Dimension theory in iterated local skew power series rings

By analysing the structure of the associated graded ring with respect to certain filtrations, we deduce a number of good properties of iterated local skew power series rings over appropriate base rings. In particular, we calculate the Krull dimension, prime length and global dimension in well-behaved cases, obtaining lower bounds to complement the upper bounds obtained by Venjakob and Wang.


Introduction
The main objects of study of this paper are iterated skew power series rings for suitable choices of k and (σ i , δ i ).By this, we mean that R 0 = k, and we iteratively define for each 1 ≤ i ≤ n and finally set R = R n .Full definitions of these objects are given below.
Skew power series extensions B = A[[x; σ, δ]] were first introduced in [18] and [20], with a view to using them to study Iwasawa algebras: it was noted in [18] that certain Iwasawa algebras can be written as iterated skew power series extensions over a base field (usually F p ) or complete discrete valuation ring (usually Z p ).
Modelled on the notion of a skew polynomial extension A[x; σ, δ], a skew power series extension A[[x; σ, δ]] is a ring which is isomorphic to A[[x]] as topological left (or right) A-modules, with multiplication given by xa = σ(a)x + δ(a) (0.2) for all a ∈ A. Here, σ is an automorphism of A, and δ is a σ-derivation of A, i.e. a linear map δ : A → A satisfying δ(ab) = δ(a)b + σ(a)δ(b).(We say that (σ, δ) is a skew derivation for short.) Unlike the case of skew polynomial extensions, it is not the case that an arbitrary skew derivation (σ, δ) on A gives rise to a well-defined ring A[[x; σ, δ]], due to possible convergence issues.For this reason, we work throughout the paper with a special class of skew power series extensions whose question of existence is already well understood.We denote this class of rings by SPS(A), and call them local skew power series extensions: we require A to have a unique maximal ideal m, with respect to which it is complete (and separated), and we stipulate that σ(m) ⊆ m, δ(R) ⊆ m and δ(m) ⊆ m 2 .These are known to exist, and while other classes have been studied (see e.g.[2] and [9, §5], dealing with the rather different case in which δ is assumed locally nilpotent ), the local case encompasses the case of interest in the Iwasawa context.
There is a growing literature on simple power series extensions S = R[[x; σ, δ]]: see, for instance, [19] and [13] for important milestones in this theory.However, very little work has been done to treat the iterated case so far.
The most basic noncommutative case, that of q-commutative power series rings (completed quantum planes), has been studied in [14].A few properties of general iterated local skew power series rings are established in [21] (though see the remarks before Theorem D below for a caveat).The author is not aware of any other developments in the theory.
As above, we restrict our work to the local case, and denote the class of n-fold iterated local skew power series extensions of A by SPS n (A) (Definition 1.10 below).Even under rather general conditions like this, we can understand much of the dimension theory of these rings: our first result, along these lines, complements the work of Wang [21].
This result follows from Theorem 4.3, Theorem 4.4 and Corollary 4.10.
The process of iteration is often rather badly behaved.Indeed, in the notation of (0.1): when n ≥ i ≥ 2, there is no guarantee that (σ i , δ i ) restricts to a skew derivation of R j for any 0 ≤ j ≤ i − 2. Hence we identify a subclass of SPS n (A), which we denote RSPS n (A), and whose elements we call rigid, whose elements are well-behaved in the above sense.See Definition 3.8 for the precise definition.
This appears to be a fairly restrictive condition: we expect that RSPS n (A) is in fact a much smaller class than SPS n (A) except in trivial cases when n and A are very small.But, in fact many well-known rings do lie in RSPS n (A), including q-commutative skew power series rings (Example 3.11), Iwasawa algebras of supersoluble uniform groups (Examples 3.12-3.13),and various other completed quantum algebras (Examples 3.14-3.15).
We are interested in the prime ideal structure of iterated skew power series rings R ∈ SPS n (A).As a first step towards a more general theory, we calculate the classical Krull dimension (prime length), as defined in [15], of elements of RSPS n (A), and show that they obtain the upper bound given by the Krull dimension, similarly to pure automorphic extensions, which are well studied and in some senses easier to understand.This perhaps indicates that, in understanding the theory of prime ideals of iterated non-pure-automorphic skew power series rings, a good first step will be to consider the rigid ones.
Theorem B. Let k be a division ring, and The proof of Theorem B is contained within Theorem 4.4 and Proposition 4.6.
So far we have not said much about the natural filtration on R, but this will be crucial for understanding its ideals in future work.If R is an iterated local skew power series ring with unique maximal ideal m, we can consider R as a complete m-adically filtered ring.We compute its associated graded ring as follows: Theorem C. Let (R, m) be a complete local ring, (σ, δ) a local skew derivation on R, and S = R[[x; σ, δ]] a local skew power series extension of R with unique maximal ideal n.Then there exists a skew derivation (σ, δ) of gr m (R) such that the inclusion of graded rings gr m (R) → gr n (S) extends to an isomorphism (gr m (R))[X; σ, δ] ∼ = gr n (S) upon mapping X to gr(x).Moreover, if δ(m) ⊆ m 3 , then δ = 0.
The proof is given in Lemma 2.3.
We give Example 2.5 to show that δ can indeed be nonzero: this corrects a small error in [20, Proposition 2.9], [21, 2.3(iii)] and [13, 3.10(i)], and a similar error can be found in [18,Lemma 1.4(iv)].This phenomenon also arises in practice when studying the Iwasawa algebras of certain non-uniform 2-valuable groups.However, we can get rid of δ by modifying the filtration, which we prove as Lemma 2.6: Theorem D. Let R and S be as in Theorem C. Then there exists a filtration f of S, cofinal with the n-adic filtration, such that the inclusion of graded rings gr f (R) → gr f (S) extends to an isomorphism (gr f (R))[X; σ] ∼ = gr f (S) upon mapping X to gr(x).
With this in mind, we note in §2.3 that the results of [21] hold even in cases such as Example 2.5.
Finally, we note some basic structural results that seem to be absent from the literature.
Theorem E. Let R and S be as in Theorem C, and let n be the maximal ideal of S. Let f be either the n-adic filtration or the filtration of Theorem D. Then: (i) If gr f (R) is prime, then gr f (S) is prime, and hence S is prime.
(ii) If gr f (R) is a maximal order, then gr f (S) is a maximal order, and hence S is a maximal order.
(iii) The natural inclusion map R → S is faithfully flat.
Part (i) is proved as Proposition 2.7, part (ii) as Lemma 2.9 and part (iii) as Lemma 1.11.

Notation and conventions
We often talk about topological rings: unless specified, the filtration on a local ring R with unique maximal ideal m is the m-adic filtration.All filtrations considered are assumed to be discrete, positive and separated.When we need to specify a filtration explicitly, we will usually denote it as a function f : R → N ∪ {∞}.

Definitions and first results
Write A for the left R-module of left formal power series over R. Suppose the multiplication in R, together with the multiplication rule (0.2), extends to a well-defined right R-module structure on A. Then there is a unique R-bimodule map A ⊗ R A → A such that r ⊗ s → rs for all r, s ∈ R and x i ⊗ x j → x i+j ; and A becomes a ring under this multiplication.In this case, we write R[[x; σ, δ]] := A for this ring.Note that R[[x; σ, δ]] contains R, and is a skew power series ring over R as described above by definition.
When A has a right R-module structure in this way, we will say as shorthand simply that the ring R[[x; σ, δ]] exists, and otherwise that R[[x; σ, δ]] does not exist.
The following computational lemma will be useful.
Lemma 1.1.Fix a positive integer n.Then, for a ∈ R, we have inside the skew polynomial ring R[x; σ, δ] or (if it exists) the skew power series ring R[[x; σ, δ]].Here, M n is the set of formal (noncommutative) monomials m = m(X, Y ) of degree n in the variables X and Y , and e(m) is the total degree of X in the monomial m.
Proof.When n = 1, this is just the multiplication rule (0.2) in S. For n > 1, this follows by an easy induction: note that all elements of M n are either of the form Xm(X, Y ) or of the form Y m(X, Y ) for some m ∈ M n−1 .

Local skew power series extensions
There are several variant notions of "locality" in the literature; for the purposes of this paper, we fix the following definition.
Definition 1.2.Let R be a ring.Then R is local if it has a unique maximal two-sided ideal m, and R is scalar local ring if additionally m is maximal among right (and hence also left ) ideals.(It is easy to see that scalar local implies local, but that this implication is not reversible.) An alternative formulation of this definition, which is sometimes useful, goes as follows.Let R be a ring with unique maximal ideal m.Then R is local (resp.scalar local ) if R/m is simple artinian (resp.a division ring).
We will sometimes say that (R, m) is a local (resp.scalar local ) ring, as shorthand for "R is a local (resp.scalar local) ring with unique maximal ideal m".
The residue ring of a local ring (R, m) is defined to be the quotient R/m.
Remark 1.3.Let R be a ring with unique maximal left ideal m.Then it is well known that m is also the unique maximal right ideal, and is hence equal to the Jacobson radical of R.
The applicability of our results is guaranteed by the lemma below.
Definition 1.4.Let (R, m) be a complete local ring.We say that (σ, δ) is a skew derivation of (R, m) (or just R) if σ is an automorphism of R and δ is a (left) σ-derivation of R. We say that the skew derivation (σ, δ) is We do not deal with the question of existence any further in this paper; all rings written are implicitly assumed to exist.
Definition 1.6.Let (R, m) be a local ring.We will say that the ring S is a local skew power series ring over R, or a local skew power series extension of R, if it satisfies the following properties: (i) S is a skew power series ring over R with respect to some automorphism σ and some σ-derivation δ, say Remark 1.7.Condition (iii) implies that δ(m n ) ⊆ m n+1 for all n ∈ N. Hence, taken together, conditions (ii) and (iii) are simply the natural requirement that the multiplication data σ, δ of R should interact "nicely" with the m-adic topology on R, i.e. σ should be a homeomorphism, and δ should be topologically nilpotent.
As the hypotheses above are cumbersome to repeat, we set up some shorthand notation that will remain in force for the rest of the paper.Definition 1.8.Let (R, m) be a complete local ring.Then we will write S ∈ SPS(R) to mean that S is a local skew power series ring R[[x; σ, δ]] over R. If σ and δ are A-linear (for some subring A ⊆ R), we may write S ∈ SPS A (R) to emphasise this fact.We may also choose to specify the maximal ideals of R or S explicitly, e.g. by writing (S, n) ∈ SPS(R, m) to mean that m and n are the maximal ideals of R and S respectively.
(i) n = m + xR is the unique maximal ideal of S, and the natural inclusion R ⊆ S induces an isomorphism Proof.(i) and (iii) follow from [20, §2].(ii) follows from (i) and the remark immediately after Definition 1.2.Definition 1.10.Let (R, m) be a complete local ring.Then, for each positive integer n, we will inductively define the notation S ∈ SPS n (R) as follows: S ∈ SPS 1 (R) if and only if S ∈ SPS(R); and, for n > 1, we write S ∈ SPS n (R) if and only if S ∈ SPS(T ) for some T ∈ SPS n−1 (R).
As before, we may write S ∈ SPS n A (R) if all the σ i and δ i are A-linear.In this way, the expression SPS n A (R) may be read as "the class of n-fold iterated A-linear local skew power series extensions of R".
If S ∈ SPS n (R), the number n is called the rank of S (over R). (Later, we will see that this number can often be recovered from the global dimensions or Krull dimensions of R and S.)

Constructing skew derivations
In the rare case when we wish to construct a local skew derivation on a ring R, the following lemma gives us a method of doing so.
Let k be a fixed local ring, and ] with maximal ideal m.Let τ be a fixed automorphism of R (necessarily preserving m).Proof.First, as the elements x 1 , . . ., x n are k-linearly independent, the mapping To extend d to a k-module homomorphism R → R, it will suffice to define d(m) for each ordered monomial m in the elements x 1 , . . ., x n .We do this inductively on the degree of the monomial as follows.If the monomial m is of degree 1, then m ∈ {x 1 , . . ., x n }, and the value of d(m) is already known.Proceeding inductively, let n > 1: if m is a monomial of degree n, then it may be written uniquely as x j m ′ , where m ′ is a monomial of degree n − 1 in x j , x j+1 , . . ., x n .Then we will define d(m To show that this is indeed a τ -derivation, we must show that the value of d(m) is well-defined regardless of how it is computed.More precisely: let m be an ordered monomial of total degree s, m = x j1 x j2 . . .x js , where j 1 ≤ j 2 ≤ • • • ≤ j s .Then, for each 1 ≤ r < s, the monomial m can be written as the product m = p r q r , where p r = x j1 x j2 . . .x jr and q r = x jr+1 x jr+2 . . .x js .Then we may define: Note that d(m) was defined to be d 1 (m).In this notation, we must show that We will do this by induction on s.There is nothing to check when s = 2. Now suppose that, for some N , we have established for all monomials m of total degree 2 ≤ t ≤ N .Take a monomial m of total degree N + 1: in the above notation, we will write it as m = x j1 x j2 . . .x jN+1 = p r q r for each 1 ≤ r ≤ N .
Fix 1 ≤ r ≤ N − 1.Then m = p r q r = p r+1 q r+1 , where q r = x r+1 q r+1 and p r+1 = p r x r+1 , i.e. m = p r x r+1 q r+1 qr = p r x r+1 pr+1 q r+1 .So, exactly as above, we may calculate Hence these d i are all equal, and in particular are equal to d.This shows that d is a τ -derivation as required.

Filtrations and their associated graded rings
Let k be a complete local ring and R ∈ SPS n (k).Then R is also a complete local ring, say with maximal ideal m.Studying the m-adic filtration and its associated graded ring is the key to understanding many basic properties of R.
(i) When k is a field, it is easy to see that the rank n of (R, m) ∈ SPS n (k) over a field is uniquely defined, and can be recovered as (iii) Let (R, m) be a complete local ring, and (S, n) ∈ SPS n (R).Then the restriction of the n-adic valuation to R is the m-adic valuation, i.e.R ∩ n i = m i .In particular, m is generated in n-adic degree 1. (iv) Let (R, m) be a complete local ring, and (S, n) ∈ SPS n (R, m).We will always write gr(R) for the graded ring associated to the m-adic filtration, and likewise gr(S) for the graded ring associated to the n-adic filtration.
We now calculate this associated graded ring inductively.
Let R be a complete local ring with maximal ideal m, take a local skew derivation (σ, δ) of R, and form S = R[[x; σ, δ]] ∈ SPS(R) with maximal ideal n.Note that σ and δ induce linear endomorphisms of the graded ring gr(R), of degrees 0 and 1 respectively, as follows: for all x ∈ m λ \ m λ+1 , where λ ∈ N, we have It is easy to check that σ is in fact a graded automorphism.
Remark 2.4.It is easy to see that δ = 0 if for all nonzero x ∈ R, where v is the m-adic filtration: v(x) = λ if x ∈ m λ \ m λ+1 .We give an example for which δ = 0.
Retain the above notation, and continue to write v for the m-adic filtration on R.
Lemma 2.6.The function f : S → R ∪ {∞} defined by is a Zariskian ring filtration on S, and the associated graded ring is The claims of the lemma now follow easily.

Lifting properties from the graded ring
Set f 1 to be the n-adic filtration on S, and f 2 to be the filtration obtained in Lemma 2.6.
The proofs in [21] were given under the erroneous assumption that δ = 0 (in the notation of Lemma 2.3), but they remain true with essentially identical proofs even after removing this assumption.(We note that they also hold on replacing f 1 by f 2 , of course.)Throughout this subsection, we fix either f = f 1 or f = f 2 , giving a filtration f on R and S. The expressions gr(R) and gr(S) will always implicitly mean gr f (R) and gr f (S) respectively.
We record some further important properties that lift from the graded ring.Proposition 2.7.Let R be a complete local ring, and S ∈ SPS n (R).If gr(R) is prime, then gr(S) is prime, and hence S is prime.
Corollary 2.8.Let R be a complete local ring, and S ∈ SPS n (R).Suppose that gr(R) is Auslander regular.Then S is AS-Gorenstein.
Proof.S is Auslander regular by [21, 2.3(iii)], and so in particular S is Auslander-Gorenstein.As S is scalar local by Lemma 1.9(ii), we deduce from [6, Lemma 4.3] that S is AS-Gorenstein.Lemma 2.9.Let R be a complete local ring such that gr(R) is a maximal order, and take S ∈ SPS n (R).Then S is a maximal order.

Rigid iterated extensions
Let (R 0 , m 0 ) be a complete local ring, and let S ∈ SPS n (R 0 ) for n ≥ 2. That is, for each 1 ≤ i ≤ n, we may inductively find We noted in Remark 1.7 that we wanted each R i to interact "nicely" with the topology of R i−1 , i.e. (σ i , δ i ) should be a local skew derivation of (R i−1 , m i−1 ) (see Definition 1.6(ii-iii)).Unfortunately, naively iterating this procedure as in Definition 1.10, we may end up in a situation like the following.Writing the above chain of local rings explicitly, and (σ, δ) is a local skew derivation of R n−1 as desired, but we are not guaranteed that it restricts to a local skew derivation (σ| Ri , δ| Ri ) of R i for any 0 ≤ i ≤ n − 2, as the following example shows.
where σ is the k-linear automorphism of R given by σ(Y ) = Z, σ(Z) = −Y , and δ = 0. Then S ∈ SPS 2 k (R 0 ), but σ(m 0 ) ⊆ m 0 .This turns out to be a very natural stipulation to make when performing this iterative construction, and this motivates the definitions we make in this section.
We slightly extend this notion as follows.
Definition 3.4.Let R be a ring and I an ideal.Let σ = (σ 1 , . . ., σ r ) and δ = (δ 1 , . . ., δ r ), where each (σ i , δ i ) is a skew derivation of R. We will say that I is a (σ, δ)-ideal if it is a (σ i , δ i )-ideal for each 1 ≤ i ≤ r.Proposition 3.5.Let R be a complete local ring, and Proof.Both claims follow from recursive application of Lemma 3.3.

Skew power series presentations
Definition 3.6.Fix a complete local ring (R 0 , m 0 ) and a ring R where R i ∈ SPS di (R i−1 ) for each 1 ≤ i ≤ ℓ, and The number ℓ is called the length of the presentation.
The next two examples are not crucial for the current paper, but we include them to illustrate the wide applicability of Theorems B and D.
Example 3.14.Completed quantised k-algebras.Let k be a field, Γ = (γ ij ) ∈ M n (k × ) a multiplicatively antisymmetric matrix, and P = (p 1 , . . ., p n ) ∈ (k × ) n , Q = (q 1 , . . ., q n ) ∈ (k × ) n two vectors with p i = q i for all 1 ≤ i ≤ n.Horton's algebra R = K P,Q n,Γ (k), a simultaneous generalisation of quantum symplectic space and quantum Euclidean 2n-space, can be written as an iterated skew polynomial ring, where the σ i (for 2 ≤ i ≤ n) and τ i (for 1 ≤ i ≤ n) are k-linear automorphisms, and each δ i (for 2 i n) is a klinear τ i -derivation.In [21, §3.2], it is proved that the I-adic completion R of R, where I = (x 1 , y 1 , x 2 , y 2 , . . ., x n , y n ), is an iterated skew power series extension of k: We do not spell out the relations in full: see [10, Proposition 3.5] for details.It is only necessary, for our purposes, to know the following: ) are scalar multiples of x j for all j < i; • σ i (y j ), τ i (y j ), δ i (y j ) are scalar multiples of y j for all j < i; ) is a k-linear combination of the elements y l x l for all l < i.
It now follows by an easy calculation that the saturated presentation associated to (3.2) is stable, and hence that R ∈ RSPS 2n k (k).Example 3.15.Completed quantum matrix algebras.Let k be a field, λ ∈ k × a scalar, and p = (p ij ) ∈ M n (k × ) a multiplicatively antisymmetric matrix (i.e.p ij p ji = 1 for all i, j).Then the multiparameter quantum n × n matrix algebra R = O λ,p (M n (k)) can be defined (see e.g.[3, Definition I.2.2]) as a skew polynomial ring in n 2 variables labelled X i,j for each 1 ≤ i, j ≤ n, in which k is central.Again, we do not spell out the relations in full, but we note: • X l,m X i,j is a linear combination of X i,j X l,m and X i,m X l,j when l > i and m > j; • X l,m X i,j is a scalar multiple of X i,j X l,m whenever either l ≤ i or j ≤ m.
In [21, §3.2] it is proved that the I-adic completion R of R, where I is the ideal generated by the n 2 variables X i,j for 1 ≤ i, j ≤ n, is an iterated skew power series extension of k satisfying the same relations: R ∈ SPS n 2 k (k).But the "obvious" saturated presentations, e.g.those associated to (where we have omitted the skew derivations for readability) are usually not stable.(For a counterexample, take R to be the usual quantum 2 × 2 matrix algebra O q (M 2 (k)), given by n = 2, p = 1 q q −1 1 , and λ = q −2 for any q ∈ k × .See [3, Definition I.1.7])for the relations in this case.) We fix this by adjoining the variables in the following order: • at the 0th stage, adjoin the "antidiagonal" elements X i,j satisfying |i + j − (n + 1)| = 0, in any order; • at the 1st stage, adjoin those X i,j satisfying |i + j − (n + 1)| = 1; • at the 2nd stage, adjoin those X i,j satisfying |i + j − (n + 1)| = 2; and so on, until finally all variables have been adjoined at the end of the (n − 1)th stage.Diagrammatically: It is easy to verify that such a presentation of R is stable.The only case in which there is anything to prove is when l > i and m > j: multiplying X l,m by X i,j results in a term involving the variables X i,m and X l,j , and we must check that each of the variables X i,m and X l,j has been adjoined before we adjoin both of X l,m and X i,j .
More precisely, let M = | max{i + j, l + m} − (n + 1)| be the first stage after which both X l,m and X i,j have been adjoined, and likewise let N = | max{i + m, l + j} − (n + 1)| be the first stage after which both X i,m and X l,j have been adjoined.Then it is easy to see that the inequalities l > i and m > j imply i + j < i + m < l + m, i + j < l + j < l + m, and hence N < M .
The existence of such a stable presentation shows that R ∈ RSPS n 2 k (k).

Non-examples
Showing that an extension is not rigid appears to involve lots of tedious calculation, but we give two examples which seem of interest to the theory.By construction, S ∈ SPS 3 k (k).Suppose that S is rigid, and denote its maximal ideal by n.Then there must exist a stable presentation for S. We will show that no such stable presentation can exist.Write I = m 1 S and J = m 2 S: then, by Lemma 1.9(i),I = sS and J = sS + tS with s, t ∈ n, and s, t ∈ n 2 by Remark 2.2(iii).
As S is rigid, I = m 1 S must in fact be a two-sided ideal.This places some restrictions on possible choices for s, which we now compute.
Henceforth, we work in S/n 3 for ease of computation.We have Multiplying out the right-hand side: Now, equating the coefficients of each monomial on both sides, some tedious case-checking shows that the only solution to this congruence is b Hence we have s = aX + ε, so that a = 0. Now, as S is rigid, J = m 2 S must also be a two-sided ideal, and so we calculate the restrictions that this places on t.
As R 1 ∈ SPS k (k), it must be a commutative power series ring over k, so that and an easy calculation shows that there must be a unit η Indeed, equating monomial coefficients on the left and right hand sides: On the one hand, multiplying the equations labelled (i = 0) and (i = p) together, we get and a very similar (but easier) calculation shows that we must have a = 0. Hence s = ε ∈ n 2 : that is, m 1 is generated in n-adic degree ≥ 2. This contradicts Remark 2.2(iii).

Dimension theory
Many of the results in this section can be slightly extended; but we will not always strive for full generality, and often work over a field or a division ring for simplicity.

Krull dimension
Let k be a division ring, and (R, m) ∈ SPS n (k).Remark 2.2(v)(c) implies that Kdim(R) ≤ n: in this subsection, we show that this is always an equality.
For any ring R, write I r (R) for the lattice of right ideals of R.
The following is the result corresponding to [13, 3.14(ii)] in the case when I is not a (σ, δ)-ideal.Finally, it is clear that, if I 1 ≤ I 2 , then θ(I 1 ) ≤ θ(I 2 ); and, if J = θ(I), then we may recover I as J/Jx.This shows that θ is a strict map of posets.Proof.The proof of this theorem closely follows some of the methods of [1].We will calculate the right Krull dimension of S, but the calculation of the left Krull dimension is identical.Let θ continue to denote the map I r (R) → I r (S) of Proposition 4.1.
When n = 0, we have S = gr(S) = k, and there is nothing to prove.We proceed by induction on the rank of S. Together with the inequality of [21, Corollary 2.9(iv)], we see that n ≤ Kdim(S) ≤ Kdim(gr(R)) + 1 = n, and the result follows.

Classical Krull dimension
m) be a complete local ring, and (σ, δ) a local skew derivation of R. Then the ring R[[x; σ, δ]] exists.

Lemma 1 .
11. S is a faithfully flat R-module.Proof.As a left R-module, we have S ∼ = ∞ i=0 Rx i , a direct product of flat R-modules, which is flat by [7, Theorem 2.1], as R is noetherian.We may now conclude that S is faithfully flat as a left R-module using [15, Proposition 7.2.3]:given any proper right ideal I of R, we have IS ⊆ mS ⊆ n = S. (The same holds for S as a right module by a symmetric argument.) 2 Filtered rings 2.

Lemma 2 . 1 .
Given any choice of b 1 , . . ., b n ∈ m 2 , the assignment d(x i ) = b i for all 1 ≤ i ≤ n extends to a unique local k-linear τ -derivation d of R.

Example 2 . 5 .
Take R = k[[x]] with skew derivation σ = id and δ(x) = x 2 : this is indeed a skew derivation by e.g.2.1.Form S = R[[y; id, δ]].It is easy to see that the graded ring has nonzero δ.Set X = x + n 2 and Y = y + n 2 inside gr(S): then the multiplication in gr(S) is determined by the rule

Definition 3 . 2 .
Let R be a ring and I an ideal.If (σ, δ) is a skew derivation of R, then I is said to be a (σ, δ)-ideal if σ(I) ⊆ I and δ(I) ⊆ I.This is a useful class of ideals because of results such as the following lemma: Lemma 3.3.Let R be a complete local ring, and S = R[[x; σ, δ]] ∈ SPS(R).Suppose that I is a (σ, δ)-ideal of R. Then IS = SI is a two-sided ideal, and S/IS ∼ = (R/I)[[x; σ, δ]] ∈ SPS(R/I).

Proposition 4 . 1 .
Let R be a complete local ring, S = R[[x; σ, δ]] ∈ SPS(R), and let I be a right ideal of R. Set I[[x; σ, δ]] := { a i x i : a i ∈ I}.Then I[[x; σ, δ]] is a right ideal of S.Moreover, the map θ : I r (R) → I r (S) sending I to I[[x; σ, δ]] is a strictly increasing poset map.Proof.I[[x; σ, δ]] is clearly an additive subgroup of S. Take r ∈ R and a = ∞ i=0 a i x i ∈ I[[x; σ, δ]].Then we may evaluate ar inside S: i m(σ, δ)(r)x e(m) in the notation of Lemma 1.1.Now, m(σ, δ)(r) ∈ R, so a i m(σ, δ)(r) ∈ I, and hence ar ∈ I[[x; σ, δ]].We need to check that this gives a well-defined right S-action on I[[x; σ, δ]] -in other words, that the right actions of the elements xr and σ(r)x + δ(r) agree for all r ∈ R.But this is already true a fortiori, as the right action of S on I[[x; σ, δ]] is just induced by multiplication inside the ring S.

Lemma 4 . 2 .
Let (R, m) be a complete local ring with residue ring k.Let Y ⊆ X be adjacent right ideals of R: then, as right S-modules, θ(X)/θ(Y ) is isomorphic to S/mS ∼ = k[[x; σ]], a skew power series ring of automorphic type.Proof.Let θ continue to denote the map I r (R) → I r (S) defined in Proposition 4.1.Y ⊂ X are adjacent if and only if X/Y is a simple right module.In particular, the annihilator ann(X/Y ) R must be the unique maximal right ideal m of R, and so we must haveX/Y ∼ = R/m ∼ = k.The proposed isomorphism is obvious on the level of abelian groups, and it is easy to see that right multiplication by x ∈ S is the same as right multiplication byx ∈ k[[x; σ]].It remains to check the R-action.Recall, from Lemma 1.1, that ∞ i=0 a i x i r = ∞ i=0 m∈Mi a i m(σ, δ)(r)x e(m)for all a i ∈ X, r ∈ R. But, for a given m ∈ M i , if m(σ, δ) contains an instance of δ (i.e. if e(m) < i), then m(σ, δ)(r) ∈ m, and hencea i m(σ, δ)(r) ∈ Xm ⊆ Y .This implies that ∞ i=0 a i x i r = ∞ i=0 a i σ i (r)x iinside X/Y , as required.Theorem 4.3.Let k be a division ring, and S ∈ SPS n (k).Then Kdim(S) = Kdim(gr(S)) = n.

Theorem 4 . 4 .
Let k be a division ring, and R ∈ RSPS n (k).Then clKdim(R) = n.Proof.It is well known that clKdim(R) ≤ Kdim(R) (e.g.[15, Lemma 6.4.5]), and here Kdim(R) = n by Theorem 4.3.To obtain a lower bound, consider the chain of ideals {m i R} n i=0 given in Proposition 3.10.This chain has length n, and again by Proposition 3.10, the quotient rings are iterated local skew power series rings over k, and are hence prime by Proposition 2.7.Hence clKdim(R) ≥ n.

Remark 4 . 5 .
Rigidity is a sufficient, but not a necessary, condition to have clKdim(R) = Kdim(R), as shown by the following proposition together with Non-example 3.16.Proposition 4.6.Let k be a division ring, and let R ∈ SPS n (k) have pure automorphic type.Then clKdim(R) = n.