_{γ}^{max} − S_{γ}^{min}, for γ = 1, …, s, where S_{γ}^{max} and S_{γ}^{min} are the maximal and minimal elements of St(U_{γ}). Note that m_{γ} does not depend on the choice of U_{γ} but only on the orbit O_{γ}.

Let

Int := {{_{σ^{j}}U_{γ}} | γ = 1, …, s; S_{γ}^{min} < j ≤ S_{γ}^{max}}.

Then every T -module B_{i}/A_{i+1} is annihilated by the homogeneous ideal [m] of T that is generated by X^{m} and Y^{m}.

The T -submodule B_{i}/A_{i+1} of A_{i}/A_{i+1} is essential and is an epimorphic image of T (V_{i}). Every other subfactor C_{j}/C_{j+1} of A_{i}/A_{i+1} belongs to one of the categories 3, 5-7 in Proposition 2.13 and is an epimorphic image of T (V_{j}), where K(V_{j}) ≤ β − 1 = K(V_{i}). Every orbit O{V_{j}} is distinct from O{V_{i}}, since in the last three categories, O{V_{j}} is finite, by Lemma 3.2, and K(V_{j}) < β − 1 = K(V_{i}), in the first case. Thus, applying Lemma 3.3, step by step, to the chain of submodules

0 ≤ B_{i}/A_{i+1} = C_{m(i)}/C_{m(i)+1} ≤ C_{m(i)−1}/C_{m(i)+1} ≤ … ≤ C_{m(i−1)+1}/C_{m(i)+1} = A_{i}/A_{i+1},

we conclude that [m](A_{i}/A_{i+1}) = 0. Then, by Lemma 3.4, all subfactors C_{j}/C_{j+1} ≃ L_{−}(_{σ^{s(j)}}V_{j}, V_{j}] are from category 3 in Proposition 2.13.

Therefore, every A_{1}/A_{i} has a finite chain of R-submodules

A_{1} = D_{1} > D_{2} > … > D_{t} = A_{i}, t = t(i)

with factors isomorphic to one of the direct summands_{σ^{k}}V_{j} of L_{−}(_{σ^{s(j)}}V_{j}, V_{j}] = ⊕_{s(j)<k≤0} _{σ^{k}}V_{j}. We use induction on i to show that [m] annihilates A_{1}/A_{i}. This is certainly true for i = 2; so assume, by induction, that [m](A_{1}/A_{i−1}) = 0, or, equivalently, that

[m]A_{1} = T X^{m}A_{1} + T Y^{m}A_{1} ≤ A_{i−1}.

Hence, E = [m](A_{1}/A_{i}) is a T -submodule of A_{i−1}/A_{i}. If E ≠ 0, then E ∩ (B_{i−1}/A_{i}) ≠ 0 and so the Krull dimension of the T -module E is β − 1. On the other hand, inside A_{i−1}/A_{i} there is a chain of R-modules X^{m}A_{1} + A_{i} = X^{m}D_{1} + A_{i} ≥ X^{m}D_{2} + A_{i} ≥ … ≥ X^{m}D_{t} + A_{i} = A_{i}, and each factor (X^{m}D_{j} + A_{i})/(X^{m}D_{j+1} + A_{i}) is an epimorphic image of the R-module _{σ^{m}}(D_{j}/D_{j+1}). Clearly, K_{R}(A_{i−1}/A_{i}) = β − 1 and the subisomorphism class of every (β − 1)-critical subfactor of the R-module A_{i−1}/A_{i} belongs to Int.

If K(D_{j}/D_{j+1}) = β − 1, then {_{σ^{m}}(D_{j}/D_{j+1})} ∉ Int, by the choice of m, hence the factor (X^{m}D_{j} + A_{i})/(X^{m}D_{j+1} + A_{i}) is a proper epimorphic image of _{σ^{m}}(D_{j}/D_{j+1}) and so has Krull dimension less than β − 1. Finally, K_{R}(X^{m}(A_{1}/A_{i})) < β − 1, and similarly, K_{R}(Y^{m}(A_{1}/A_{i})) < β − 1. Now,

β − 1 = K_{T} ([m](A_{1}/A_{i})) ≤ K_{R}([m](A_{1}/A_{i})) = K_{R}(T · U),

where U = X^{m}(A_{1}/A_{i}) + Y^{m}(A_{1}/A_{i}) is an R-submodule of A_{i−1}/A_{i} of Krull dimension < β − 1. Consider the R-module epimorphism

T ⊗_{R} U = ⊕_{j∈ℤ} (v_{j} ⊗ U) ↠ T · U, t ⊗ u ↦ tu.

The R-module A_{i−1}/A_{i} is noetherian, so there is a finite sum F = ⊕ (v_{j} ⊗ U) which maps onto T · U. However, K_{R}(F) = K_{R}(⊕_{σ^{j}}U) < β − 1. Hence K_{R}(T · U) < β − 1, a contradiction, so that [m]A_{1} ⊆ A_{i}. This now holds for each i, and so [m]A_{1} ⊆ ∩_{i=1}^{∞}A_{i} = C. Hence, [m] annihilates the module A_{1}/C.

We conclude that

β = K_{T} (A_{1}/C) = K_{T} (L_{−}(_{σ^{k}}W, W]) = K_{R}(W),

by Lemma 3.4 and Lemma 2.6.(2), where L_{−}(_{σ^{k}}W, W] is isomorphic to a subfactor of A_{1}/C. We have shown above that W is isomorphic to a subfactor of some _{σ^{r}}M. Since both of the R-modules W and _{σ^{r}}M are critical and have the same Krull dimension, β, this forces W to be a submodule of _{σ^{r}}M, leading to O{W } = O{M}, which is contradiction, since M is a-torsionfree, whereas aW = 0.

4 Krull dimension formulae

The formulae given in Proposition 2.8 and Theorem 3.5 will be used as an inductive step by which the Krull dimension of an induced module T ⊗_{R} M may be computed in terms of the Krull dimension of the modules T ⊗_{R} A, V_{±}(A), where A runs through the simple subfactors of M.

Definition. Let A be a simple R-module and let M be an arbitrary R-module.

1. If A is a-torsionfree, then h(A : M) is defined to be the supremum of those non-negative integers n for which there exists a sequence A = A_{0}, A_{1}, …, A_{n} of T -clean R-modules such that A_{i} is isomorphic to a minor subfactor of A_{i+1}, for i = 0, …, n − 1, while A_{n} is isomorphic to a subfactor (not necessarily minor) of M.

2. If A is a-torsion, then h_{+}(A : M), (h_{−}(A : M)), is defined to be the supremum of those non-negative integers n for which there exists a sequence A = A_{0}, A_{1}, …, A_{n} of R-modules such that each of the R-modules A = A_{0}, A_{1}, …, A_{i}, for some i ≥ 0, is (T, +)-clean, ((T, −)-clean), while each of the R-modules A_{i+1}, …, A_{n} is T -clean; and A_{j} is isomorphic to a minor subfactor of A_{j+1}, for j = 0, … n − 1, while A_{n} is isomorphic to a subfactor (not necessarily minor) of M.

Remark. Any simple a-torsionfree R-module is T -clean. However, a simple a-torsion R-module may fail to be either (T, +)-clean or (T, −)-clean. So, h_{±}(A, M) is only defined for the simple a-torsion modules which are (T, ±)-clean.

The sequence of R-modules A = A_{0}, A_{1}, …, A_{n} is called a (T, •)-clean sequence associated with A in M, where • stands for ∅, + or −. The corresponding h_{•}(A : M) is called the •-height of A in M. If M has Krull dimension, then h_{•}(A : M) ≤ K(M), since K(A_{n}) > K(A_{n−1}) > … > K(A_{0}) = 0, so K(M) ≥ n.

Observe that if A = A_{0}, A_{1}, …, A_{n} is a T -clean (or (T, ±)-clean) sequence associated with A in M, then _{σ^{i}}A = _{σ^{i}}A_{0},_{σ^{i}}A_{1}, …,_{σ^{i}}A_{n} is a T -clean (or (T, ±)-clean) sequence associated with _{σ^{i}}A in _{σ^{i}}M, for i ∈ ℤ (or ±i ≥ 0). Thus, h(A : M) = h(_{σ^{i}}A : _{σ^{i}}M), for i ∈ ℤ (or h_{±}(A, M) ≤ h_{±}(_{σ^{i}}A,_{σ^{i}}M), for ±i ≥ 0).

Theorem 4.1 Let M be a nonzero noetherian R-module with finite Krull dimension. Suppose that the ideal Ra of R has the left AR property.

Then 1.

K(T ⊗_{R} M) = μ(M) := max{μ_{−}, μ_{0}, μ_{+}}

where

μ_{0} = max{K(T ⊗_{R} A) + h(A : M) | A ∈ M}

and M is the set of a-torsionfree simple subfactors of M;

μ_{±} = max{K(V_{±}(A_{i})) + h_{±}(A_{i} : _{σ^{±i}}M) | A_{i} ∈ M_{±,i}, i ≥ 0}

and M_{±,i} is the set of (T, ±)-clean a-torsion simple subfactors of _{σ^{±i}}M.

2. Let A = R/I be a simple R-module, where I is a maximal left ideal of R. (i) If A is a-torsionfree, then K(T ⊗_{R} A) = 1 if and only if the orbit, O{A}, is finite;

that is, ^{σ^{n}}A ≃ A, for some n ≥ 1, or, equivalently, σ^{n}(I)x ⊆ I, for some x ∈ R\I, and some n ≥ 1.

(ii) If A is a-torsion, then K(V_{±}(A)) = 1 if and only if the set

St_{±}(A) := {0 < ±i ∈ ℤ | a_{σ^{i}}A = σ^{−i}(a)A = 0}

is infinite.

Remark. If A is a simple a-torsion R-module, then _{σ^{±i}}A is (T, ±)-clean for i ≫ 0, by Lemma 2.5.(2,3).

Proof. Set n = K(M). If n = 0, then every (T, •)-clean subfactor of _{σ^{j}}M, for j ∈ ℤ, is a simple R-module of zero height, so the result follows from Lemma 2.7.(2).

Now, let n ≥ 1, and assume that the result holds for noetherian R-modules of Krull dimension less than n. If A ∈ M and h(A : M) = m, then there exists a T -clean sequence A = A_{0}, A_{1}, …, A_{m} associated with A in M. Then each T ⊗_{R} A_{i} is isomorphic to a minor subfactor of the critical T -module T ⊗_{R} A_{i+1}, hence

K(T ⊗_{R} M) ≥ K(T ⊗_{R} A_{m}) > K(T ⊗_{R} A_{m−1}) > … > K(T ⊗_{R} A_{0}).

Thus,

K(T ⊗_{R} M) ≥ K(T ⊗_{R} A_{0}) + h(A : M)

and so K(T ⊗_{R} M) ≥ μ_{0}.

If A ∈ M_{±,i}, for some i ≥ 0, and h_{±}(A : _{σ^{±i}}M) = m, then there exists a (T, ±)-clean sequence A = A_{0}, A_{1}, …, A_{m} associated with A in _{σ^{±i}}M, with each of the R-submodules A_{0}, …, A_{j}, for some 0 < j ≤ m, being (T, ±)-clean, while each of the R-modules A_{j+1}, …, A_{m} is T -clean (if j < m). Then each V_{±}(A_{k}), for 0 ≤ k < j, is isomorphic to a minor subfactor of the critical T -module V_{±}(A_{k+1}); while V_{±}(A_{j}) is isomorphic to a minor subfactor of the T -critical module T ⊗_{R} A_{j+1}, and each T ⊗_{R} A_{l}, for j + 1 ≤ l < m, is isomorphic to a minor subfactor of the T -critical module T ⊗_{R} A_{l+1}. Hence,

K(T ⊗_{R} M) ≥ K(T ⊗_{R} A_{m}) > … > K(T ⊗_{R} A_{j+1})
> K(V_{±}(A_{j})) > K(V_{±}(A_{j−1})) > … > K(V_{±}(A_{0})).

Thus, we obtain

K(T ⊗_{R} M) ≥ K(V_{•}(A_{m})) ≥ K(V_{±}(A_{0})) + m = K(V_{±}(A)) + h_{±}(A : _{σ^{±i}}M),

where V_{•}(A_{m}) = T ⊗_{R} A_{m}, if j < m, while V_{•}(A_{m}) = V_{±}(A_{m}), if j = m. Consequently, K(T ⊗_{R} M) ≥ μ_{±}, and, finally, K(T ⊗_{R} M) ≥ μ.

Without loss of generality, we may suppose that M is (T, •)-clean, by Lemma 2.7.(2).

If M is (T, ±)-clean then, by Proposition 2.8, there exists a (T, ±)-clean minor subfactor, N say, of _{σ^{±i}}M, for some i ≥ 0, such that

K(V_{±}(M)) = K(V_{±}(N)) + 1.

Note that here we are using the fact that if N is (T, ±)-clean then so is _{σ^{±i}}N, for all i ≥ 0.

Since M is critical, so is _{σ^{±i}}M, and so K(N) < n. Hence, using the inductive hypothesis, there exists a simple subfactor A of _{σ^{±j}}N, for some j ≥ 0, such that

K(V_{±}(N)) = K(V_{±}(A)) + h_{±}(A : _{σ^{±j}}N).

Consequently,

K(V_{±}(M)) = K(V_{±}(A)) + h_{±}(A : _{σ^{±j}}N) + 1.

In addition, A ∈ M_{±,(i+j)}, and h_{±}(A : _{σ^{±(i+j)}}(M)) ≥ h_{±}(A : _{σ^{±j}}N) + 1, since N is a minor subfactor of the (T, ±)-clean module _{σ^{±i}}M. Hence,

K(V_{±}(M)) ≤ K(V_{±}(A)) + h_{±}(A : _{σ^{±(i+j)}}(M)) ≤ μ(M).

If M is T -clean, then by Theorem 3.5, there exists either a T -clean minor subfactor N of M such that K(T ⊗_{R} M) = K(T ⊗_{R} N) + 1, or a (T, ±)-clean minor subfactor N_{±} of σ^{±i}M, for some i ≥ 0, such that K(T ⊗_{R} M) = K(V_{±}(N)) + 1. Note that K(N) < n, since M is critical.

In the latter two cases, we use the same arguments as above to show that K(T ⊗_{R}M) ≤ μ_{±}. In the first case, K(T ⊗_{R} N) = μ(N), by the inductive hypothesis. Clearly, μ(M) ≥ μ(N) + 1, hence K(T ⊗_{R} M) = μ(N) + 1 ≤ μ(M).

2.(i) As an R-module, T ⊗_{R} A is the direct sum of simple modules ⊕_{i∈ℤ} _{σ^{i}}A. Thus, if the orbit, O{A}, of A is infinite (that is, _{σ^{i}}A ≄ _{σ^{j}}A, for all i ≠ j), then T ⊗_{R} A is a simple T -module and so K(T ⊗_{R} A) = 0. If the orbit O(A) is finite, and contains n elements, then ^{σ^{n}}A ≃ A, and there exists x ∈ R\I with σ^{n}(I)x ⊆ I. The nonzero element u := 1 ⊗ (1 + I) + (Y^{n} ⊗ x) is annihilated by I; moreover, the submodule T u of T ⊗_{R} A is proper and isomorphic to T ⊗_{R} A, by using the ℤ-grading of T ⊗_{R} A and the fact that A is a-torsionfree. Therefore, 1 ≤ K(T ⊗_{R} A) ≤ K(A) + 1, and so K(T ⊗_{R} A) = 1.

(ii)If St_{±}(A) is infinite, then K(V_{±}(A)) = 1, by (4) or (5). If St_{±}(A) is finite, then the orbit O{A} is infinite. Let h be the maximal hole in Ho_{+}(A), and let s be the minimal star in St_{−}(A). Then K(V_{+}(A)) = K(V_{+}(_{σ^{h}}A)) and K(V_{−}(A)) = K(V_{−}(_{σ^{s}}A)).

As R-modules V_{+}(_{σ^{h}}A) = ⊕_{j≥h} _{σ^{j}}A and V_{−}(_{σ^{s}}A) = ⊕_{j≤s} _{σ^{j}}A are the direct sums of nonisomorphic R-modules, hence, as T -modules, they are simple, and so K(V_{±}(A)) = 0.

Corollary 4.2 Let R be a left noetherian ring with finite Krull dimension. Then

K(T) = max{K(V_{•}(A)) + h_{•}(A : R) | A ∈ M, • ∈ {∅, ±}}

where M is the family of simple (T, •)-clean R-modules, and V_{•}(A) is equal to T ⊗_{R} A when A is a-torsionfree, and is equal to V_{±}(A) when A is a-torsion.

Proof. Observe that T ≃ T ⊗_{R} R, and that _{σ^{i}}R ≃ R, for all i ∈ ℤ.

Corollary 4.3 Let R be a left noetherian ring with finite Krull dimension. Then

K(T) = K(R)

unless there exists a simple (T, •)-clean R-module A such that K(V_{•}(A)) = 1 and h_{•}(A : R) = K(R), in which case

K(T) = K(R) + 1.

Corollary 4.4 Let R be a left noetherian ring with finite Krull dimension. Then

K(T) = K(R)

unless there exists either

(i) a simple a-torsionfree R-module A such that h(A : R) = K(R) and O{A} is finite; or (ii) a simple (T, +)-clean R-module A such that h_{+}(A : R) = K(R) and the set St_{+}(A) is infinite; or (iii) a simple (T, −)-clean R-module A such that h_{−}(A : R) = K(R) and the set St_{−}(A) is infinite.

5 Fully Bounded Noetherian rings

In this section, we specialize our results to the case of Generalized Weyl Algebras, T = R(σ, a), where R is a fully bounded noetherian ring. In this case we recover the same formula as that obtained in the case of a commutative noetherian base ring in [7].

Lemma 5.1 Let R be a left noetherian ring, and let A be a simple R-module. Set p = Ann_{R}(A), and assume that R/p is a simple artinian ring. Then

1. If A is a-torsionfree, then K(T ⊗_{R} A) = 1 if and only if p is invariant under some nonzero power of σ.

2. If A is a-torsion, then K(V_{±}(A)) = 1 if and only if the set Z_{±}(p) := {i ≥ 0 | a ∈ σ^{±i}(p)} is finite.

Proof. 1. By Theorem 4.1.2(i), K(T ⊗_{R} A) = 1 if and only if the orbit O{A} is finite, or, equivalently, σ^{n}(p) = p, for some n ≥ 1, since R/p is a simple artinian ring.

2. By Theorem 4.1.2(ii), K(V_{±}(A)) = 1 if and only if the set St_{±} is infinite, or, equivalently, if and only if the set Z_{±}(p) is infinite, since R/p is a simple artinian ring.

Proposition 5.2 Let R be a fully bounded noetherian ring, and let A be a simple Rmodule. Set p = Ann_{R}(A).

If A is a-torsionfree, then h(A : R) = height(p).

If A is a-torsion, then

height(p) = max{h_{+}(_{σ^{j}}A : R) | j ≫ 0} = max{h_{−}(_{σ^{j}}A : R) | j ≪ 0}.

Proof. If A is a-torsion, then _{σ^{±j}}A is (T, ±)-clean, for j ≫ 0; and h_{±}(_{σ^{±j}}(A : R)) ≤ h_{±}(_{σ^{±k}}(A : R)), for k ≥ j ≫ 0. Hence, without loss of generality, in the case that A is a-torsion, we may assume that A is (T, ±)-clean. Suppose that there is a (T, •)- clean sequence A = A_{0}, A_{1}, …, A_{n} associated with A in R. Each of the annihilators p_{i} := Ann_{R}(A_{i}) is a prime ideal of R and A_{i} is a nonsingular R/p_{i}-module, so that K(A_{i}) = K(R/p_{i}), see, for example [10, Theorem 2.5, Proposition 1.4]. Now, p_{i} > p_{i+1}, for each i, since p_{i} ≥ p_{i+1} and K(p_{i}) > K(p_{i+1}). Thus, height(p) ≥ n, and so height(p) ≥ h_{•}(A : R), By combining this with the facts that Ann_{R}(_{σ^{j}}A) = σ^{j}(p) and height(σ^{j}(p)) = height(p), for all j ∈ ℤ, we obtain, in the case that A is a-torsion,

height(p) ≥ h_{±} := max{h_{±}(_{σ^{±j}}A : R) | j ≫ 0}.

Next, consider a strictly descending chain of prime ideals p = p_{0} > p_{1} > … > p_{n}. Set A = A_{0}. By repeating the argument in [9, Proposition 6.2], we can construct cyclic critical R/p_{i}-modules A_{i}, for each i < n, such that A_{i} is isomorphic to a minor subfactor of A_{i+1}. Since A_{n} is cyclic, it is isomorphic to a subfactor of R. Each A_{i} is compressible, by [10, Theorem 2.5]. If A_{i} is a-torsionfree, then A_{i} is T -clean, by Lemma 2.5, and so _{σ^{j}}A_{i} is T - clean, for each j ∈ ℤ. If A_{i} is a-torsion, then, by using Lemma 2.5.(2), or Lemma 2.5.(3), we conclude either that _{σ^{j}}A_{i} is (T, +)-clean, for almost all positive integers j, or that _{σ^{j}}A_{i} is (T, −)-clean, for almost all negative integers j.

Hence, if A is a-torsionfree, then h(A : R) ≥ n, and so h(A : R) ≥ height(p); while if A is a-torsion then there exist a positive integer, j say, such that each _{σ^{j}}A_{i} is either T -clean or (T, +)-clean, so that h_{+}(_{σ^{j}}A : R) ≥ n, and h_{+} ≥ height(p); similarly, h_{−} ≥ height(p).

Theorem 5.3 Let R be a fully bounded noetherian ring with finite Krull dimension. Then K(T) = K(R) unless there exists a maximal ideal p of R such that height(p) = K(R) and either p is invariant under some nonzero power of σ, or there are infinitely many i ∈ ℤ with σ^{i}(a) ∈ p.

Proof. This follows from Corollary 4.4 and the above Proposition.

Corollary 5.4 Let R be a fully bounded noetherian ring with finite Krull dimension. Then the left and right Krull dimensions of T coincide.

Proof. This follows immediately form the above result, by using the ±-symmetry.

6 Examples

In this final section, we present a class of examples that demonstrates that the methods we have introduced and the theorems we have proved do, indeed, enable one to calculate Krull dimensions.

Let R be a ring. Suppose that we are given the following data: σ ∈ Aut(R) and b, ρ ∈ Z(R), the centre of R, with ρ being a σ-stable unit; that is, σ(ρ) = ρ. We form an overing, E = R⟨σ; b, ρ⟩, by adjoining symbols X and Y to R subjected to the relations:

Xr = σ(r)X, Y r = σ^{−1}(r)Y, for all r ∈ R; XY − ρY X = b.

As an example, observe that in the case that R = K[H], ρ = 1 and σ is defined by σ(H) = H − 1, while b = 2H, then E is the enveloping algebra U(sl(2)).

We may view E as the iterated skew polynomial ring E = R[Y; σ^{−1}][X; σ, ∂], where ∂ is a σ-derivation of R[Y; σ^{−1}] and ∂R = 0, while ∂Y = b; moreover σ is extended from R to R[Y; σ^{−1}] by setting σ(Y) = ρY.

When R is left Noetherian it is known, [15, 6.5.4], that

K(R⟨σ; b, ρ⟩) = K(R) + 1 or K(R) + 2.

It is not trivial to decide which of the two cases actually happens.

The rings of type E fall within in the scope of this paper because, as the lemma below shows, the rings E = D⟨σ; b, ρ⟩ are generalized Weyl algebras, see [4].

Lemma 6.1 Let R be a ring; then R⟨σ; b, ρ⟩ ≃ R[H](σ, H) and σ is extended from R to R[H] by σ(H) = ρH + b.

If b = 0, then ∂ = 0 and E = R[Y; σ^{−1}][X; σ], thus K(E) = K(R) + 2, by [15, 6.5.4.(i)]. From here on, we will assume that b ≠ 0.

Let K be an algebraically closed field of characteristic zero, and let A_{1} = A_{1}(K) be the first Weyl Algebra; so that A_{1} = K[p, q] with pq − qp = 1. We aim to compute the Krull dimension of the algebra

E = A_{1}⟨σ; b(≠ 0), ρ⟩, σ ∈ Aut_{K}(A_{1}).

We may view E in the form E = A_{1} ⊗K[H](σ, H), where σ(H) = ρH +b, by the previous lemma. The algebra A_{1} is central simple, since the characteristic is zero; so both ρ and b are nonzero scalars. If ρ ≠ 1, then σ(H − γ) = ρ(H − γ), where γ = b(1 − ρ)^{−1}, while if ρ = 1, then σ(b^{−1}H) = b^{−1}H + 1. Thus, by making the obvious change of variables, we may assume that either (i) σ(H) = ρH, with ρ ≠ 1 and a = H + γ, or (ii) σ(H) = H + 1 and a = bH.

Lemma 6.2 Let K be an algebraically closed field of characteristic zero. Then

K(A_{1}⟨σ; b(≠ 0), ρ⟩) = 3

if and only if at least one of the following two cases holds:

1. ρ ≠ 1 and there is a simple A_{1}-module M such that ^{σ^{n}}M ≃ M, for some n ≥ 1, or 2. ρ ≠ 1, but ρ^{n} = 1, for some n ≥ 2.

In all other cases, K(A_{1}⟨σ; b(≠ 0), ρ⟩) = 2.

Proof. (⇐) The algebra A_{1}⟨σ; b, ρ⟩ is isomorphic to E = A_{1}⊗K[H](σ, H), where either (i) σ(H) = ρH, with ρ ≠ 1 and a = H + γ, or (ii) σ(H) = H + 1 and a = bH. Suppose first that case 1 above holds; so that ρ ≠ 1, and σ(H) = ρH. Let M be a module as in case 1 above. The simple K[H]-module K_{0} := K[H]/(H) is isomorphic to ^{σ}K_{0}. The simple A_{1} ⊗ K[H]-module M_{0} := M ⊗ K_{0} is a-torsionfree, while ^{σ^{n}}M_{0} ≃ M_{0}, as A_{1} ⊗ K[H]- modules. Also, h(M_{0} : A_{1}⊗K[H]) = 2, because the sequence {M_{0}, M ⊗K[H], A_{1}⊗K[H]} is a T -clean sequence associated with M_{0} in A_{1} ⊗ K[H], since all the modules involved are compressible. It follows, from Theorem 4.1, that

K(E) = K(A_{1} ⊗ K[H]) + 1 = K(A_{1}) + 2 = 3.

Suppose next that case 2 holds. Let N be a simple A_{1}-module, and let K_{−γ} = K[H]/(a) = K[H]/(H + γ). Then the simple A_{1} ⊗ K[H]-module N_{−γ} := N ⊗ K_{−γ} is a-torsion, and both of the sets St_{±}(N_{−γ}) are infinite, since σ^{n}(H) = ρ^{n}H = H. By Lemma 2.5, there is a positive integer i such that _{σ^{±i}}N_{−γ} is (T, ±)-clean. Also, h_{±}(_{σ^{±i}}N_{−γ} : A_{1} ⊗ K[H]) = 2, since the sequence {_{σ^{±i}}N_{−γ}, _{σ^{±i}}N_{−γ} ⊗ K[H], A_{1} ⊗ K[H]} is a (T, ±)-clean sequence associated with _{σ^{±i}}N_{−γ} in A_{1} ⊗ K[H], by noting that the second and third modules in the sequence are compressible. The result follows as in the first case.

(⇒) We now prove the converse. The field K is algebraically closed and the first Weyl algebra is almost commutative, hence, by Quillen’s Lemma, [15, 9.7.3(a)], the simple A_{1} ⊗ K[H]-modules are exactly the modules which are isomorphic to modules of the form M_{λ} := M ⊗ K_{λ}, where M is a simple A_{1}-module and K_{λ} := K[H]/(H − λ), for some λ ∈ K.

Now assume that neither case 1 nor 2 holds. There are two possibilities: either ρ = 1, or the orbit O{M} is infinite for every simple A_{1}-module M and ρ is not a root of unity. In either case, it can easily be checked that the orbit O{M_{λ}} is infinite and the set St(M_{λ}) is finite, for every simple A_{1} ⊗ K[H]-module M_{λ}. Thus, K(E) = K(A_{1} ⊗ K[H]) = 2, by Theorem 4.1.

It is easy to give examples of the two possibilities. If ρ ≠ 1 is a root of unity then K(E) = 3, while if we choose σ to be the identity automorphism and b = ρ = 1, the algebra E is isomorphic to the second Weyl algebra A_{2} = A_{1} ⊗ A_{1}. In this case, we have K(A_{2}) = 2, since the characteristic is zero.

In fact, the analysis of this example can also be used to establish the following corollary. First, recall that a K-algebra A is said to have the endomorphism property over K if, for each simple A-module M, the endomorphism ring End_{K}(M) is algebraic over K, see, for example [15, 9.1.4]. Every countably generated algebra over an uncountable field has the endomorphism property, [15, 9.1.7].

Corollary 6.3 Let K be an algebraically closed field of characteristic zero, and let A be a left noetherian K-algebra which is a domain, has Krull dimension equal to one, and has the endomorphism property over K. Let E = A⟨σ; b, ρ⟩, with σ ∈ Aut_{K}(A) and b, ρ ∈ K^{∗} = K\{0}. Then K(E) = 3 if and only if at least one of the following cases holds.

1. ρ ≠ 1 and there is a simple A_{1}-module M such that ^{σ^{n}}M ≃ M, for some n ≥ 1, or 2. ρ ≠ 1, but ρ^{n} = 1, for some n ≥ 2.

In all other cases, K(A⟨σ; b, ρ⟩) = 2.

Proof. Quillen’s Lemma is used in the proof of the previous lemma to establish that the first Weyl algebra has the endomorphism property. Here we given the endomorphism property and the proof is then the same.

A particular class of algebras to which we could apply this corollary would be the McConnell-Pettit algebras, [14]. Let K be an uncountable algebraically closed field of characteristic zero, and let Λ = (λ_{ij}) be a multiplicatively antisymmetric matrix of nonzero elements of K. The McConnell-Pettit algebra, P (Λ), associated with this data, is the K-algebra generated by X_{1}^{±1}, …, X_{n}^{±1}, subject to the commutation relations X_{i}X_{j} = λ_{ij}X_{j}X_{i}. In the case that the multiplicative subgroup of K^{∗} generated by the λ_{ij} has maximal rank, n(n − 1)/2, the algebra P (Λ) is known to have Krull dimension one, [14, Corollary 3.10]; and so the Corollary above applies.

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