Ring Theoretic Properties of Iwasawa Algebras

Ring theoretic properties of Iwasawa algebras

Parham Hamidi November 1, 2017

Abstract Let p be a fixed but arbitrary prime number unless it is stated otherwise. In the firstpartof this project, we briefly discuss profinite groups and pro-p groups. Moreover, we develop and prove necessarily tools required in the following sections. The reader who feels comfortable with the notions of profinite and pro-p groups can skip this section and start off by reading section 2. In the second part, we move on to introduce Iwasawa algebras. We discuss a few important results and machinery given in [11] and we finish by noting some properties of Iwasawa algebras as rings, as in [4]. In the last section, we summarize [7], which gives an interesting result regarding reflexive Iwasawa algebras, called the control theorem, for short. We also noteafew consequences of the control theorem. We adopt the notation given in [11], in particular, N denotes the set of all non-negative integers which includes 0. All rings are assumed to be unital.

Contents

1 Preliminary 2 1.1 Profinite groups ...... 2 1.2 Inverse limits ...... 4 1.3 Pro-p groups ...... 7

2 Iwasawa algebra 8 2.1 Uniform groups ...... 8 2.2 Powerful Zp-Lie algebras ...... 10 2.3 p-adic analytic groups ...... 12 2.4 Structure of Iwasawa algebras ...... 13

3 Reflexive ideals in Iwasawa algebra 20 3.1 Introduction to reflexive ideals ...... 20 3.2 A few important results about reflexive ideals and derivations ...... 21 3.3 Frobenius pairs and the derivation hypothesis ...... 26 3.4 The control theorem for principal reflexive ideals generated by a normal element .. 28 3.5 Control theorem for reflexive ideals ...... 30 3.6 Consequences of the control theorem of reflexive ideals for Iwasawa algebras ..... 33

Bibliography 37

1 Motivation

Recall that, if K is a number field, then an elliptic curve E over K is a smooth projective curve of genus 1 with a K-rational point. If K does not have characteristic 2 or 3, then E can expressed in the form

y2 = x3 + ax + b a, b ∈ K.

The group E(K), which is defined to be the set of K-rational points of E, has a central importance in number theory and the Mordell-Weil theorem states that E(K) is a finitely generated abelian group1 and the rank of E(K) is called the Mordell-Weil rank. On the other hand, for a tower of number fields it is usually a hard question to find a nice pattern of how their class numbers change. However, Iwasawa noticed that the p-th Sylow subgroups of the class groups of such towers behave much more nicely. Iwasawa studied the growth of class groups of certain towers of number fields such as Zp-extensions of number fields (that is the Galois group of the extension is isomorphic to Zp). This technique was extrapolated to find the Mordell-Weil rank of p-adic Lie extensions of number fields (again this means that the Galois group of the extension isa p-adic Lie group where charts are over Zp instead of R). While the classical Iwasawa theory is usually concerned with abelian extensions of number fields (that is the Galois groups of the extensions are abelian), onemight want to work in more generalized settings. Among the key players in the non-commutative Iwasawa theory are objects called Iwasawa algebras and also finitely generated modules over them (see [10] and [22]). Although, they were rigorously defined and some fundamental properties of them were found by Lazard [12] more than 60 years ago, they may still behave in ways that are mysterious to us and there is much more to learn about them.

1 Preliminary 1.1 Profinite groups Let us recall that by a topological group we mean a group G endowed with a topology such that the group inversion and multiplication are continuous with respect to the given topology; that is the maps

m : G × G → G (g, h) 7→ gh, where G × G has product topology and

i : G → G − g 7→ g 1 are continuous. For example any group G endowed with the discrete topology is a topological group. The Galois group of any Galois extension endowed with the Krull topology is another example of a topological group. Then we have the following definition. Definition 1.1. Let G be a topological group such that

(i) G is compact and Hausdorff; (ii) the set of open subgroups of G forms a local basis for the open neighbourhoods of the identity element 1 in G. 1The rank of the group E(K) is called the Mordell-Weil rank which the Birch and Swinnerton-Dyer conjecture (also known as the BSD conjecture) claims it is the same as the analytical rank of E coming from the order of vanishing of the associated L-function at 1. Iwasawa’s main conjecture tries to give an account for this deep relation.

2 Then we call G a Profinite group. We note that any finite group endowed with the discrete topology is a profinite group. Moreover, one can show that in any countable topological group G that is Hausdorff and locally compact, G has a singleton that is isolated, i.e. it is clopen, and by continuity of translation in a topological group we see that G must have the discrete topology2 and being compact forces G to be finite. Thus any profinite group is either finite with the discrete topology or has uncountable cardinality. For example, the ring of integers Z is not a profinite group. Let us take a closer look at each ofthe conditions stated above:

1. Since G is compact, if H ≤o G, where we mean H is an open subgroup of G, then G = ⊔ggH (disjoint union) where g runs over the left-coset representatives of H in G and since H is open, gH is open for any g ∈ G. The compactness of G implies that every open subgroup of G has finite index. Furthermore, G\H = ⊔ggH where g runs over the left-coset representatives except for the coset H i.e gH ̸= H and so if H ≤o G, then H is also closed in G. A similar argument as above shows that if H ≤c G (closed subgroup) and [G : H] is finite, then H is open in G.

2. For a topological group G, being T1 (being accessible) is enough to show that G is Hausdorff and by continuity of translation, it is enough that {1} is closed in G. Indeed, suppose G is a topological group for which {1} is closed and suppose g, h ∈ G are such that g ̸= h. Then − − {g 1h} is closed and thus, G\{g 1h} is an open neighbourhood of 1. Since the map

− − f : (x, y) 7→ (x, y 1) 7→ xy 1

is a composition of continuous maps, then f is a continuous map and hence the pre-image of − G\{gh 1} is open and contains U × V where U and V are nonempty neighbourhoods of 1 in G. We see that gU ∩ hV = ∅, otherwise, we have gu = hv for some u ∈ U and v ∈ V and − − − − − since f (U × V) = UV 1 ⊆ G\{g 1h} we see g 1h = uv 1 ∈ G\{g 1h} which is impossible. ≤ ∃ ⊆ ⊆ ∪ 3. If H G and U o G (open subset) with U non-empty such that U H, then H = h∈HhU where hU is open for all h ∈ H and so H is open. 4. Any open subgroup H of G contains an open normal subgroup in G. To see this, consider the intersection of all conjugates of H in G; It is a closed subgroup of H and by the tower law of groups it has finite index and hence it is open and by construction it is normal. So condition (ii) can be replaced by ”open normal subgroups of G form a local basis around 1”. This implies that, every open set in G can be written as a union of cosets of normal open subgroups of G. 5. One can show that condition (ii) can be replaced by requiring that G has a totally disconnected topology, that is, any set with more than one element in G is disconnected (see Appendix B, [11]). Using the above remarks the following proposition is easy to prove.

Proposition 1.2. Suppose G is a profinite group and H ≤c G is endowed with the subspace topology. Then H is also a profinite group. Moreover, if L ≤o H, then L = H ∩ K for some K ≤o G. Proof. Since G is compact and Hausdorff and since H is closed in G, then H is also compact and Hausdorff. Note that by the definition of the subspace topology, weknowthat L = H ∩ U where ⊆ ∪ ≤ α ∈ U o G is an open neighbourhood of 1. Hence U = α∈I Kα where Kα o G for all I and ∩ ∪ ∩ ≤ thus, L = H U = α∈I H Kα. By compactness of L, we see that for any L o H, we have

2Another way to see this is that, since G is a compact Hausdorff group, we can find a normalized Haar measure on G. By sigma-additivity of Haar measure, we see that if G is countable but not finite then by the translation invariance property of Haar measure, all elements of G should have the same measure. But the measure of each element cannot be zero as they should sum to 1 and it cannot be positive as they would sum to infinity.

3 ∪n ∩ ∈ N 3 L = i=1 H Ki for some finite n + . Without loss of generality, suppose n is taken to be minimal and let us prove the claim by the induction on n. For n = 1, this is the claim. Suppose ≤ ∪N+1 ∩ we know the claim holds for all n N for some N and suppose L = i=1 H Ki. Then we note ∪N ∩ that i=1 H Ki is an open subgroup of H and hence by the induction hypothesis, we have that ∪N ∩ ∩ ′ ′ ≤ ∩ ′ ∪ ∩ i=1 H Ki = H K for some K o G. This implies that L = H (K KN+1) = H K for some K ≤o G by the induction hypothesis. In particular, since {K | K ≤o G} forms a local basis for neighbourhoods of 1 in G, then {H ∩ K | K ≤o G} = {L | L ≤o H} forms a local basis for neighbourhoods of 1 in H.

1.2 Inverse limits An alternative definition of profinite groups can be given using the concept of inverse limits.We begin with the following definition. Definition 1.3. Suppose (I, ≤) is a partially ordered set with some relation ≤ and suppose that for any i, j ∈ I there exists k ∈ I such that i ≤ k and j ≤ k. Then we call (I, ≤) (or just I) a directed set. Intuitively, we can think of directed sets as generalizations of linearly ordered sets, where although we may not be able to compare any two given objects, they have a common upper bound. Let us, in view of the next definition, denote i → j whenever i ≤ j for i, j ∈ I, and so (I, ≤) := (I, →). In particular, i → i for any i ∈ I and if i → j → k, then i → k. → { } Definition 1.4. Let (I, ) be a directed set and suppose Xi i∈I is a family of topological spaces (or groups, topological groups, rings, etc.) such that whenever i → j for i, j ∈ I, there exists a π → π ∈ continuous map (or group homomorphism and etc.) ij : Xj Xi such that ii = idX for all i I → → π π π { π } i and whenever i j k in I, then ij jk = ik. We call Xi , ij I an inverse system. Now for any inverse system of topological spaces (or other structures given above) we can define the inverse limit (also known as projective limit) as follows. Definition 1.5. Suppose {X , π } is an inverse system, then its inverse limit, denoted4 by lim X , i ij I ←−i∈I i is { ∈ | π → } lim←− Xi := (xi)i∈I ∏ Xi ij(xj) = xi whenever i j in I , i∈I i∈I ∏ where i∈I Xi is endowed with the product topology and lim←− Xi is endowed with subspace topology in ∏i∈I Xi. The operator lim←− may be seen as a generalization of intersection in the following sense. Suppose { } S = Xi i∈I is a family of subspaces of a space X with the property that for any Xi and Xj in S, we ∩ ∈ → have Xi Xj S. Thus the index set I is a directed set, where S is ordered by inclusion and i j in ⊆ ∈ ∩ ∈ I whenever Xj Xi (reverse inclusion relation). For any i, j I, if Xk := Xj Xi for some Xk S, ⊆ ⊆ → → π then Xk Xi and Xk Xj. So i k and j k and hence I is a directed set. Let ij be reverse { π } ∼ ∩ inclusion, then Xi , ij I is an inverse system and we see that lim←− Xi = i∈I Xi. In particular, if ∩ ∈ ∈ → i∈I Xi = Xk for some k I, then this means that for any i I, i k. In other words, k is the common upper bound for all elements in I and lim←− Xi = Xk. In the language of categories, we may think of I as a category where objects are the elements of I and the partial ordering given by the morphism between two objects. Then definition 1.4. describes a contravariant functor from I to the category of topological spaces Top (or groups and so on) and this is why we prefer i → j to i ≤ j. It is also worth mentioning that the construction lim X ←−i∈I i comes with a universal property. 3 By N+ we mean positive integers, that is N+ = {1, 2, ... }. 4 π This is a standard notation; however, it does not specify the maps ij’s, even though the resulting inverse limit depends on them.

4 We see that there is a morphism ϕ : lim X → X (called projection) for each j ∈ I and π ϕ = ϕ . j ←−i∈I i j ij j i If X is any other object with these properties, then there is a unique morphism h : X → lim X ←−i∈I i that makes the following diagram commute.

h ϕ′ ϕ′ i j lim X ←−i∈I i

ϕ ϕ j i X π X j ij i

{ π } Proposition 1.6. For an inverse system Xi , ij I of non-empty Hausdorff compact topological spaces, the inverse limit lim←− Xi is also Hausdorff compact and non-empty. Proof. The following proof is from [17], Chapter IV Proposition 2.3. By Tychonoff’s theorem (which is equivalent to the axiom of choice) ∏i∈I Xi, endowed with the product topology, is compact and it is non-empty; it is also Hausdorff. The important part of the proof relies on the factthat X := lim←− Xi ∈ → is closed in ∏i∈I Xi which we shall show first. Fix i, j I with i j and let { ∈ | π } Xij := (xi)i∈I ∏ Xi ij(xj) = xi . i∈I ∩ ∏ → Then X = lim←− Xi = i→jXij. So it is enough to show that Xij is closed in i∈I Xi for any i j in φ → I. Consider the natural projection i : ∏i∈I Xi Xi which is continuous in the product topology. π π φ φ ∈ We see that the condition ij(xj) = xi can be expressed as ij j(x) = i(x) for x ∏i∈I Xi. If we π ◦ φ φ φ let g := ij j then the relation g(x) = i(x), where i and g are continuous maps on X to Xi, → defines a closed set in a Hausdorff topology. Hence Xij is closed in ∏i∈I Xi for any i j in I and thus X is closed in ∏i∈I Xi. Therefore X is compact and Hausdorff. It remains to show that X is ∩ ∅ non-empty. Suppose towards a contradiction that X = i→jXij = . Then, since X is compact, this ⊆ → ∩ ∅ means that there exists a finite set S I of i j such that SXij = . As I is a directed set the elements in S have a common upper bound, say n in I; that is for all i ∈ S we have i → n for some ∈ ∈ n I. Note this is possible since S is finite. Then if we consider some x := (xi) ∏i∈I Xi with the π → property that in(xn) = xi for all i n in I and arbitrary otherwise. Then by the definition of Xij ∈ → ∈ ∩ we have x Xij for all i j in S and so x SXij. This is a contradiction and hence we have the result. We have the following important proposition which tells us there is a correspondence between profinite groups and the inverse systems of finite groups, governed by the inverse limit.Note that for a profinite group G, if N ◁o G (normal open subgroup), then G/N is a finite group and hence profinite. If we consider the set N := {N : N ◁o G} then we can make N a directed set by considering the reverse inclusion. We say N → M whenever M ⊆ N; we note that the intersection of two open normal subgroup is again normal and open. Thus N is a directed set. For N → M we π → π have the natural continuous group homomorphism NM : G/M G/N given by NM(gM) = gN. { π } So G/N, NM N is an inverse system. { π } Proposition 1.7. Suppose G is a profinite group then if G/N, NM N is the inverse system discussed above. We have topological isomorphism ∼ G = lim←− G/N. N∈N Conversely, if {G , π } is an inverse system of finite groups, then lim G is a profinite group. i ij I ←−i∈I i

5 b Proof. To show the first part of the proposition, let G := lim G/N and we consider the natural ←−N∈N group homomorphism

ϕ : G → ∏ G/N N∈N 7→ g (gN)N∈N . ∈ ϕ ∈ → ⊆ N π If (gN)N∈N (G) for some g G then for any N M (M N) in , we have NM(gM) = gN. ϕ ≤ b ϕ ∩ { } N Hence (G) G. On the other hand, ker = N∈N N = 1 . This is because forms a basis for the neighbourhoods of 1 in G; if g ̸= 1 then there exists N ◁o G such that g ̸∈ N. Hence ∈ ∩ ϕ ϕ g / N∈N N = ker , as G is Hausdorff. Therefore is injective. ϕ b ∈ b Now we show that (G) = G. Suppose (gN N)N∈N G, we need to find some g in G such that ∈ ∈ N (gN)N∈N = (gN N)N∈N , that is, g gN N for all N . Suppose towards a contradiction that ∩ ∅ such g does not exist in G. This implies that N∈N gN N = . Note that gN N is closed in G. ∩ ⊆ N Thus N∈N gN N is closed and compact in G and hence there exists a finite subset S such that ∩ ∅ N∈SgN N = . We argue similar to the proof of Proposition 1.6 to see that there exists a common ∈ N λ λ ∈ upper bound for the elements of S, say M . So there exists = ( N) ∏N∈N G/N such π λ → λ ∈ ∩ that NM( M) = gN N for all N M. This implies that N∈SgN N, which is a contradiction. ∈ ∩ ϕ Hence there exists some g N∈SgN N and consequently (g) = (gN N)N∈N . It remains to show that ϕ is continuous and open. Let us first show continuity. For any N ∈ N , consider the open subgroup H(N) of ∏N∈N G/N defined by

H(N) = ∏ {1} × ∏ G/M, N→M N̸→M where by 1 we mean the identity element in the appropriate group. Note that G/M is finite with the discrete topology and therefore {1} is open in G/M for any M ∈ N . Note that {H(N) : N ∈ N } { ∩ b} forms a basis for the neighbourhood of 1 in the product topology ∏N∈N G/N. Thus H(N) G b forms a local basis around 1 in G. So it is enough to show that H(N) has open preimage in G for all N ∈ N , that is, N ◁o G. By construction of H(N), we have that H(N) = ϕ(N). This is because N → M means that M ⊆ N and hence NM/M = N/M in G/M. If M ⊈ N, then N contains a − representative for each coset of M in G/M and so NM/M = G/M. Therefore ϕ 1(H(N)) = N which is open in G by construction. The openness of ϕ is due to a more general phenomenon where every bijective continuous map from a compact space to a Hausdorff space is a open and thusa homeomorphism. Here with our notation, let U be open in G and let P := ∏N∈N G/N. Then G\U is closed and hence compact. Since ϕ is continuous, ϕ(G/U) is also compact in P. Moreover, ϕ(G/U) is also closed, as P is a Hausdorff space. But ϕ(G/U) = P/ϕ(U) and we conclude that ϕ(U) is open. ∈ Conversely, as Gi is finite with the discrete topology for all i I, Tychonoff’s theorem implies that ∏i∈I Gi is compact; it is also Hausdorff. It remains to verify condition (ii). Similar to our argument above, consider a finite subset S of I and define

( ) = ∏{ } × ∏ ≤ ∏ U S : 1Gi Gi o Gi, i∈S i∈/S i∈I which forms a local basis around 1 in ∏i∈I Gi as S varies. This means that ∏i∈I Gi is a profinite ∏ group and by Proposition 1.6 we know lim←− Gi is a closed subgroup of i∈I Gi. Thus Proposition 1.2 implies that ∏i∈I Gi is a profinite group. Recall that any finite group is a Galois group of some field extension. Note that wecanendowa Galois group with the Krull topology. Then Galois theory states that there is a correspondence between intermediate field extensions and closed subgroups of Galois group. Furthermore, finite

6 subextensions correspond to open subgroups. It is worth mentioning that, due to Galois correspondence, any profinite group can also be seen as the Galois group of some abstract Galois field extension([17], Chapter IV). Example 1.8. Consider Z and fix some prime number p. Then we can consider the family of normal subgroups of the form piZ for i ∈ N. Note that if i ≤ j (i → j) in N, then pjZ ⊆ piZ. There is a natural homomorphism π Z jZ → Z iZ ij : /p /p a + pjZ 7→ a + piZ.

Z iZ {Z iZ π } N Note that /p is a finite abelian group and /p , ij N is an inverse system (note that is a linearly ordered set and so it is directed). Then Z := lim Z/piZ is a profinite group called p ←−i∈N the (ring of) p-adic integers. Less abstractly, an element of Zp can be seen as a formal power series ∑∞ i ∈ { − } in one variable p of the form i=0 ai p where ai 0, 1, ... , p 1 . Example 1.9. More generally, suppose G is a group (or a ring) and Λ is a family of normal subgroups (or ideal) with finite index in G, such that Λ is a directed set with reverse inclusion. Then the b profinite group GΛ := lim G/N is called the profinite completion of G with respect to Λ. ←−N∈Λ Example 1.10. Again consider Z but this time consider Λ, consisting of all non-trivial normal subgroups (ideals) of Z. Since Z is an abelian group, every subgroup is normal. Moreover, Z is a principal ideal domain, so Λ = {nZ : n ∈ N}. We say n → m whenever mZ ⊆ nZ which means that n | m. We have the natural surjective homomorphism πnm : Z/mZ → Z/nZ. Thus {Z/nZ , πnm}N is an inverse system and b Z := lim←− Z/nZ n∈N

Z Z Z vp(n)Z is a profinite group (a Prüfer ring). By the Chinese remainder theorem, /n = ∏p /p where p varies over all prime numbers and vp measures the highest power of p dividing an integer. Zb ∏ Z Then we see that, looking at lim←− as a functor, = p p. Example 1.11. More generally, if G is a group and Λ consists of all normal subgroups of G with b finite index, then G := lim G/N is called the profinite completion of G. ←−N∈Λ

1.3 Pro-p groups Now we focus our attention on special type of profinite groups called pro-p groups, where p is a fixed but arbitrary prime number. Recall that a finite p-group G is a group which has order equal to a power of p. In particular, by Lagrange’s theorem, every element of a p-groups also has order equal to a power of p.

Definition 1.12. Suppose G is a profinite group with the property that for any N ◁o G, G/N is a p-group. Then we call G a pro-p group.

Recall that in a profinite group any open subgroup H ≤o G contains a normal open subgroup N ◁o G. Since both H and N have finite index then, by the tower law of group index, [G : H] | [G : N]. Hence any open subgroup of G also has a power of p as its index. Example 1.13. Any finite p-group is a pro-p group. Example 1.14. Recall Z = lim Z/piZ. One can show that Z is a discrete valuation ring and p ←−i∈N p i if N is an open non-trivial ideal of Zp with finite index, then it is of the form N = p Zp for some i ∼ i i ∈ N. Moreover it is easy to show that Zp/p Zp = Z/p Z as rings. Therefore Zp is a pro-p group and it can be considered as the most fundamental example of a pro-p group.

7 We show that similar results, as in the case of profinite groups, hold true in the case ofpro-p groups.

Proposition 1.15. Suppose G is a pro-p group and H ≤c G (closed subgroup). Then H is also a pro-p group.

Proof. By Proposition 1.2, we know that H is a profinite group. Suppose N ◁o H. Then N = H ∩ L | for L <0 G. As discussed above, L has p-power index in G. Since [H : N] [G : L] we see H/N is a p-group. Proposition 1.16. G is a pro-p group if and only if G is topologically isomorphic to an inverse limit of an inverse system of finite p-groups.

Proof. If G is a pro-p group then, by definition of a pro-p group, for all N ◁ G we have G/N is a ∼ o finite p-group; Proposition 1.7 implies that G = lim G/N, as required. ←−N◁o G Conversely, suppose {G : π } is an inverse system of finite p-groups and consider G = lim G . i ij I ←−i∈I i Π Π Then G is a profinite group. Let us denote ∏i∈I Gi by and let U(S) be the open subgroup of associated to a finite subset S of I, discussed in the proof of Proposition 1.7. Then we have seen that { ⊆ } Π { } U(S) : S I, S finite forms a local basis of the neighbourhoods of 1 in . Moreover H(S) S∈F forms a local around 1 in G, where F is the set of all finite subsets of I and H(S) := U(S) ∩ G for all S. Subsequently | Π | | n [G : H(S)] [ : U(S)] = ∏ Gi = p , i∈S for some non-negative n. Thus H(S) has index equal to a power of p. For any open subgroup N of G we can find some S ∈ F such that N contains H(S). The tower law of group index implies that N also has index equal to a power of p in G. Consequently, G is a pro-p group. 1

2 Iwasawa algebra 2.1 Uniform groups Suppose G is a profinite group. Analogous to the Jacobson radical for rings, we can consider the Frattini subgroup of G, denoted by Φ(G). If G contains a proper maximal open subgroup, then Φ(G) is defined to be the intersection of all proper maximal open subgroups of G, otherwise Φ(G) := G . The Frattini subgroup of a profinite group carries many interesting properties. We can show using Zorn’s lemma that Φ(G) consists of all non-generators of G. By a non-generator we mean an element g ∈ Φ(G) such that, whenever X ⊆ G is such that ⟨X ∪ g⟩ = G (X ∪ {g} generates G topologically), then ⟨X⟩ = G.

Moreover, suppose M ◁o G is a maximal open subgroup of G, and suppose H ≤ G is such that − − − g 1 Mg < H for some g ∈ G. Thus M < gHg 1, which implies that gHg 1 = G and hence H = G. − − Therefore g 1 Mg is also a maximal open subgroup of G and g 1Φ(G)g = Φ(G) for any g ∈ G. So Φ(G) is a normal closed subgroup of G. Similarly one can show that Φ(G) is a characteristic subgroup of G, that is, it is invariant under all automorphisms of G. In the case of a pro-p group G, the Frattini subgroup of G turns out to be of the form (Proposition 1.13 in [11])

Φ(G) = Gp[G, G],

− − − where Gp := ⟨gp : g ∈ G⟩, [G, G] := ⟨[g, h] : g, h ∈ G⟩ and [g, h] := g 1gh = g 1h 1gh. For a pro-p group G we have the following important descending series, known as the lower p-series of G, defined recursively as follows.

8 ≥ Definition 2.1. Suppose G is a pro-p group. Then we let P1(G) := G and for all n 2

p Pn(G) := Pn−1[Pn−1(G), G].

p Φ ⊆ ≥ Note that P2(G) = G [G, G] = (G) and Pn+1(G) Pn(G) for all n 1. Moreover, note that Pn(G) is a closed subgroup in G and hence it is also a pro-p group. Since [Pn(G), Pn(G)] ≤ [Pn(G), G], we Φ ≤ see that (Pn) Pn+1(G). ≥ { }∞ If G is finitely generated, then Pn(G) is open for all n 1 and Pn(G) n=1 forms a basis around the identity element in G (Proposition 1.16 in [11]). In particular, we do not need to take the p ≥ closure in the definition 1.17, that is, Pn(G) = Pn−1[Pn−1(G), G] for all n 2. This can be used to show that for a finitely generated5 pro-p group, a subgroup is open if and only if it has finite index. Consequently, the topology is completely fixed by the group structure (Theorem 1.17, [11]). Definition 2.2. Suppose G is a pro-p group. If p ̸= 2, then G is said to be a powerful group whenever [G, G] ≤ Gp. If p = 2, then G is powerful, when [G, G] ≤ G4.

Let us for simplicity denote Pn(G) by Gn. If G is a powerful pro-p group, then

Φ p p G2 = (G) = G [G, G] = G . p ∈ N If, furthermore, G is finitely generated, then G2 = G and we can show that for all n pn { pn ∈ } Gn+1 = G = g : g G ,

pk ≥ and Gk+n = Pk+1(Gn) = Gn for all n 1 (Theorem 3.6, [11]). In particular, each Gn is a finitely generated powerful group for all n ∈ N. Definition 2.3. A powerful finitely generated pro-p group G is uniform (or uniformly powerful), if ≥ ∼ [G : G2] = [Gn : Gn+1] for all n 2, which implies that G/G2 = Gn/Gn+1. Equivalently, a powerful finitely generated pro-p group is uniform if and only if it is torsion free ⊇ ⊇ (Theorem 4.5 [11]). If we consider the chain G G2 G3 ... for a powerful finitely generated pn group G where Gn+1 = G , since there is a surjective map, we see that → f : Gn/Gn+1 Gn+1/Gn+2, p ≥ ≥ sending xGn+1 to x Gn+2. Then [Gn : Gn+1] [Gn+1 : Gn+2] for all n 1 and hence we have a decreasing sequence which should terminate, as each index is finite (and in particular is of the form d p from some d ≥ 0). It follows that for n large enough, Gn is a uniform group, which implies that every finitely generated pro-p group contains a uniform subgroup.

Example 2.4. Recall that Zp is a pro-p group. It is also true that Zp is a uniform group (and again the most fundamental one). Notice that every Cauchy sequence in a profinite group G converges6 in G, as G is compact and Hausdorff. Suppose G is a pro-p group and suppose a ∈ Zp and hence ∑∞ i ∈ { − } a = i=0 ai p where ai 0, ... , p 1 is a unique representation of a in the p-adic expansion. Then ∞ ∑k i ∈ Z → → ∞ { sk } let us denote sk := i=0 ai p and note that sk a as k . Then the sequence g k=0 is d d Cauchy, since if N ◁o G, then G/N = p from some d ≥ 0. Therefore sm = sn mod p for all m, n large enough and so gsm = gsn mod N. This means that gsk → h for some h ∈ G as k → ∞ and it makes sense to talk about

ga := lim gsk , k→∞

5By a finitely generated group G in this section, we always mean finitely generated topologically, that is, there { } ⊆ ⟨ ⟩ exists a finite set g1, ... , gn G such that g1,, ... , gn = G, unless it is stated otherwise. 6 { }∞ That is, if gi i=0 is a sequence G such that for any N ◁o G there exists N = N(i) such that gm = gn mod N or −1 ∈ → ∈ → ∞ equivalently gm gn N, then gi g G as i .

9 for any a ∈ Zp and g ∈ G. We can show that Zp is topologically generated by {1}, in other words, Zp n Zp = ⟨1⟩ = Z = 1 . Note that Zp is abelian and in particular powerful. Also Gn = p Zp and hence

nZ n+1Z ∼ nZ n+1Z Z Z Gn/Gn+1 = p p/p p = p /p = /p , for all n ≥ 1. Thus Zp is a uniform group. Zd ∈ N { }d Example 2.5. Similarly G = p for any d + is a uniform group generated by ei i=1, with d ≥ ei = (0, ... 1, ... 0) where 1 is in the i-th position and we have [Gi : Gi+1] = p for all i 1. Let G be a pro-p group and suppose g ∈ G. Investigating our construction in example 2.4., the a 7 Zp map ϕ : Zp → G with a 7→ g is a continuous homomorphism. Obviously ⟨g⟩ ≤ g and by the Z Z Z continuity of ϕ, g p is compact in G. Since G is Hausdorff, g p is closed and so ⟨g⟩ ≤ g p . On the Z a sk ∈ Z sk ∈ ⟨ ⟩ p ≤ ⟨ ⟩ other hand, g = limk→∞ g or any a p where g g , hence g g . This implies that Z g p = ⟨g⟩ in any pro-p group. By Proposition 3.7 in [11], If G is a finitely generated pro-p group, then ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ G = g1, ... , gn = g1 ... gn .

Z Z p p 8 Therefore G = g1 ... gn . In particular, if G is a uniform group and n is assumed to be the Φ Zn → Φ a1 an least number of generators of G. Then the map : p G where (a1, ... , an) = g1 ... gn is a homeomorphism. We call integer n described above the dimension of the uniform group G.

2.2 Powerful Zp-Lie algebras Suppose G is a uniform pro-p group of dimension d. We already know that by the above argument G Zd is homeomorphic to p and we call the homeomorphism described above the multiplicative structure of G. However, note that this map does not necessarily respect the group structure of G. There is a way to turn G into a free Zp-module of rank d, by giving G an addition structure. Recall that since pn { pn ∈ } ∈ G is a powerful finitely generated pro-p group Gn+1 = G = g : g G . Moreover, if a Gn+1, n −n then there exists a unique g ∈ G such that a = gp , that is, ap makes sense and is an element − → 7→ p n of G. Thus the map f : Gn+1 G given by a a is a bijection. Now we define a new group structure on G, given by

− pn pn p n x +n y := (x y ) , for any x, y ∈ G. It is not hard to verify that for any integer n ≥ 1 (and of course for n = 0) this defines a group structure on G and it turns the map f to an isomorphism of Gn+1 to G. Then Lemma 4.11 in [11] implies that for any x, y ∈ G and all m ≥ n > 1

x +n y = x +m y mod (Gn+1).

Since {Gn}n is a local basis of G around 1, we see that (x +n y)n is a Cauchy sequence and hence converges in G. Therefore for all x, y ∈ G, we can define an additive structure on G by

x + y := lim x +n y. n→∞

Proposition 4.13 in [11] tells us that (G, +) is an abelian group with identity element 1 and inversion −1 x → −x = x . If x, y ∈ G are such that xy = yx, then x +n y = xy for all n and so x + y = xy. m n−1 This implies that mx = x for all m ∈ Z and thus, Gn = p G ≤ (G, +). Furthermore, Gn has

7More generally, any homomorphism from a finitely generated pro-p group to a profinite group is continuous. 8Note that this depends on the ordering of the generators as often G is not abelian.

10 → the same cosets in (G, +) and in G and the identity map Gn/Gn+1 Gn/Gn+1 from G to (G, +) is an isomorphism. This can be used to show that (G, +) is also a uniform group of dimension d, { } where any set of topological generators of G also generates (G, +). Moreover, if g1, ... , gd is a set λ λ of topological generators of , then for any ∈ , we have 1 d in for λ ∈ Z for all G g G g = g1 ... gd G i p 1 ≤ i ≤ d and hence λ ··· λ g = 1g1 + + dgd in (G, +). We have the following important theorem (Theorem 4.17, [11]). Theorem 2.6. Given a uniform pro-p group G of dimension d with a set of topological generators { } Z g1, ... , gd , (G, +) is a free p-module of rank d. In particular Z ··· Z G = pg1 + + pgd. We can also endow (G, +) with a Lie algebra structure. Suppose x, y ∈ G and n ∈ N, we define

− pn pn p 2n (x, y)n := [x , y ] . pn pn ∈ ≤ Note that [x , y ] [Gn+1, Gn+1] G2n+2, so the above definition makes sense. We again see that for all m ≥ n > 1

(x, y)m = (x, y)n (mod(Gn+2)).

So ((x, y)n)n∈N is a Cauchy sequence in G and it converges to an element of G. Similar to our definition for x + y, we define

(x, y) := lim (x, y)n. n→∞ We note the following theorem (Theorem 4.30, [11]). · · Z Theorem 2.7. If G a uniform pro-p group of dimension d, then LG := (G, +, ( , )) is a p-Lie algebra of rank d. Note that since G is a uniform (and so powerful) pro-p group, we see that for all x, y ∈ G and n ∈ N ϵ ∈ p ϵ ∈ N ϵ ≥ ϵ we have that (x, y)n G = p LG for all n where = 1 if p 3 and = 2 if p = 2 and ⊆ ϵ therefore, (LG, LG) p LG. ϵ Generally, if L is any Zp-Lie algebra of finite rank (as Zp-module) such that [L, L] ⊆ p L with ϵ as described above, we call L a powerful Zp-Lie algebra. It turns out that any powerful Zp-Lie algebra can be endowed with a uniform pro-p group structure. To see this we take advantage of Campbell-Hausdorff series9. By Theorem 6.28 in [11], the Campbell-Hausdorff series Φ(X, Y) can be expressed as 1 Φ(X, Y) = X + Y + [X, Y] + ∑ ∑ qe(X, Y)e, 2 ≥ ∈Nn n 3 e + ⟨e⟩=n−1 ∈ Nn ⟨ ⟩ ··· where for any e = (e1, ... , en) +, e := e1 + + en and qe is a constant rational number. Moreover, by (X, Y)e we mean a repeated Lie commutator, defined by

(X, Y)e := (X, Y| , ...{z , Y}, |X, ...{z , X}, ... ) e1 e2 = [... [... [... [[[X, Y], Y] ... , Y], X], ... , X], ... ].

By Corollary 6.38 in [11], if x, y ∈ L where L is a powerful Zp-Lie algebra L, then Φ(x, y) ∈ L. If we define x ∗ y := Φ(x, y) in L, we have the following (Theorem 9.8, [11]).

9For definition and details see §6.3 and §6.5, [11].

11 Z { } Z ∗ Theorem 2.8. Suppose L is a powerful p-Lie algebra with a1, ... ad as a p-basis. Then (L, ) − is a uniform pro-p group with identity element 0 and the inversion x → x 1 := −x. Furthermore, { } ∗ a1, ... ad is a set of topological generators and (L, ) has dimension d. Z Finally, we see that if we turn a uniform pro-p group G into a powerful p-Lie algebra LG, and then ∗ consider (LG, ) we get G back, and vice versa. This correspondence also respects the morphisms. This is better said in the language of category theory (Theorem 9.10, [11]): Theorem 2.9. The category of uniform pro-p group and the category of powerful Lie algebras over Zp are equivalent, where if G is a uniform pro-p group and L is a powerful Zp-Lie algebra, then the maps 7→ G LG (L, ∗) ←[ L are mutually inverse isomorphisms and any morphism (in any one of the categories) is sent to itself as the map of the underlying set.

2.3 p-adic analytic groups

Recall that Zp is metric space where the metric is induced by p-adic norm (which is a non-archimedean | | | | 1 vp(a) norm), and if a = 0, then 0 p = 0, otherwise a p = ( p ) where vp(a) measures the highest power 10 of p dividing a ∈ Zp . Therefore the p-adic distance is defined by d(a, b) = |a − b|p and it induces Zn ∈ N Q a metric on p for any n +. Also, p the field of p-adic numbers, is the p-adic completion of Q where vp(a/b) = vp(a) − vp(b) for a, b ∈ Z and the norms on Q and Qp are defined similarly as above. Then Qp is also the field of fractions of Zp and less abstractly every element in Qp is exactly ∑∞ i ∈ { − } the from i=−m ai p where m is a non-negative integer and ai 0, ... , p 1 for all i, and then vp measures them least integer i for which ai is non zero. Zn ∈ N Suppose V is a non-empty open subset of p and consider the function where n, m + → Zm f = ( f1, ... , fm) : V p We say f is analytic at v ∈ V if there exists a ball −h { ∈ Zn | − | ≤ −h } { h ∈ Zn} ⊆ B(v, p ) = x p : xi vi p p for i = 1, ... , n = v + p x : x p V, ∈ N ∈ Q for some h and formal power series F1(X), ... , Fm(X) p[[X]] where X := (x1, ... , xn) such that h ∈ Zn fi(v + p x) = Fi(x) for all x p and i = 1, ... , m . We say f is analytic on V if it is analytic at every point of V. Hence being analytic is a local property. Definition 2.10. A topological group G is said to be p-adic analytic group if: (i) It has a p-adic analytic manifold structure. In other words, we can find an atlas (up to the A { ϕ } ∈ ϕ equivalent classes of compatible atlases) = (Ui, i, ni) i∈I, where for each i I, (Ui, i, ni) ϕ → Zni { } is chart on G, such that Ui is open in G, : Ui p is a homeomorphism and Ui i∈I is ϕ ϕ ∈ an open cover of G. Furthermore, any two chart (Ui, i, ni) and (Uj, j, nj) for i, j I must be compatible, that is,

ϕ ◦ ϕ−1 ϕ ◦ ϕ−1 j ϕ ∩ and i ϕ ∩ i i(Ui Uj) j j(Ui Uj) are analytic functions.

10 1 | · | φ p in the definition of p can be replaced by any real number 0 < < 1.

12 (ii) The multiplication map and the inversion map are analytic. It turns out that for any two p-adic analytic groups, any continuous homomorphism between them is analytic (Theorem 9.4, [11]). This implies that for a topological group G, there is at most one p-adic analytic group structure (up to equivalence) and if G is not a discrete group, then the prime number p is uniquely determined. Moreover, if G is a p-adic analytic group and H ≤c G and N ◁c G, then H with subspace topology and G/N with quotient topology are again p-adic analytic groups (Theorem 9.6, [11]). The above definition is rather abstract, however, there are more intrinsic characterizations ofa compact p-adic analytic group. A pro-p group G is of finite rank11 if and only if G is a finitely generated pro-p group and it has a powerful open normal subgroup (Theorem 3.13, [11]). Hence such G contains an open normal uniform subgroup. We have the following important theorem (Theorem 8.32, its corollaries in [11] and also see [12]) which among all of the characterizations mentioned below, we use (iv) most often. Theorem 2.11. Suppose G is a topological group. The following are equivalent: (i) G is compact p-adic analytic group. (ii) G is a profinite group and it has an open pro-p subgroup of finite rank. Z ≥ (iii) G is isomorphic to a closed subgroup of GLd( p) for some d 1. (iv) G has an open normal uniform pro-p subgroup of finite index. The proof, on one hand, uses the fact that if a topological group G has an open subgroup H which is p-adic analytic group (note that a uniform group is a p-adic analytic group with a global chart, − with homeomorphism Φ 1 described in Example 2.5) such that H contains a local basis for the identity element and conjugation by elements of G land in H and conjugations are analytic, then the p-adic analytic structure of H can be extended uniquely to G (Proposition 8.15, [11]). On the other hand, it constructs an open subgroup called a standard group (Definition 8.22, [11]) for which the p-adic analytic manifold structure is simple and it shows that every standard group is uniform.

2.4 Structure of Iwasawa algebras Definition 2.12. Suppose G is a compact p-adic analytic group. We define the Iwasawa algebra of Λ Z 12 G, denoted by G, to be the completed group algebra of G over p, that is Λ Z Z G = p[[G]] := lim←− p[G/N] N◁o G we also call the completed group algebra of G over Fp, Ω F Λ Λ F G = p[[G]] := G/p G = lim←− p[G/N], N◁o G the Iwasawa algebra of G. This hopefully should not cause any confusion and it should be clear from the context which version of Iwasawa algebras we are talking about. Ω Ω Λ Here we mostly concentrate on G, knowing that many of the properties of G are shared with G. We often want to focus our attention on the Iwasawa algebra of a specific open normal subgroup of G, specially on a uniform subgroup of G, where the structure of Iwasawa algebra is simpler and more well-understood. In order to do this, we take advantage of a concept which is a generalization of a group algebra called a crossed product.

11The rank of G is defined to be r(G) := sup {d(N)}, where d(N) is the dimension of N which measures the N◁o G cardinality of a minimal generating set of N. r(G) and d(G) may not coincide but for the case of finitely generated powerful pro-p groups they do. 12 When M ≤ N or N → M, then G/N ← G/M, and this can be extended Zp-linearly to get Zp[G/N] ← Zp[G/M].

13 Definition 2.13. Suppose R is a ring and G is a group. Then the crossed product of R and G, denoted by R ∗ G, is an associative ring which contains R as a subring and there exists a set × G = {g : g ∈ G} ⊂ (R ∗ G) (groups of units of R ∗ G) which is bijective to G. We require gR = Rg for all g ∈ G and R ∗ G to be a free right (and by last condition left) R-module with basis G. Furthermore for any g, h ∈ G we must have g · hR = ghR. Note that if H ≤ G, then R ∗ H is naturally a subring of R ∗ G. We can show that if N ◁ G, then ∼ R ∗ G = (R ∗ N) ∗ G/N (see Chapter 1, section 5, in [15] for details). Similarly R[G] = R[N] ∗ G/N and since G/N = (G/H)/(N/H) for any H ≤ N such that H, N ◁o G, we see that

Zp[G/N] = Zp[N/H] ∗ G/H.

Now suppose G is a compact p-adic analytic group. Note that G is a profinite group and by compactness, open subgroups have finite index. Let H ◁o G and consider S = {M ◁o G : M ≤ H}. Λ′ Z Then if G := limM∈S p[G/M], we see that by universal property of inverse limits, there exists Λ → Λ′ ≤ a unique ring homomorphism h : G G. On the other hand, note that M H means exactly that H → M (morphisms are reverse inclusion); if N ◁o G, then there exists L ◁o G such that N → L and M → L. In particular H → L and hence L ∈ S. This means that Λ′ → Z → Z G p[G/L] p[G/N]. ′ Λ′ → Λ Similarly there exists a unique morphism h : G G which is the inverse of h and therefore, Λ Λ′ G = G. Thus Λ F Z Z ∗ Λ ∗ G = lim←− p[G/N] = lim←− p[G/M] = lim←− p[H/M] G/H = H G/H. N◁o G M∈S M∈S Λ Hence if a property of H is known for a particularly nice H ◁o G (e.g. H is a uniform subgroup of Λ G), we hope to be able to extend it to G, given that G/H is finite. The explanation above justifies our attempt to focus on the case where G is a uniform group. So from now on let us assume G is a uniform group, unless it is stated otherwise.

Definition 2.14. We define the augmentation ideal of N ◁o G to be W F −→π F N := ker( p[G] p[G/N]), where the map π is the Fp extension of the surjective (quotient) map G → G/N, that is, π λ λ λ ∈ F ∈ (∑g∈G gg) = ∑g∈G ggN where g p for all g G. {W } F The family of ideals N N◁o G is a directed system of ideals in p[G] with respect to the reverse inclusion. This is because, for any N, M ◁o G there exists L ◁o G such that N, M → L, and then W ⊆ W ∩ W F ∼ F W Ω F W L N M. Since p[G/N] = p[G]/ N, under this identification, G = lim←− p[G]/ N. { } As Gi i∈N+ forms a local basis for the neighbourhoods of 1 in G, where Gi = Pi(G). In particular ∈ N any N ◁o G contains Gi for some i +. Subsequently Ω = F [ ] W = F [ ] W G lim←− p G / N lim←− p G / Gi . N◁o G i∈N+ { }∞ Definition 2.15. We say Ii i=0 is a descending chain of ideals of a ring R, if I0 = R, Ii ◁ R and ⊆ ≥ ∩ ∈ N Ii+1 Ii for all i 0. Furthermore, we require i∈N+ Ii = 0 and for all i, j , we have that ⊆ Ii Ij Ii+j. { }∞ Given any descending chain of ideals Ii i=0 on a ring R, we can define a norm associated to it, where − if r = 0, then ||r|| = 0, otherwise, ||r|| = ϕ v(r), where ϕ > 1 and v(r) measures the highest integer ∈ \ || · || || || ≥ || || ⇐⇒ k such that r Ik Ik+1. Then (R, ) is a normed ring, that is r 0 with r = 0 r = 0, || || || || ≤ || || || || || || ≤ {|| || || ||} ∈ 1R = 1, ab a b and a + b max a , b for all a, b R.

14 W { i}∞ 0 F Let J := G and note that J i=0, where J = p[G] is a descending chain of ideals. Therefore F we can define a norm on p[G], called the J-adic norm, where vJ(a) measures the highest power k ∈ k\ k+1 | | −k F of J such that a J J . Then a J = p . Naturally the J-adic norm induces a metric on p[G], called the -adic metric. Similarly we can define a norm for F [ ] associated to {W }∞ where J p G Gi i=0 13 W = F [ ]. It turns out that the descending chain of ideals {W } ∈N is cofinal with { i} ∈N G0 : p G Gi i J i (see Section 7.1, [11]). What is important to us here is that cofinality implies that the metrics defined by these two descending chains give rise to the same concept of Cauchy sequences and hence same topological completion. On the other hand, the topological completion Fp[G] of Fp[G] using the J-adic metric and the inverse limit completion lim R/Ji are isomorphic, as for any Cauchy sequence (r + ji) ←−i∈N i i F F in lim←− R/J , we have that (ri) is Cauchy in p[G] and conversely, if (ri) is Cauchy in p[G], then there exists a subsequence such that (r + ji) is Cauchy in lim R/Ji. This implies that n(i) ←−i∈N Ω = F [ ] W ∼ F [ ] i ∼ F [ ] G lim←− p G / Gi = lim←− p G /J = p G . i∈N i∈N Λ Z → Z Z The same result holds for G, where J := ker( p[G] p) + p p[G]. More generally, for Ω Λ F Z any compact p-adic analytic group G, G and G are J-adic completions of p[G] and p[G], respectively, for a suitable choice of J. Note that the J-adic norm and J-adic metric on Fp[G] naturally ∈ F ∈ F || || || || extend to the completions. If a = lim ai p[G] with ai p[G] for all i, then a := lim ai . F ⊆ F ∼ Ω In particular, p[G] p[G] = G, which can also be seen using the map F → Ω i : p[G] G λ 7→ λ ∑ gg ( ∑ ggN)N◁o G. g∈G g∈G ∈ ∈ ∩ W { } Ω Then x ker(i) whenever x N◁o G N = 0 , which is due the fact that G is Hausdorff. Thus F Λ i is injective. We can identify p[G] by its image. The same result holds for G. Also notice that

J = ker(Fp[G] → Fp) = { ∑ λgg : ∑ λg = 0}, g∈G g∈G λ − ∑ λ ∈ which means that 1 = g∈G\{1} g where 1 G is the identity element. ∑ λ ∑ λ − So we have g∈G gg = g∈G\{1} g(g 1). Thus ⊕ J = ∑ Fp[G](g − 1) = Fp(g − 1). g∈G\{1} g∈G\{1} { } Now, suppose g1, ... , gd is a minimal generating set for the uniform pro-p group G. Then we see − ∈ ∈ { } α α α ∈ Nd that bi := gi 1 J for all i 1, ... , d . Let := ( 1, ... , d) . For simplicity we write α α α 1 d b := b1 ... bd . We have the following important theorem (Theorem 7.20, 21 and 23, [11]) which states that { α α ∈ Nd} Ω Λ F Z b : forms a topological basis for G and G over p and p, respectively. Theorem 2.16. Suppose G is a uniform pro-p group of dimension d. Then ∈ Ω (i) Every element a G is uniquely of the form α a = ∑ λαb , α∈Nd λ ∈ F α ∈ Nd Ω where α p for all . Conversely, every such series converges in G. 13 { } { } ⊆ We say two descending chain of ideals Fi and Li are cofinal, if for each i there exists n(i) such that Ln(i) Fi ⊆ and Fn(i) Li.

15 ∈ Ω (ii) For a G, as described in (i), we have − α ··· α ||a|| = sup {p ( 1+ + d)}, α∈Nd λα̸=0

|| · || F Ω ∼ F where denotes the extension of J-adic metric on p[G] to G = p[G]. ∈ Λ (iii) Every element A G is uniquely of the form α A = ∑ γαb , α∈Nd

γ ∈ Z α ∈ Nd Λ where α p for all . Conversely, every such series converges in G. ∈ Λ (iv) For A G, as described in (iii), we have

−(α +···+α ) ||A|| = sup {p 1 d |γα|p}, α∈Nd γα̸=0

|| · || Z Λ ∼ F | · | where denotes the extension of J-adic metric on p[G] to G = p[G] and p is the p-adic norm on Zp. ′ It is important to notice that the above theorem requires us to first fix an ordering of bi s, since G Ω Λ is often non-commutative and hence G and G are non-commutative. Ω Now, if G is a uniform pro-p group, then G is a local ring with Jacobson radical (or the maximal ideal)

J Ω ··· Ω { ∈ Ω || || ≤ −1} = b1 G + + bd G = a G : a p , Ω J ∼ F ∈ J − and the residue field G/ = p. To see this, recall that a if and only if 1 ra is a unit in Ω ∈ Ω Ω ··· Ω G for all r G. Since b1 G + + bd G is a two-sided ideal it is enough to show that for any ∈ Ω − || || ≤ −1 a G such that the constant term of a is zero, then 1 a is invertible. Note that a p , so − i 1 a has the inverse ∑i≥0 a , since

(1 − a) ∑ ai = lim 1 − ai+1, →∞ i≥0 i

|| − − i+1 || || i+1|| ≤ −i−1 → ∞ − i and 1 (1 a ) = a p , which tends to zero as i . We have (1 a) ∑i≤0 a = 1. i − − Ω Similarly (∑i≤0 a )(1 a) = 1 and 1 a is a unit. Therefore J is the Jacobson radical of G.

Example 2.17. In particular, G = Zp has only one generator (for example taking the generator to − Ω ∼ F Λ ∼ Z be g1 = 1). We see G = p[[X]] and G = p[[X]] for X an indeterminate variable. One of the most fruitful ways of approaching Iwasawa algebras is through the concept of filtration. { } Definition 2.18. Suppose R is a ring and suppose FR = FiR i∈Z is an ascending chain in R (that ⊆ ∈ Z is, FiR Fi+1R for each i ) such that ≤ + ∈ Z (i) FiR R , that is, FiR is an additive subgroup of R for all i , ∈ (ii) 1 F0R, ⊆ ∈ Z (iii) FiRFjR Fi+jR for all i, j , ∪ (iv) i∈ZFiR = R. Then FR is a filtration on the ring R.

16 Note that the above conditions imply that F0R is a subring of R. { } Definition 2.19. Let R be a ring with filtration FR = FiR i∈Z and suppose M is an R-module. Let { } ⊆ FM = Fi M i∈Z be an ascending chain of additive subgroups of M, such that FiRFj M Fi+jR for ∈ Z ∪ all i, j and i∈ZFi M = M. Then M is said to be a filtered R-module with filtration FM. { } A filtration induces a topology on M, where Fi M i∈Z defines basic local neighbourhoods of M around 0, turning M into a topological group. We say that the filtration FM is separated14 if ∩ ∈ ∈ Z i∈ZFi M = 0. Given a filtration, for an element m M, either there exists a unique n , which ∈ \ we call the degree of m and denote by deg(m), such that m Fn M Fn−1 M, or such integer does not exist and we let deg(m) = −∞ by convention. We define the principal symbol of m, denoted by gr(m), by ∈ gr(m) := m + Fn−1 M Fn M/Fn−1 M, when deg(m) = n, otherwise we let gr(m) = 0. Furthermore, we define ⊕ gr(M) := i∈ZFi M/Fi−1 M, which is a gr(R)-module with group action taken to be the componentwise group action. Note gr(m) ∈ gr(M) for all m ∈ M. By a homogeneous element a in gr(M), we mean a nonzero element ∈ ∈ Z such that a Fn M/Fn−1 M for some n . Since any ring R is a module over itself, all of the above ⊆ definitions apply to a filtered ring R. Furthermore, By our construction, since FiRFjR Fi+jR, we ∈ ∈ can define multiplication for homogeneous elements. If a Fn+1R/FnR and b Fm+1R/FmR for ∈ Z ′ ∈ ′ ∈ m, n , suppose a Fn+1R and b Fm+1R are representatives of a and b, respectively, then ′ ′ ∈ ab := a b + Fm+n+1R Fn+mR/Fn+m+1R.

We extend this linearly (and using distributivity) to gr(R). Therefore gr(R) is a ring called the associated (Z-) graded ring of R with respect to filtration FR. Note that if gr(r)gr(s) ̸= 0, then we have gr(rs) = gr(r)gr(s). In particular, this is always the case for all r, s ∈ R\{0} if gr(R) is { } a domain. If N is an R-submodule of M, then the induced subfiltration on N, FN = Fi N i∈Z , ∩ ∈ Z is defined by Fi N := Fi M N for all i . The induced quotient filtration on N is defined by { } ∈ Z F(M/N) = Fi(M/N) i∈Z where Fi(M/N) := Fn(M) + N/N for all i . For more details see §I.3 in [16].

Although, gr(R) with respect to a given filtration may not capture the structure of R completely, gr(R) is often easier to work with and some proprieties of gr(R) may be lifted up to R depending on { } ≥ the filtration. A filtration FR = FiR i∈Z is said to be negative if FiR = R for all i 0 and positive { −i} if FiR = 0 for all i < 0. Here we are mostly concerned with the filtrations of the form FR = I i∈Z − where I ◁ R is a proper non zero ideal of R and I i = R for i ≥ 0. So we have an ascending chain

· · · ⊆ I2 ⊆ I ⊆ R ⊆ R ⊆ ... .

This is a negative filtration, called the I-adic filtration of R and we often denote its associated graded ring by grI R. Now we can see that in the Definition 2.15, a descending chain of ideals corresponds to a negative separated filtration consisting of ideals in R. W { −i} 15 As in our discussion above, J = G defines a filtration FR = J i∈Z, the J-adic filtration , on F Ω F p[G]. For this reason, G is also said to be the completion of p[G] with respect to the J-adic filtration and we can identify

Ω ∼ F ∼ F i G = p[G] = lim←− p[G]/J . i∈N

14Separation can be thought of as saying that the topology induced by the filtration is Hausdorff. 15Separation is implied by the Krull intersection theorem.

17 Ω Ω −n ≤ n This filtration induces a filtration on G such that Fn G = J for n 0, where J is the closure n Ω Ω Ω of J in G and Fn G = G for all n > 0. More explicitly, n { ∈ Ω || || ≤ −n} n i i J = a G : x p = lim←−(J + J )/J , i∈N

n n which implies that J = J ∩ Fp[G] for all n. We have a natural map

J ↠ J/J2 i 7→ 2 (xi + J ) x1 + J .

i ∈ ∈ 1 2 2 ∼ 2 Since (xi + J ) J, we have x1 J . Then the kernel of this map is J and J/J = J/J . Similarly ∼ Jn/Jn+1 = Jn/Jn+1, for all n. Thus Ω ∼ F ⊕ n n+1 grJ( G) = grJ( p[G]) = n≥0 J /J . ∈ \ 2 2 We see that bi J J and so gr(bi) = bi + J . Lemma 7.10 in [11] implies that bi and bj commute 3 − ∈ 3 mod J , that is, bibj bjbi J for all i, j = 1, ... , d. This leads us to the following useful theorem (Theorem 7.22 and 24, [11]). { }d Theorem 2.20. Suppose G is a uniform pro-p group with a minimal generating set gi i=1 and let { }d F bi i=1 be as described above. The p-algebra homomorphism Φ F → F : p[y1, ... , yd] grJ( p[G]) 7→ 2 yi gr(bi) = bi + J for all i = 1, ... , d, Λ ∼ Z ∼ F is an isomorphism. Similarly grJ( G) = grJ( p[G]) = p[y0, y1, ... , yd]. F Now we can use Hilbert basis theorem to conclude that grJ( p[G]) is noetherian and it is a domain F Ω (does not contain any zero divisors). These properties can be lifted up to p[G] and G (Proposition Ω F 7.27, [11]). Furthermore, since G is J-adically complete and grJ( p[G]) is noetherian, then the Ω F J-adic filtration is a Zariskian filtration (page 83,[13]) which implies that G and grJ( p[G]) share Ω many ring theoretic properties. In particular, G is a Auslander-Gorenstein (for definition see §3.6) noetherian scalar local16 domain and it is a maximal order in its division ring of fractions.

We can extend these results to the case where G is a compact p-adic analytic group, not necessarily Ω Λ uniform, depending on the structure of the group G. We already know that G and G are complete. Ω Ω ∗ Note that G = H G/H, where we take H to be a uniform normal subgroup of G with finite Ω Ω Ω index. Since H is noetherian and G is a (free) finitely generated module over H, we see that Ω Λ G is also noetherian. Similarly G is noetherian. An application of Maschke’s theorem for crossed products (Theorem 1.4.1, [19]) and Nakayama’s lemma shows that J(R) ⊆ J(R ∗ N) for a finite group N, where J(R) and J(R ∗ N) denote the Jacobson radical of associative rings R and R ∗ N, respectively (Theorem 1.4.2, [19]). This, together Ω Λ with our knowledge of the uniform case, shows that G and G are always semi-local. One can show that for a finite p-group N, Fp[N] and Zp[N] are scalar local rings. Therefore if G is assumed Ω Λ Z to be a pro-p group, then G and G are scalar local. Furthermore, since p is an integral domain, Λ 17 G is always semi-prime (see [18]). Ω Λ We have the following theorem that relates a few properties of G to the structure of G and G (see [5] and [4]).

16We say R is semi-local if R/J(R) is a semi-simple artinian ring with J(R) denoting the Jacobson radical of R. If R/J(R) is a simple artinian ring, we say R is a local ring. R is said to be scalar local if R/J(R) is a division ring. 17Recall that a ring R is semi-prime if and only if it does not have any non zero nilpotent two-sided/left/right ideal.

18 Theorem 2.21. Suppose G is a uniform pro-p group. Then the following hold: Ω Λ (i) G and G are prime if and only if G has no trivial finite normal subgroup. Ω (ii) G is semi-prime if and only if G has no trivial finite normal subgroup of order divisible by p. Ω Λ (iii) G and G are domains if and only if G is torsion free. Let us see why these conditions are necessary. Suppose H ◁ G is a non-trivial finite normal subgroup of G. Consider b ∈ F ⊂ Ω H := ∑ h p[G] G. h∈H

Since hH = H for all h ∈ H, we have that hHb = Hb and so Hb2 = |H|Hb. On the other hand, as gH = Hg for all g ∈ G as H is a normal subgroup, then gHb = Hgb for all g ∈ G and thus, b H ∈ Z(Fp[G]), where Z(Fp[G]) denotes the centre of Fp[G]. This in turn implies that

b ∈ F ⇒ b ∈ Ω H Z( p[G/N]) for all N ◁o G = H Z( G).

b b − | | bΩ b − | | b b − | | Now we have H(H H ) = 0 and hence H G(H H ) = (0). But H and H H are non zero, Ω b ∈ Z Λ and hence G cannot be prime. Since H p[G], a similar argument shows that G cannot be a prime ring. | | b2 F bΩ 2 b ̸ Ω If H is such that p H , then H = 0 in p[G]. Thus (H G) = 0, but H = 0 and so G cannot be semi-prime. n Finally, if g = 1 where g ∈ G and n ∈ N+ is the order of g, then

− ··· n−1 F ⊂ Ω (1 g)(1 + g + + g ) = 0 in p[G] G, Ω and as a result, G is not a domain in this case. Again, a similar argument can be applied to show Λ that G is not a domain when G is has a torsion element. Let us finish this section by noting some interesting results about the two-sided ideals of anIwasawa Ω algebra. Given a compact p-adic analytic group G, let Z( G) denote the centre of the Iwasawa Ω algebra of G. One approach to construct a two-sided ideal in G, is to use central elements, that is, Ω if I is an ideal that is generated by a subset of Z( G) as a right or left module, then it is trivially a two-sided ideal. However the crossed product allows us to concentrate on a subgroup of G with finite index. As usual, we focus on the case where G is a uniform pro-p group and we recall that any p-adic analytic group has a finite index uniform pro-p subgroup. Then Corollary A in [1] states that the following holds. Ω Ω Λ Λ Proposition 2.22. For a uniform pro-p subgroup G we have Z( G) = Z(G) and Z( G) = Z(G), where Z(G) is the centre of G. It may well happen that Z(G) = {e} where e is the identity element of G, which makes using central elements to construct two-sided ideals not useful. Another approach is what we have seen before. Given a normal subgroup we can consider its Ω ↠ Ω augmentation ideal. Using the universal property we get a map G G/H where H is a closed normal subgroup of G. Note that when H ◁c G, we have G/H with the quotient topology induced by Ω G is a compact p-adic analytical group and therefore G/H is an Iwasawa algebra. Then wH,G := Ω ↠ Ω Ω ker( G G/H), which is often shortened by wH,G = wH, is a two-sided ideal of G associated Λ ↠ Λ Λ Ω ↠ Ω to H. Similarly IH = IH,G := ker( G G/H) is a two-sided ideal of G. Since G G/H is Ω ∼ Λ surjective, we have G/wH = G/H. So to check whether wH is prime or semi-prime, we can use Theorem 2.21. As a result, we only need to look at G/H. In particular, IH is always semi-prime and wH is prime if and only if IH is prime.

19 π Λ ↠ Ω ∼ Λ Λ We can also consider the inverse image of wH under the natural map : G G = G/p G. π−1 Λ We denote it by vH := (wH) which is also a two-sided ideal of G. Recall that a semi-prime ideal I of a noetherian ring R is said to be right localizable if the set C { ∈ } R(I) := c R : c + I is regular in R/I , where by regular we mean not a zero divisor, satisfies the right Ore condition. Left localizable is defined similarly. We say I is localizable if it is right and left localizable. Then Theorem G in [5] states the following:

Theorem 2.23. Let G be a compact p-adic analytic group and suppose H ◁c G. Suppose F is the largest finite normal subgroup of H with order coprime to p. Then

1. wH and vH are localizable if and only if H/F is pro-p. 18 2. IH is localizable if and only if H is finite-by-nilpotent . It is interesting to note that The authors in [4] (Question G) ask whether there is any other way not involving central elements or ideals associated to closed normal groups, discussed above, to construct a two-sided ideal of an Iwasawa algebra19.

3 Reflexive ideals in Iwasawa algebra

In the last section of this project we follow [7]. We talk about the reflexive ideals in Iwasawa algebras. One of the reasons one might care about the reflexive ideals as we will see in §3.6, is given bythe Lemma 4.12 in [10] which implies that they can help us to study the prime ideals in Iwasawa algebras. Moreover, [7] makes use of many properties of Iwasawa algebras that we have talked about so far. Therefore this is a good chance to put these ideas into the practice.

3.1 Introduction to reflexive ideals ∗ Suppose R is a ring and suppose M is a left R-module. Then M := HomR(M, R), the set of all φ ∈ left R-module homomorphism from M to R, is naturally a right R-module. If HomR(M, R) and ∈ φ · ∈ φ · φ ∈ ∈ r R we define r HomR(M, R) by ( r)(m) = (m)r for any m M. Note that if s R then (φ · r)(s · m) = φ(s · m)r = sφ(m)r = s(φ · r)(m). Similarly ∗∗ ∗ ∗ M = (M ) = HomRop (Hom(M, R), R) ∗ is a right R-module (note that M is a right R-module and hence it is a left Rop-module, where Rop is the opposite ring of R). ∼ ∗∗ Definition 3.1. Suppose M is a left R-module. We say M is reflexive as a left R-module if M = M , with the isomorphism given by the natural left R-module homomorphism m 7→ Φm for any m ∈ M, ∗ where Φm(φ) := φ(m) for any φ ∈ M . We can define reflexive right R-modules similarly. If M is a two-sided R-module then M is said to be reflexive if it is reflexive as a left and right R-module. Example 3.2. Recall that an element w of a ring R is normal if wR = Rw. If w ∈ R is a normal element then Rw is a two-sided reflexive ideal. By symmetry it is sufficient to showthat Rw is ∗ reflexive as left R-module. First note that any φ ∈ Rw = Hom(Rw, R) is completely determined by φ(w). So let φa denote the left R-module homomorphism φa(w) := a for some a ∈ R and consider Ψ → : Rw HomRop (Hom(Rw, R), R) rw 7→ Φrw,

18 H is said to be finite-by-nilpotent, if H has a finite normal subgroup N such that H/N is a nilpotent group. 19There might have been some development regarding this question, after [4] was published.

20 where Φrw(ϕa) = ϕa(rw) = rϕa(w) = ra for any φa ∈ Hom(RW, R) and a ∈ R. We need to show that Ψ is bijective. Note that if Ψ(rw) = Φrw = 0 (Φrw(φa) = 0 for all a ∈ R). Then let a = 1, Ψ Φ ∈ ∗∗ Φ φ we see that r = 0 and so is injective. Now suppose Rw and suppose ( 1) = x for some ∈ Φ φ Φ φ · Φ φ Φ ≡ Φ Ψ x R, then ( a) = ( 1 a) = ( 1)a = xa. Thus xw. Therefore is surjective. Now suppose R is a noetherian domain. Then S := R\{0} is the set of regular elements of R. Since − R is noetherian, S is an Ore set. Let Q := RS 1 be the division ring of fractions of R. Definition 3.3. Suppose I is a non zero right R-submodule of Q. If I ⊆ uR for some u ∈ Q\{0} then I is said to be a fractional right R-ideal. Equivalently aI ⊆ R and bR ⊆ I for some a, b ∈ Q.A fractional left R-ideal is defined in a same manner. Note that a fractional right (or left) ideal is not necessarily an ideal of R. However any right (or left) ideal of R is a right (resp. left) fractional ideal. Suppose I is a fractional right ideal. Then − I 1 := {a ∈ Q : aI ⊆ R} ∼ η −1 −→= ∗ is a fractional left ideal. Moreover, I : I I = Hom(I, R), where the left R-module isomorphism η 7→ ϕ ϕ ∈ −1 ∈ I is given by a a with a(i) := ai (left multiplication by a) for all a I and i I . This − − ∼ ∗∗ implies that the fractional ideal (I 1) 1 = I and motivates the following definition.

Definition 3.4. Suppose I is a fractional right R-ideal for a noetherian domain R. Then we call − − I := (I 1) 1 the reflexive closure of I. The reflexive closure of a fractional left ideal is defined similarly. Note that I ⊆ I and I is reflexive exactly when I = I.

3.2 A few important results about reflexive ideals and derivations Let us list here, a few important tools that will help us throughout the rest of this section. We first note the following crucial fact (see Chapter 7 of [9] for proof). Lemma 3.5. Every reflexive ideal in a commutative UFD (unique factorization domain) is principal.

Suppose R is a ring and suppose M is an R-module. We say M is a pseudo-null module if for any i R-submodule N of M we have ExtR(N, R) for i = 0, 1. For simplicity of notation we may denote i i ExtR(N, R) by E (N), which should not cause confusion in the context. Here we are mostly concerned with the case where R is a noetherian domain in which case if M is finitely generated. Then M is −1 ∈ { ∈ } pseudo-null exactly when annR(m) = R for all m M, where annR(m) := r R : mr = 0 . If, furthermore R is assumed to be commutative, then since M is finitely generated, we see that M is −1 { ∈ ∀ ∈ } pseudo-null if and only if AnnR(M) = R, where AnnR(M) := r R : mr = 0 m M . Proposition 3.6. Suppose R is a commutative noetherian UFD and I is a non zero ideal of R with I = xR for some x ∈ R. Then xR/I is pseudo-null and if R is a graded ring and I is graded ideal, then x is a homogeneous element of R. − Proof. Note that x is non zero and let us denote {a ∈ R : ax ∈ I} := x 1 I ⊆ R. Then { ∈ ∈ ∀ ∈ } AnnR(xR/I) = a R : axr I r R .

∈ ∈ −1 ∈ −1 ∈ If a AnnR(xR/I), letting r = 1, we see that a x I. On the other hand, if a x I then a −1 ∈ −1 −1 −1 ⊆ AnnR(xR/I). This means that AnnR(xR/I) = x I. For q (x I) , we have that qx I R, − − − − − − which implies that qx 1 ∈ I 1. Note that if c ∈ I 1 and i ∈ I = (I 1) 1 then iI 1 ⊆ R and so − −1 −1 − − −1 ci = ic ∈ R. Thus I 1 ⊆ I . Since I ⊆ I we see that I ⊆ I 1 and hence I 1 = I . Also note that I = xR so

−1 − I = {q ∈ Frac(R) : qxR ⊆ R} = x 1R,

21 − − −1 − and qx 1 ∈ I 1 = I = x 1R.Therefore q ∈ R. This implies that

−1 AnnR(xR/I) = R.

Thus xR/I is pseudo-null. Now given R is a graded ring and I is a graded ideal, I is generated by homogeneous elements. Consequently there exists a homogeneous non zero element y ∈ I ⊆ I = xR. Therefore y = xr for some non zero element r ∈ R. But R is a domain and x must also be homogeneous.

Next we consider the relation between the reflexive ideals of S and the reflexive ideals of a subring R of S. It turns out that given some properties are satisfied, the reflexive ideals in S lie over reflexive ideals in R and we also have a type of going up from a reflexive ideal of R to a reflexive ideal S. Proposition 3.7. Suppose R is a noetherian ring and it is a subring (with identity) of ring S. Suppose S a noetherian domain and S is a flat as a left and right module of R. Then:

(i) If J ◁r R is a right (or left) ideal of R, then J.S = J.S.

(ii) If I ◁r S is a reflexive right (or left) ideal of S, then I ∩ R is a reflexive right (resp. left) ideal of R. Proof. Consider the following two functors that act on finitely generated right R-modules and have left-S modules as outputs ⊗ − F := S R HomR( , R) − ⊗ G := HomS( R S, S).

For any finitely generated right R-module M we have Φ → M : F(M) G(M) ⊗ 7→ Φ ⊗ s R f M(s R f ), ∈ ∈ Φ ⊗ Φ ⊗ ⊗ for any s S and f HomR(M, R), where M(s R f ) is defined by M(s R f )(m t) = s f (m)t ∈ ∈ Φ ≡ Φ for any m M and t S. Then ( M)M defines a natural transformation from F to G. Note that for any n ∈ N we have n ⊗ n ∼ ⊗ n ∼ n F(R ) = S R HomR(R , R) = S R R = S . Similarly

n n ⊗ ∼ n ∼ n G(R ) = HomS(R R S, S) = HomS(S , S) = S . Φ We see that P is an isomorphism of modules for any free finitely generated right R-module P. Now suppose P• is a free projective resolution of M where P• consists of finitely generated free right R-modules (note that it is not hard to show such resolution exists, see Proposition 6.2 [20]). Then we have

i ⊗ i ⊗ ExtS(M R S, S) : = H (HomS(P• R S, S)) ∼ i ⊗ • Φ ∀ = H (S R HomR(P , R)) as Pn is an isomorphism for n ⊗ i = S R H (HomR(P•, R)) by flatness of S ⊗ i = S R ExtR(M, R). Therefore ∼ Φi ⊗ i −→= i ⊗ M : S R ExtR(M, R) ExtS(M R S, S). (⋆)

22 For all finitely generated right R-modules M and i ≥ 0. Note that since S is a domain, R is also a domain and they are both noetherian. Therefore we can ⊆ ⊆ talk about their division rings of fractions and as R S and Q := QR QS. Given I, a non zero fractional right R-ideal in Q, we have I ⊆ uR for some u ∈ Q\{0}. Then IS ⊆ uS and so IS is a non zero fractional S-right ideal in QS. On the other hand − (SI 1)IS ⊆ SRS = S.

− − This implies SI 1 ⊆ (IS) 1. To see the other containment, we consider the following commuting diagram ⊗ −1 α −1 i −1 S R I SI (IS) ⊗η η 1 I IS ⊗ ∗ ⊗ ∗ ∗ S R I Φ (I R S) β (IS) I η η α ⊗ where i is the inclusion map, I and IS are the module isomorphisms given before, (s a) := sa − for s ∈ S and a ∈ I 1. It is obvious that α is surjective, and injectivity is implied by the assumption that S is a flat left R-module (Corollary 3.59, [20]). Similarly β : Hom(IS, S) → Hom(I ⊗ S, S) is given by β( f )(i ⊗ s) = f (is) and it is an isomorphism as S is a flat left R-module. Moreover by Φ (⋆) we see that I is also an isomorphism. This implies that i must also be an isomorphism and − − − − SI 1 = (IS) 1 for any non zero fractional right R-ideal I. By symmetry (SI) 1 = I 1S for any non zero fractional left R-ideal I. Now let us show (i). Note that the result is obvious for the case where J is the zero ideal. So suppose J ◁r R is any non zero right ideal of R, then − − − − − − JS = ((JS) 1) 1 = (SJ 1) 1 = (J 1) 1S = JS. ∩ To see (ii), let I ◁R S be a reflexive right (or left) ideal of S. If I R = (0), then the result is obvious. If not, since (I ∩ R)S ⊆ I, we have

I ∩ R ⊆ (I ∩ R)S ⊆ I = I, and trivially I ∩ R ⊆ R = R. Thus I ∩ R ⊆ I ∩ R. The other direction of containment is trivial and hence I ∩ R = I ∩ R. As a result I ∩ R is a reflexive right ideal of R. By symmetry the same results hold true for the case where I and J are assumed to be left ideals. Our last tool arises through the concept of derivations. This would be our first encounter with ideals satisfying certain conditions in a ring B, which implies that they are ”controlled” by a subring of B. Let us for now define what we mean by a derivation of aring. Definition 3.8. Suppose B is a commutative ring. We say the additive map d : B → B is a derivation, ∈ ⊆ if it satisfies the Leibniz’s law, that is, d(ab) = d(a)b + ad(b) for all a, b B. Suppose B1 B is a ′ ′ ′ ′ ∈ ∈ subring of B, then we say d is B1-linear if d(b b) = b d(b), or d(b ) = 0 for all b B1 and b B. The set of all -linear derivations of is denoted by Der ( ). B1 B B1 B Now suppose B is a commutative K-algebra with K a field of characteristic p, for some prime number p, and let B1 be a subring of B. Recall that we have the Frobenius map, which is a ring endomorphism given by

φ : B → B b 7→ bp.

In particular, for r ∈ Fp, we have φ(rb) = rφ(b). If B is assumed to be reduced (has no non zero nilpotent elements) then φ is injective. We denote the image of φ by B[p] = {bp : b ∈ B}. If B is reduced (we usually assume that B is a domain) then B[p] is isomorphic as a ring to B.

23 Notice that any derivation d : B → B is B[p]-linear. This is because for any a ∈ B one can show, using induction and by commutativity of B, − − − d(ap) = d(a)ap 1 + ad(ap 1) = ··· = pap 1d(a) = 0,

p p ∈ [p] ⊆ as B has characteristic p. Thus d(a b) = a d(b) for any b B. Suppose B B1 and suppose { } B is a finitely generated free B1-module. Specifically, suppose y1, ... , yt is a set of non zero ∈ N α α α ∈ Nt distinct elements of B where t + and suppose = ( 1, ... , t) . Following our notation, if y = (y1, ... , yt) , we denote α 1 αt α y1 ... yt := y .

Suppose [p − 1] denotes the set {0, 1, ... , p} and let T := [p − 1]t which is a finite set. Then we consider ⊕ α B = B1y . α∈T D The reason for such a specific structure will hopefully become clear later. Let := DerB (B) and ∂ → 1 suppose j : B B for j = 1, ... , t, be the formal B1-linear derivative with respect to yj, defined as follows

α α−ϵ ∂ α j j( ∑ uαy ) := ∑ juαy , α∈T α∈T α j>0

∈ α ∈ ϵ ∈ Nt where uα B1 for all T and j = (0, ... , 1, ... , 0) is the element with all but the j-th ∂ { }t component zero. Since j is by definition B1-linear and since yi = generate B as B1-algebra (since i 1 ′ [p] ⊆ ∂ ∈ D α α B B1), to show j for any j = 1, ... , t it is enough to check Leibniz’s law for y y whenever ′ ′ α, α ∈ T such that α + α ∈ T. But ′ ′ ∂ α α ∂ α+α j(y y ) = j(y ) { ′ ′ α α −ϵ ′ (α + α )y + j if α or α ̸= 0, = j j j j 0 otherwise ′ ′ ∂ α α α∂ α = j(y )y + y j(y ). ∂ ∈ D So j for all j = 1, ... , t. ∂ Proposition 3.9. Suppose B, B1 and j for j = 1, ... , t be as above. Then ⊕ D t ∂ (i) = j=1 B1 j. ∈ D ∈ D ∈ (ii) For all x B, (x) = 0 (for all d we have d(x) = 0) if and only if x B1. (iii) Suppose I is an ideal of B. Then I is D-stable, which we denote by D(I) ⊆ I, if and only if I is controlled by B1. In other words ⊆ ∀ ∈ D ⇐⇒ ∩ d(I) I d I = (I B1)B.

∑t ∂ ⊆ D Proof. (i): Notice that C := j=1 B1 j and let us first show the linear independence. Iffor ∈ ∑t ∂ ∑t ∂ ∈ some bj B1 for j = 1, ... , t, we have j=1 bj j = 0, that is j=1 bj j(b) = 0 for all b B, then ∂ δ δ ∑t ∂ since j(yi) = ij (where ij is the Kronecker delta), then we see that j=1 bj j(yi) = bi = 0 for

24 ∈ D ∑t ∂ ∈ any i = 0, ... , t. Now suppose d and consider j=1 d(yj) j C. Then as we have seen before, n n−1 ∈ N d(yj ) = nd(yj)yj for n , and so

t α α−ϵ α j d(y ) = ∑ jd(yj)y , j=1

∑t ∂ α α ∈ ∑t ∂ which agrees with j=1 d(yj) j(y ) for all T. Since d and j=1 d(yj) j are B1-linear, they agree ∑t ∂ ∈ on B which implies that d = j=1 d(yj) j C. ∈ D ⇐ (ii): Since d is B1-linear for any d then ” =” is obvious. Conversely suppose α ∈ \ x = ∑ xαy B B1. α∈T ̸ α ∈ \{ } α ̸ α ̸ Thus xα = 0 for some T 0 . In particular j = 0 for some j = 1, ... , t as = 0 and so ∂ ̸ j(x) = 0. ⇐ ∈ D (iii): To see “ =”, suppose I is an ideal of B that is controlled by B1. Then for any d we have ∩ d(I) = d((I B1)B) ∩ = (I B1)d(B) as d is B1-linear ⊆ ∩ ⊆ (I B1)B d(B) B = I. ∩ D Conversely if J = I B1, we want to show that if I is -stable then JB = I. Notice that → −→π B1 , B B/JB, ∩ and the kernel of B1 these two maps is JB B1 which contains J. So we have ′ → ′ B1 := B1/J , B := B/JB. On the other hand ⊕ ⊕ α α JB = J B1y = Jy . α∈T α∈T Therefore ⊕ / ⊕ ⊕ ⊕ ′ α α α ′ α B = B/JB = B1y Jy = B1/J y = B1y , α∈T α∈T α∈T α∈T ⊕ α where y is the image of y modulo JB (note that y ∈/ JB = α∈ Jy ). This can be used to easily ′ ′ T ′ [p] ⊆ p ∈ [p] ⊆ ′ ′ check that, since B B1, then y B1 and also we have B B1. Then B1 and B satisfy the D π → same structure as B1 and B. Since⊕ I is -stable the image of I under : B B/JB is also stable ′ ′ ′ t ∂′ ∂′ ′ under D = Der ′ (B ) (as D = B1 with defined as the formal B -linear derivation with B1 j=1 j j 1 respect to yj). So without the loss of generality, we may assume that J = 0. Thus we want to show ∩ that if I B1 = 0, then I = 0. Suppose not. We get a contradiction using a minimality argument. Let ρ : B → N be a map defined by α ρ( ∑ uαy ) := max{⟨α⟩ = α + ··· + α : uα ̸= 0}, α∈ 1 t α∈T T α and let i ∈ I\{0} be such that ρ(i) is minimal in I. Since i = ∑ uαy is non zero then uα ̸= 0 for α ∈ α ̸ α α α α ∂ ̸ some T. So j = 0 for some j with = ( 1, ... , j, ... , t). Then j(i) = 0 and by assumption ∂ ∈ ρ ∂ ρ ρ j(i) I. However ( j(i)) < (i), which contradicts the hypothesis that (i) is minimal in I, and thus I = (0).

25 3.3 Frobenius pairs and the derivation hypothesis We first introduce an abstract structure that is tailored for our purpose of proving the “control theorem” for specific ideals. We move on to talk about an important condition on such structures, which we need in order to show that the ideals are controlled in the sense we saw in the last section. From now on all filtrations are assumed to be separated. Definition 3.10. Suppose A is a complete filtered (with separated filtration) K-algebra, where K is a field of characteristic p, and suppose A1 is a subalgebra of A endowed with the subspace filtration. Then we say the pair (A, A1) is a Frobenius pair if ≤ (i) A1 c A, that is A1 is closed in the topology of A given by the filtration. Note that since A is complete, this is equivalent to say that A1 is complete with respect to the subspace topology. (ii) B := gr(A) is a commutative domain. [p] ⊆ (iii) B1 := gr(A1) is such that B B1. ∈ ∈ N (iv) There exists y1, ... , yt B homogeneous elements for some t with (adopting our notation as before) ⊕ α B = B1y . α∈T Ω F Assuming that G is a uniform pro-p group of dimension d, A := G = p[[G]] and A1 := Ω F p ∼ F Gp = [[G ]], then we see (A, A1) is a Frobenius pair. Moreover B = p[y1, ... , yd] and ∼ F p p ∈ N p F B1 = p[y , ... , y , yt+1, ... , yd] for some t which is the dimension of G/G as an p vector 1 t ⊕ α space. Therefore B = α∈T B1y . The following proposition that we include here will come in handy later. ∈ Proposition 3.11. Let ui A be such that gr(ui) = yi for all i = 1, ... , t and let u = (u1, ... , ut). { α α ∈ } Then A is a finite free left and right A1-module with basis u : T , that is ⊕ ⊕ α α A = A1u = u A1. α∈T α∈T

Proof. We only show this for A as a left A1-module and we argue by symmetry to see the result also α ⊆ holds for A as right A1-module. Suppose M := ∑α∈T A1u A an A1-submodule of A. Let us first see that the sum is direct. Suppose not and take a minimal dependence in M, that is, a nonempty ′ ⊆ α ∈ \{ } α ∈ ′ set T T such that ∑α∈T′ aαu = 0 where aα A1 0 for all T . Recall the degree map on { α } {α α } a filtered ring and consider n := maxα∈T′ deg(aαu ) . Let S := : deg(aαu ) = n . So for all α ∈ α ∈ \ α ∈ ′\ α ∈ S we have aαu Fn A Fn−1 A and for all T S we must have that aαu Fn−1 A. Hence α α ∑ aαu + Fn−1 = ∑ aαu + Fn−1 α∈T′ α∈S α α = ∑ gr(aαu ) as all aαu have degree n α∈S α = ∑ gr(aα)y ∈ B as B is a commutative domain. α∈S ⊕ ∈ α Note that gr(aα) B1; since B = α∈T B1y we must have gr(aα) = 0. In particular this means ∈ F α ∈ ′ α ∈ ′ aα n−1 for all ⊕T which is a contradiction and so aα = 0 for all T . α ⊆ Now we know that α∈T A1u A and by construction of M and also the fact that B is a commutative domain gr(M) = B = gr(A). Suppose towards a contradiction that a ∈ A\M. Since a ̸= 0, let deg(a) = n. Then ∈ ⇒ ∈ gr(a) = a + Fn−1 A gr(M) = gr(a) = gr(a1) for some a1 M ⇒ = a = a1 mod Fn−1 A,

26 − ∈ − − and so a a1 Fn−1 A. If a a1 = 0 then we are done; if not we know deg(a a1) < n and we can − ∈ repeat the argument above to get gr(a a1) = gr(a2) for some a2 M. If this process stops then we are done; if not we get a sequence {an} with strictly decreasing degree and since A is complete → → ∞ ∑N → → ∞ this means that an 0 as n and n=1 an a as N . Since A1 is also complete and M ∑∞ ∈ is a finite free A1-module, a = n=1 an M. This is a contradiction and consequently M = A. Now we move to the second part of this section. Motivated by part (iii) of Proposition 3.9, we want a criterion that helps us verifying the D-stability of an ideal in commutative graded noetherian domain B. First we need to introduce some machinery. Given our notations above, note that A is a an associative ring and we can define a Lie bracket on A using the commutator. Therefore we are able to turn the K-algebra A into a Lie K-algebra. Suppose ∈ ∈ Z ∀ ∈ Z ⊆ a A and there exists n such that k . We have [a, Fk A] Fk−n A. In other words, for all k ∈ Z we have the following linear map: − → [a, ]n : Fk A Fk−n A b 7→ [a, b]. This induces a linear map { −} → a, n : Fk A/Fk−1 A Fk−n A/Fk−n−1 A 7→ b + Fk−1 A [a, b] + Fk−n−1 A.

We extend the above map linearly to get {a, −}n : B → B which is a derivation on B as [a, −] is a Lie algebra derivation on A. However {a, −}n is not necessarily in D, that is, {a, −}n may not be B1-linear. To fix this, we consider the following definition. ∈ N Definition 3.12. Suppose (A, A1) is a Frobenius pair. Let a := (ai)i∈N A be a sequence such θ θ θ θ → N that there exist two maps and 1 where , 1 : a with ⊆ ∈ N ∈ Z (i) [ai, Fk A] Fk−θ(i) A for all i and k ,

(ii) [a F A ] ⊆ F −θ A for all i ∈ N and k ∈ Z, i, k 1 k 1(i) θ − θ → ∞ → ∞ (iii) 1(ai) (ai) as i .

Then we call such sequence a source of derivations. The set of all sources of derivations for (A, A1) S is denoted by (A, A1).

As we discussed above {a , −}θ is a derivation on B. Since the filtration on A is separated, by i (ai) abuse of notation, we may assume that F−∞ A = {0}. The extra conditions (ii) and (iii) ensure that for large enough N ∈ N we have that {a , b}θ ∈ F−∞ A = {0} for b ∈ B . So {a , −}θ ∈ D N (aN ) 1 i (ai) for all i ≥ N. ∈ { −} ∈ N Example 3.13. If a A is such that a, n is a B1-linear derivation on B for some n , then we θ θ ∞ can take a = (a)i∈N. This trivially defines a source of derivations with (a) = n and 1(a) = . Ω Ω Example 3.14. In practice we will see that when A := G and A1 := Gp with G a uniform pro-p ∈ { pi } group, for g G we have gi i∈N is a source of derivations. Moreover we have the following map on A which can be regarded as a measure of how well an element of A can be approximated by elements of A1. δ → N ∪ {∞} Definition 3.15. Suppose (A, A1) is a Frobenius pair. We define : A given by { max{k ∈ Z : w ∈ F A + F − A} if w ∈/ A δ(w) := n 1 n k 1 , ∞ otherwise ∈ \ δ and n = deg(w) whenever w A A1. We call the delta function on (A, A1).

27 To see that this is well defined, first note that for w ∈ A with deg(w) = n implies that w ∈ Fn A and so δ(w) ≥ 0 for all w. On the other hand, by definition of the topology induced by the filtration, Fn A is open in A for all n ∈ Z. Since in a topological group any open subgroup is also closed, we ∈ Z ∩ have that Fn A is also closed in A for all n . As Fn A1 = Fn A A1 and A1 is closed in A, we have ∈ Z { } that aFn A1 is closed in A for all n . On the other hand, since Fi A i∈Z defines a local basis for the open neighbourhoods around zero in A, the closure of Fn A1 in A is given by ∩ Fn A1 = Fn A1 = (Fn A1 + Fi A) ∈Z i∩ ∩ = Fn A1 + Fn−i A as Fn A1 = Fn A A1. i∈N ∈ \ ∈ N ∈ As a result, if w A A1 with degree n, there exists some k such that w / Fn A1 + Fn−k A. This implies that δ(w) is finite and hence it is well-defined. Also notethat δ(w) > 0, that is δ(w) ≥ 1, ∈ ∈ when w Fn A+Fn−1 A which we can show is the same as saying gr w B1. ∈ \ δ δ Consequently for any w A A1, if we let (w) := and n = deg(w), we have w = x + y where ∈ ∈ \ δ δ x Fn A1 and y Fn−δ A Fn−δ−1 A by the maximality of . Note that if (w) = 0 we can take x = 0. δ Let us denote Yw := gr(y) = y + Fn−δ−1 A and so if = 0 then Yw = gr(w). ∈ S Definition 3.16. Suppose (A, A1) is a Frobenius pair with a (A, A1) and let I ◁ B be a graded ideal. Given a homogeneous element Y ∈ B, we say Y is in the a-closure of I if {a , Y}θ ∈ I for i (ai) all sufficiently large i ∈ N. Before we state the derivation hypothesis we need consider the following proposition. ∈ \ Proposition 3.17. Given a Frobenius pair (A, A1) and I ◁ A. Let w I A1. Then Yw is in the ∈ S a-closure of gr(I) for all a (A, A1).

Proof. Suppose w = x + y, where x and y are as discussed above. Let a = (ai)i∈N. Then there exists N > large enough such that θ (a ) − θ(a ) > δ = δ(w) for all i ≥ N. So [a x] ∈ F −θ A ⊆ 0 q i i : i, n 1(i) F −δ−θ − and [a , y] ∈ F −δ−θ for all r ≥ N. Thus, since [a , w] = [a , x] + [a , y], we have n (ai) 1 i n (ai) i i i

[a , w] + F −δ−θ − = [a, y] + F −δ−θ − = {a , Y }θ , i n (ai) 1 n (ai) 1 i w (i) for all i ≥ N. Then [a , w] + F −δ−θ − ∈ gr(I) ◁ B as w ∈ I and I is a two-sided ideal. So i n (ai) 1 { } ∈ ai, Yw θ(i) gr(I). We are now ready to state the derivation hypothesis that ensures the derivations coming from the sources of derivations are of some value in the sense that we need here.

Definition 3.18. (Derivation hypothesis) We say a Frobenius pair (A, A1) satisfies the derivation hypothesis if for any X ∈ B = gr(A) a homogeneous element, if Y ∈ B is another homogeneous ∈ S D ⊆ element that lies in the a-closure of XB for all a (A, A1), then (Y) XB.

3.4 The control theorem for principal reflexive ideals generated by a normal element This section can be regarded as the first step in our proof of the control theorem for thesome Frobenius pairs.

Lemma 3.19. Suppose (A, A1) is a Frobenius pair that satisfies the derivation hypothesis, with ∈ \ ∈ B1 a UFD, and let w A A1 be a normal element. Then there exists a unit u A such that δ δ δ ∈ − (wu) > (w). Furthermore, if (w) > 0, then there exists c F−δ(w) A such that u = 1 c.

Proof. Let w = x + y with x and y as discussed above and let X := gr(w) and Y := Yw. Then we let I := wA. By Proposition 3.17, Y is in the a-closure of gr(wA) = gr(w) gr(A) = XB (B is a

28 ∈ S domain and w is non zero) for all a (A, A1). Then by the derivation hypothesis we have that D(Y) ⊆ XB. If δ = δ(w) = 0, then this means that Y = X. Thus D(X) ⊆ XB which implies that D ∩ since XB is -stable, we have XB = (XB B1)B. On the other hand, XB is reflexive as XB = BX (B is commutative). Note that B is a free B1-module and hence it is also a flat B1-module. As B ∩ and B1 are both commutative noetherian domains, we see that XB B1 is also reflexive. Since B1 ∩ ∈ is a UFD, we see that XB B1 = X1B1 is principal in B1 where X1 B1 is some homogeneous ∩ ∈ element. However, then we have XB = (XB B1)B = X1B, so there exists a unit U B such ∈ −1 that X1 = UX. Let u, v A be such that gr(u) = U and gr(v) = U , since B is a domain, ∈ ∈ 1 = gr(u) gr(v) = gr(uv). Note that 1 F0 A by definition of filtration and 1 / F−1 A, otherwise, ∈ ∈ Z ̸ 1 Fi A for all i but the filtration is assumed to be separated (1 = 0 as we assume our rings ∈ are not trivial). Therefore uv = 1 mod F−1 A, or in other word, uv 1 + F−1 A. But, since A is a complete filtered ring with respect to FA, 1 + F−1 A consists of units of A. To see this note that for ∈ − i ∈ any a F−1 A. Then 1 a has inverse ∑i∈N a A. So u is a unit and we have ∈ gr(uw) = gr(u) gr(w) = XU = X1 B1.

This means that δ(uw) > 0. For the second part of the proof we refer the reader to Proposition in §3.5, [7]. Now we are ready to prove the control theorem in the case when a reflexive ideal is principal generated by a normal element.

Theorem 3.20. Suppose (A, A1) is a Frobenius pair that satisfies the derivation hypothesis, with B1 a UFD. Let w ∈ A be a normal element. Then ∩ wA = (wA A1)A1. ∈ ∈ ∩ ⊆ ∩ ∩ Proof. Note that if w A1 then w wA A1. Thus wA wA A1 and so wA = wA A1. ∈ \ ∈ ∈ Therefore let us assume that w A A1. We find u A such that wu A1 and a similar argument ∩ ∈ shows that wA = wA A1. By Lemma 3.19, we know there exists a unit u0 A such that δ δ ≥ −1 (wu0) > (w) 0. Note that wu0 is again a normal element as au0 = u0(u au0). Let w0 := wu0. ∈ 0 The same procedure gives us w1 = w0u1 = wu0u1 for some unit u1 A. Continuing in this way, we ∈ ≥ ∈ ∈ ∈ N have wi := wi−1ui for some ui A unit for all i 0 and wi A normal. If wi A1 for some i δ δ ≥ δ ≥ then we are done; so suppose not. We get (wi+1) > (wi) (w0) > 0 for all i 0. Thus, by part 2 of the Lemma 3.19, we can assume that u = 1 − c for all i ≥ 1 where c ∈ F−δ A. Since i i i (wi−1) { } δ → ∞ Fi A i∈Z defines a local basis of open neighbourhoods around zeroin A and since (wi) as → ∞ → → ∞ ∏n i , then ci 0 as i in A by completeness of A. Now consider the sequence ( i=0 ui)n∈N. Suppose n > m and note that

m n m − − ∏ ui ∏ ui = ∏ ui(1 um+1 ... un). i=0 i=0 i=0 − ∈ ∏n ∈ ∈ N Notice that ui = 1 ci F0 for all i and hence i=0 ui F0 for all n . Moreover

n − − − ··· ··· ∈ 1 um+1 ... un = 1 ∏ (1 ci) = cm+1 + cm+2 + + cn + + cm+1cm+2 ... cn F−δ(m) A, i=m+1 ∏m − ∏n ∈ ∏n and so i=0 ui i=0 ui F−δ(m) A. Thus ( i=0 ui)n∈N is a Cauchy sequence and converges in A. − − →∞ ∏n 1 →∞ ∏n 1 Let u := limn i=0 ui which is again a unit (with u = limn i=0 ui ). Furthermore

n wu = lim w ∏ ui = lim wn. n→∞ n→∞ i=0

29 ∈ ∩ It remains to show that wu A1. Let us write A1 = A1 = k∈Z A1 + Fk A which is the closure of ∈ ∈ ∈ Z A1 in A. Therefore to show that wu A1, it is enough to show that wu A1 + Fk A for all k . ∈ N Since deg(ui) = 0 for all i and gr(A) is a domain, we have that

n ∀ ∈ N deg(wn) = deg(w0 ∏ ui) = deg(w0) + 0 = deg(w0) n . i=0 ∈ δ → ∞ → ∞ ∈ Z By construction wi Fn A1 + Fn−δ(w ) A. Since (wi) as i , for any k we have that ∈ ⊆ i − ∈ → → ∞ wi Fn A1 + Fk A A1 + Fk A for sufficiently large i and wu wi Fk A. Since wi wu as i ∈ ∈ Z ∈ then wu A1 + Fk A for all k and so wu A1 as required.

3.5 Control theorem for reflexive ideals Having the result in section §3.4, we move on to prove the control theorem in a more general setting. To do this, we take advantage of the concept of microlocalization. Here we give a brief summary of the tools we need in microlocalization. We refer the reader to §4.1 of [7] for more details20. Suppose e n ⊆ −1 R is a filtered ring with filtration FR. Let R := ∑n∈Z t FnR R[t, t ] be the Rees ring of R with π e → respect to FR. The Rees ring comes with two obvious surjective maps 1 : R R sending t to π e → n 7→ e 1 and 1 : R gr(R) defined by t a gr(a) and extending linearly. Then assuming that R is ⊆ noetherian implies that R and gr(R) is noetherian. If furthermore we assume J(F0R) F−1 then we call such FR a Zariskian filtration. Here we are mostly concerned with complete filtered rings with gr(R) noetherian which implies that FR is Zariskian. Suppose T ⊆ gr(R) is a right Ore set π π consisting of regular homogeneous elements. Then using 1 and 2, T induces two sets e e T := {a ∈ R : a is homogenous and π(r) ∈ T} π e { ∈ ∈ } S := 1(T) = r R : gr(r) T .

Then it turns out that Te and S are both right Ore sets in Re and R, respectively. Since elements of T are assumed to be regular, S also consists of regular elements. We can form the Ore localization e e e and it has an induced (Z-) grading which we denote by = ⊕ ∈Z( ) with ,→ . RTe RTe n RTe n R RS Then we extend π to πf e → in the natural sense where the induced filtration on is 1 1 : RTe RS RS given by = πf(( e ) ). We can show = { −1 ( ) − ( ) ≤ } is a Zariskian FnRS : 1 RTe n FnRS rs : deg r deg s n filtration. Here we assume deg(0) = −∞. Then by the microlocalization of R at T, denoted by QT(R), we mean the completion of RS with respect to the induced filtration FnRS. Moreover suppose M is a finitely generated right R-module. Then by the microlocalization of M at T we mean ⊗ QT(M) := M R QT(R). We have the following result (for proof see §4.3 of [7]): Lemma 3.21. Assume the notation above and suppose M is a finitely generated right R-module endowed with a good filtration21 (here we may take the filtration on M to be the tensor filtration). Then we have

(i) QT(R) is a complete filtered ring, where FnQT(R) is the closure of FnRS in QT(R). Furthermore, R embeds in QT(R). ∼ ∼ (ii) QT(R) is a flat right R-module and gr(QR(R)) = gr(RS) = (gr(R))T. ∼ 22 (iii) gr(QT(M)) = (gr(M))T .

(iv) MS = MRS is a dense in RS-submodule of QT(M).

(v) If N is a submodule of M, then QT(N) can be identified as QT(R)-submodule of QT(M).

20Note that the notation we use here is from [7] and it is “slightly non-standard”. 21 ⊕ e A filtration on M is said to be good if the associated Rees module i∈Z Fi M is finitely generated over R. 22 By (gr(M))T we mean the right Ore localization of gr(R)-module gr(M).

30 ∩ ∩ (vi) If L, N are submodules of M, then QT(N) QT(L) = QT(N L). ∼ Note that since R is noetherian, for any right ideal I of R, we have gr(Q (I)) = (gr(I)) is an ∼ T T ideal of gr(QT(R)) = (gr(R))T by the above lemma. We can construct normal elements in R in the following sense: Lemma 3.22. Suppose R and T are as above and I ◁ R is a two-sided ideal of R such that

(gr(I))T = X(gr(R))T, for some central regular homogeneous element X ∈ gr R. Then there exists a normal element ∈ w QT(R) with gr(w) = X such that

QT(I) = wQT(R). Proof. Note that since I ◁ R is a two-sided ideal and R is a noetherian ring then, by part (vi) of Proposition 2.1.16 in [15], IS = IRS is a two-sided ideal of RS. By part (iv) of the above lemma, QT(I) is the closure of IS in QT(R). Then QT(I) is a two-sided ideal of QT(R). Let w be any ∈ ⊆ element in QT(I) such that gr(w) = X. Since w QT(I), we have that wQT(R) QT(I). We may suppose that X is not zero, as the case where X = 0 is trivial. Hence w is non zero and

gr(QT(I)) = (gr(I))T = X(gr(R))T = gr(wQT(R)).

Thus QT(I) and wQT(R) have the same grading. Since QT(R) is a complete filtered noetherian ring by part (i) of the above lemma, same argument as Proposition 3.11 shows that QT(I) = wQT(R). Finally w is a normal regular element in R by Lemma 4.5 in [7].

Note that given a Frobenius pair (A, A1), A is complete and B = gr(A) is noetherian, and hence the filtration on a Frobenius pair must be Zariskian. Therefore we can apply microlocalization to Frobenius pairs. Consider a homogeneous element Z ∈ B\{0}. Since B is a noetherian commutative { i} domain, we see that T := Z i∈N is a denominator set in B. Let us, by abuse of notation, denote the microlocalization of R at T by QZ(R), and call it microlocalization at Z. We adopt similar notation for the Ore localization, as it is standard in the literature. p ∈ [p] ⊆ ∼ Note that Z B B1 and so QZp (A1) makes sense. Also note that gr QZ(A) = (gr(A))Z = BZ ∼ [p] ⊆ [p] [p] ⊆ and similarly, gr QZp (A1) = (B1)Zp . Moreover, since B B1, we see that (BZ) = BZp (B1)Zp . p Zp Finally, note that when Z is inverted in B, then Z = − also becomes a unit. Thus B = B p . Zp 1 Z Z This implies that ⊕ ⊕ α α BZ = BZp = ( B1y )Zp = (B1)Zp y . α∈T α∈T

Then it is fairly straightforward to check that if (A, A1) is a Frobenius pair, then (QZ(A), QZp (A1)) is also a Frobenius pair. By Proposition 5.1 in [7], they share our desired properties. In particular ∈ S ∈ S if a (A, A1) then a (QZ(A), QZp (A1)). Moreover if B is a UFD and (A, A1) satisfies the derivation hypothesis so does (QZ(A), QZp (A1)). Now that we have microlocalization in our arsenal we aim to generalize Theorem 3.20. First consider the following proposition:

Proposition 3.23. Suppose (A, A1) is a Frobenius pair that satisfies the derivation hypothesis and ∩ is such that B and B1 are UFDs. Let I ◁ A be a two-sided ideal in A. If J := (I A1)A, then gr(I)/ gr(J) is pseudo-null. Proof. Note that since J ⊆ I, we have gr(J) ⊆ gr(I). If I = 0 then the result is trivial. So suppose I ̸= 0. Let (gr(I)) be the reflexive closure of gr(I) in B. Then

gr(J) ⊆ gr(I) ⊆ gr(I) ⊆ B.

31 Since B is a UFD, gr(I) = XB for some homogeneous non zero element X ∈ B. Then by definition of the reflexive closure, there exists Z ∈ B a homogeneous element such that ZX ∈ gr(I). As the ′ ′ notation suggests, we can microlocalize at Z to get A := QZ(A) and A := QZp (A1). We know ′ ′ ′ ′ 1 that (A , A1) is a Frobenius pair. Let I := QZ(I) = IA . Then ∈ ⇒ −1 ∈ ZX gr(I) = X = (ZX)Z (gr I)Z = gr(QZ(I)).

−i ∈ ∈ ∈ N Also note that for any YZ (gr(I))Z where Y gr(I) = XB for i , we have that Y = XP ∈ −i −1 ∈ where P B. Thus YZ = XPZ XBZ and so XBZ = (gr(I))Z. Since B is a commutative domain, X is not only homogeneous but trivially regular and central. Hence by Lemma 3.22, ′ ′ ′ I = wA for some normal element w ∈ A with gr(w) = X. Furthermore, by Proposition 3.11, ⊕ α ⊕ α ′ ⊕ α ′ A = α∈Tu A1. Similarly, since BZ = α∈T(B1)Zp y , the same process shows that A = α∈Tu A1. Thus ⊕ ′ α ′ ′ AA1 = u A1 A1 = A . α∈T ′ ′ ′ ′ Therefore I = IA = IAA1 = IA1 = QZp (I) and by part (vi) Lemma 3.21, since A is a finitely generated A1-module we have that ′ ∩ ′ ∩ ∩ ∩ ′ I A1 = QZp (I) QZp (A1) = QZp (I A1) = (I A1).A1. On the other hand, using Theorem 3.20 regarding reflexive ideals generated by a normal element, ′ ′ ′ ′ ∩ ′ ′ since I = wA , we have I = (I A1)A . Consequently, ′ ′ ′ ∩ ′ ′ ∩ ′ ′ ∩ ′ ′ IA = I = (I A1)A = ((I A1)A1)A = (I A1)AA = JA . ′ ′ ′ So IA = JA and we have (gr(I))Z = gr(IA ) = (gr(J))Z in BZ. Since BZ is noetherian, (gr(J))Z is finitely generated and hence we can choose an appropriate n ∈ Z such that Zn gr(I) ⊆ gr(J). Note that we can apply this argument for any Z ∈ B such that ZX ∈ gr(I). Let L := {Z ∈ B : ZX ∈ gr(I)} which is an ideal of B and again since B is noetherian L is finitely generated. Since for any Z ∈ L, Zn gr(I) ⊆ gr(I) for some n ∈ Z, there exists N ∈ Z N ⊆ N ⊆ such that L gr(I) gr(J), or in other words, L AnnB(gr(I)/ gr(J)). Now, we note that the natural surjective map B → BX/ gr(I) given by Z 7→ ZX + gr(I) has kernel L. Therefore ∼ B/L = BX/ gr(I) = gr(I)/ gr(I) which, by Proposition 3.6, is pseudo-null. Given an exact sequence of R-modules ( [10])

0 → M → N → P → 0 for some ring R, N is pseudo-null if and only if M and P are pseudo-null. Therefore any quotient of B/L is again pseudo-null. Thus if we let C := B/LN, we see that C is pseudo-null. Since gr(I) is finitely generated as B-module, so is gr(I)/ gr(J). Hence / k ↠ k ↠ C (B (AnnB(gr(I)/ gr(J)))) gr(I)/ gr(J), for some k ∈ N. Note that using the result about the exact sequences mentioned above we have Ck is pseudo-null which implies that gr(I)/ gr(J) is also pseudo-null, as required. Now we are ready to prove the control theorem in a more general setting.

Theorem 3.24. Suppose (A, A1) is a Frobenius pair that satisfies the derivation hypothesis and it is such that B and B1 are UFDs. Let I ◁ A be a reflexive two-sided ideal in A. Then ∩ I = (I A1)A.

32 Proof. Following our notations from Proposition 3.23, since A is a free (and hence flat) left and right ∩ A1-module, we see that I A1 is also a two-sided reflexive ideal of A1. Thus again by Proposition ∩ ⊆ 3.7, J = (I A1)A is also a reflexive right ideal of A. It is obvious that J I, and so we need to show that I ⊆ J, which we show by first showing that I/J is pseudo-null. Let N be a right A-submodule of I/J. Then gr(N) is a B-submodule of gr(I)/ gr(J) equipped with subquotient filtration. By Proposition 3.23, gr(I)/ gr(J) is a pseudo-null module and therefore, gr(N) is also pseudo-null. Then since the filtration on A is Zariskian and, then by Proposition 3.1 in [8], there is an 1 1 1 1 induced good filtration on E (N) = ExtR(N, R) such that gr(E (N)) is a subquotient of E (gr(N)) which is (0) as gr(N) is pseudo-null. Thus gr(E1(N)) = 0, which Lemma 9 in §1.2 Chapter II of [14] implies that E1(N) = 0. A similar argument shows E0(N) = 0. Consequently, I/J is also ∈ { ∈ ∈ } pseudo-null. Now for any x + J I/J, let (J : x) := annA(x + J) = a A : xa J . In other words, (J : x) = ker(A → xA/J). Then since I/J is finitely generated pseudo-null A-module and − A is a noetherian domain, by the remarks in the beginning of section 3, we see that (J : x) 1 = A. On the other hand, x(J : x) ⊆ J by the definition of J. Therefore − − − − J 1x(J : x) ⊆ J 1 J = A =⇒ J 1x ⊆ (J : x) 1 = A. − − As a result, x ∈ (J 1) 1 = J = J as J is reflexive and so I ⊆ J, which completes the proof.

3.6 Consequences of the control theorem of reflexive ideals for Iwasawa algebras We first give a brief summary on how we can construct Frobenius pairs in Iwasawa algebras, which arises naturally using powerful Zp-Lie algebras. We then move on to note some of the important consequences of the theory developed in the last section.

Let L be a powerful Zp-algebra of rank d and let G = (L, ∗) be its associated uniform pro-p group which also has dimension d. Then by §6.5 and §7.2 of [11], we have a bijection map exp : L → G ∑ 1 i where exp(x) := i∈N i! x . By construction of the Campbell-Hausdorff series and Proposition 6.27, exp respects the structure in the sense that exp(u) exp(v) = exp(Φ(u, v)) = exp(u ∗ v). Then by Lemma 6.2 in [7]

k exp(pk L) = Gp ∀k ≥ 0. ⊆ ⊆ { }d Let L1 L be a subalgebra of L such that pL L1, and suppose that vi i=1 is a subset of L is such { }d F { }t F that vi + pL = is a p-basis of L/pL and vi + pL1 = is a p-basis of L/L1 for some integer ≤ ≤ i 1 ≤ i 1 1 t d. Then G1 := exp(L1) G = exp(L) is a subgroup of G. Moreover, if gi = exp(vi) for all ≤ ≤ { }d { p p } 1 i d, then gi i=1 is a topological generating set of G and g1 , ... , gt , gt+1, c ... , gd is a set of topological generators of G1 (see Lemma 6.3 in [7]). Suppose K is a field of characteristic p. We can extend the scalars in the definition of Iwasawa algebras by defining

K[[G]] := lim←− K[G/N]. N◁o G Ω F Note that the filtration on G = p[[G]] induces the tensor filtration on K[[G]]. Moreover, the same argument as we presented in section 2.4 can be used to show that K[[G]] is the completion of K[G] with respect to J := ker(K[G] → K). In particular, by extension of scalars, we can see that ∼ Theorem 2.16 and 2.20 also hold for K[[G]] and so gr(K[[G]]) = K[y1, ... , yd]. Therefore from now Ω F F on we work with K[[G]] instead of G = p[[G]], knowing that K in a special case can be p and Ω so all results will also hold for G.

We already know that A := K[[G]] is compete and B is a commutative noetherian UFD. If G1 is as given above, then Lemma 6.5 in [7] implies that ∼ p p gr(K[[G1]]) = gr(K[G1]) = K[y1 , ... , yt , yt+1, ... , yd].

33 Then if A1 := K[[G1]]. It is not hard to check that all of the conditions in Definition 3.10 are Φ 23 satisfied and (K[[G]], K[[G1]]) is a Frobenius pair. Furthermore, suppose is a root system and n suppose Φ(Zp) denotes the Zp-Lie algebra associated to Φ. So p Φ(Zp) is a powerful Zp-Lie algebra for some n ≥ ϵ with ϵ defined as before depending on prime p. We say prime p is nice for Φ if p ≥ 5 and p ∤ n + 1 whenever Φ has an indecomposable component of type An (where An is as in the Dynkin diagram, for more details and notation see [6]). Then the derivation hypothesis is taken care of in the case of “nice primes p” for a root system Φ by Theorem A in [6] which we mention below.

24 n Theorem 3.25. Retain the notation above and suppose p is a nice prime for Φ. Let L := p Φ(Zp) and let G := exp(L) be the associated uniform group of L. Then (K[[G]], K[[Gp]]) satisfies the derivation hypothesis. Now we move on to our final part and begin to note some ramifications of the control theorem for reflexive ideals. Recall that a ring A has finite right (resp. left) injective dimension d, which we denote by injdim(A) = d, if d is the minimal length among all finite injective resolution for A (assuming they exist), whenever A is considered to be a right (resp. left) A-module. If A is noetherian and A has finite right and left injective dimension then the coincide. Wesaya noetherian ring is Gorenstein if it has finite injective dimension on both sides. Similarly if M is a right A-module, then the projective dimension of M, when it is finite, is the minimal length among all finite projective resolutions of M. The right global dimension of A is the supremum of the projective dimension of right A-modules which we denote by gd(A); left global dimension is defined similarly. If A is assumed to be noetherian and left and right injective dimension, and left and right global dimension are all finite, then they all coincide.

Recall that if M is a left A-module, then Ei(M) = Exti(M, A) is naturally a right A-module. Then for M a finitely generated right/left A-module, we define the grade of M by

j(M) := inf{n : En(M) = 0}.

Note that j(M) ∈ {0, ... , d} ∪ {∞} where j(0) = ∞ (for more details see §5 in [4]). We say a noetherian Gorenstein ring A is Auslander-Gorenstein, if A is such that for any finitely generated right/left A-module M and any n ∈ N, whenever N is a finitely generated A−submodule of Ei(M), then j(N) ≥ i. That is, En(N) = 0 for all n < i. Moreover, if A has finite global dimension, then A is said to be Auslander-regular.

Given an Auslander-Gorenstein ring A, if M is a finitely generated left/right A-module, then we can define the canonical dimension of M by − CdimA(M) := injdim(A) j(M). ′ ′ Moreover, if A = A ∗ H for some finite group H, then A is also Auslander-Gorenstein (see §5.2 in [4]). In particular, given G a compact p-adic analytic group and H ≤ G a uniform subgroup of it, since K[[H]] is Auslander-Gorenstein, then so is K[[G]]. Furthermore, by Proposition 7.1 in [7], ∈ ≥ there exists an ideal I A such that CdimA(A/I) = n for some n 0 if and only if there exists an ∈ ′ ′ ideal J A with CdimA′ (A /J) = n.

Then M is said to be pure if for any A-submodule N of M, CdimA(M) = CdimA(N). By Lemma 4.12 in [10], if A is Auslander-regular domain and if I is a proper non zero right (or left) ideal of A, then I is reflexive if and only if A/I is pure of grade 1. Note that by Theorem J in [5], when G is a compact p-adic analytic group with dimension25 d, then K[[G]] has global/injective dimension d. Thus A = K[[G]] is an Auslander-regular ring and if we show that there exists no proper non zero

23We refer the reader to [6] and §2.8 in [2] for more details on the usage of root systems and Chevalley groups here. 24However, [6] conjectures that the condition for p to be nice is ”superfluous”. 25By the dimension of a compact p-adic analytic group, we mean the dimension of any open uniform subgroup of it. This is well-defined by Lemma 4.6 [11].

34 − two-sided ideal such that CdimA(A/I) = d 1, then this would mean that A has no proper non zero reflexive ideal. We can do this using reflexive ideals for the cases where we know the derivation hypothesis is satisfied.

Theorem 3.26. Suppose K is a field characteristic p and G is a compact p-adic analytic group of Q L ⊗ Q dimension d. Suppose the p-Lie algebra of G, denoted by (G) := LG Zp p, is split semisimple over Qp. Let p ≥ 5 and p ∤ n in the case sln(Qp) occur as a direct summand of L(G). Then there − exists no two sided ideal I in K[[G]] such that CdimA(K[[G]]/I) = d 1. Proof. If U is any uniform pro-p subgroup of G with finite index, then if we let A = K[[U]] and ′ A = K[[G]] = K[[U]] ∗ G/U, it is enough to show the result for A = K[[U]], by Proposition 7.1 in [7] mentioned above. So let us assume, without loss of generality, that G is a uniform group. we can find a Zp-subalgebra L inside L(G), by a suitable choice of Chevalleys basis, such that ∼ n L = p Φ(Zp) where Φ is the root system of L(G). Let L = exp(H) be the associated uniform pk pk k group of L. Note that the associated powerful Zp-Lie algebra of H is exp(H ) = p L for all k k+1 k ≥ 0 and (K[[Hp ]], K[[Hp ]]) is a Frobenius pair satisfying derivation hypothesis for all k ≥ 0. Again by Proposition 7.1 in [7], since H has finite index in G, it is enough to show the result for − H. Suppose for a contradiction that there exists I ◁ K[[H]] such that CdimA(K[[H]]/I) = d 1. ̸ ̸ − Note that I = (0), otherwise CdimA(K[[G]]/I) = CdimA(K[[G]]) = d = d 1, similarly I is a proper ideal. Suppose M is a maximal among pseudo-null ideals of K[[H]]/I. M exists as K[[H]]/I is noetherian. Then by definition of pseudo-null modules if N is any submodule of/ M, we have − E0(M) = E1(N) = 0. If m = π 1(M) where π : K[[G]] → K[[G]]/I, we see that A/I m/I = A/m is pure as its grade is bounded above by 1 and by the maximality of m it cannot be 0. So replacing I by m, we may assume that A/I is pure and so I is reflexive. Consequently, by the control theorem we have that I = (I ∩ K[[Hp]])K[[H]]. By continuing to apply the control theorem to the Frobenius k k+1 k pair (K[[Hp ]], K[[Hp ]]) we see that I = (I ∩ K[[Hp ]])K[[H]] for k ≥ 0. Since I is a proper ideal, k k k k so is I ∩ K[[Hp ]] in K[[Hp ]]. Therefore it is contained in the maximal ideal (Hp − 1)K[[Hp ]] for all k ≥ 0. Hence ∩ ∩ ∩ k k k k I ⊆ ((Hp − 1)K[[Hp ]]K[[H]]) = ((Hp − 1)K[[H]]) = ((H − 1)p K[[H]]). k∈N k∈N k∈N ∩ − pk ⊆ pk − pk However, note that (H 1) J and thus, any element in k∈N((H 1) K[[H]]) converges to 0 and since K[[H]] is complete, so I = (0), which is a contradiction. Thus by our argument before Theorem 3.26, we see that the following holds.

Corollary 3.27. Let K, G, and L(G) be as in the Theorem 3.26. Then K[[G]] has no non-trivial reflexive two-sided ideals. Recall that given any normal element we can construct a two-sided reflexive ideal. Then the above corollary implies that any reflexive two-sided ideal generated by a non zero normal element should be all K[[G]]. So we have the following corollary. Corollary 3.28. Let K, G and L(G) be as in the Theorem 3.26. Then every non zero normal element in K[[G]] is a unit. Z ⊆ ∈ Let L be a powerful p-Lie algebra and L1 is a subalgebra of L with pL L1. Suppose u L is such that [u, L] ⊆ pk L and [u, L] ⊆ pk+1L for some k ≥ ϵ (where ϵ = 1 if p ≥ 3 and ϵ = 2 if p = 2). Let a := exp(g) ∈ G with G is the associated uniform pro-p group. Then, by Corollary 6.7 in [7], pi ∈ S Z a = (a )i∈N (K[[G]], K[[G1]]) where G1 = exp(L1). In particular, since L is a powerful p-Lie ϵ ϵ algebra, [L, L] ⊆ p L and [L, pL] ⊆ p +1L. Thus for any g ∈ G, since g = exp(u) for some u ∈ L, i ∈ S p the sequence (g )i∈N (K[[G]], K[[G ]]).

35 ∈ ≥ ϵ ⊆ k ̸⊆ k+1 ⊆ k+1 Given u L, suppose k is such that [u, L] p L while [u, L] p L and [u, L1] p L. In ⊆ k+1 ∈ particular, if L1 = pL, [u, L1] p L is trivially satisfied. Given such u L we can define a non zero Fp-linear map φ → u : L/L1 L/pL 1 v + L 7→ [u, v] + pL. 1 pk ⊆ On the other hand, let a := exp(u), then Proposition 6.7 in [7] implies that [a, Fn A] Fn−pk+1 A, { −} where A = K[[G]]. Therefore a induces a derivation Du := a, pk−1 on B = gr(A) = K[y1 ... , yd]. φ { }t { }d Let (cij)i,j be the matrix of u with respect to the basis vj + L1 j=1 for L/L1 and basis vi + pL i=1 for L/pL, as discussed before. Then it turns out that we have the following result (for proof see Theorem 6.8 in [7]): Theorem 3.29. Retain the notation above. Then d pk ∀ Du(yj) = ∑ cijyi j = 1, ... , t . i=1 This gives us an easier way to compute the source of derivations induced by an element of L. Z As an important corollary to the above theorem we can verify the derivation hypothesis for SL2( p). Z We define the l-th congruence subgroup of SL2( p) by (see §5 of [11] for more details) Γ Z Z → Z lZ l(SL2( p)) := ker(SL2( p) SL2( /p ). Z lZ lZ Consider the powerful p-Lie algebra L = sl2(p p), the traceless matrices with coefficients in p p, then { [ ] [ ] [ ]} 0 pl 0 0 pl 0 e := , f := , h := 0 0 pl 0 0 −pl is a basis for L as Zp-module. Also G = exp(L), the associated uniform group of L, is isomorphic Γ Γ Z p Γ to l := l(SL2( p)). If we let L1 = pL then G1 = exp(pL) = G = l+1. Then using Theorem 3.29, Proposition 6.9 in [7] shows the following. ≥ ≥ lZ Proposition 3.30. Suppose l 1, p 3 and let G := exp(sl2(p p)). Then the Frobenius pair (K[[G]], K[[Gp]]) satisfies the derivation hypothesis. On the other hand for p = 2, we have that the Frobenius pair (K[[G]], K[[Gp]]) does not satisfy the derivation hypothesis. However, in order to get the derivation hypothesis, we can consider an ≥ lZ intermediate powerful group. Suppose l 2 and let L0 := sl2(p p) and L2 := pL0, as before, but Z ⊕ Z ⊕ Z let L1 := pe p p f p h p which is a subalgebra of L0 and it contains L2. Then let Gi be the associated uniform pro-p group of Li for i = 0, 1, 2. Then (K[[G0]], K[[G1]]) and (K[[G1]], K[[G2]]) are a Frobenius pair and by Proposition 8.1 in [7] satisfy the derivation hypothesis. Z Now suppose K is a field of characteristic 2. Note that since SL2( 2) has dimension 3 (as a uniform Z group) any open subgroup G of SL2( 2), we have that injdim(K[[G]]) = 3 (Proposition 4.4 in [11]). Z Since SL2( 2) satisfies the derivation hypothesis, a similar proof as Theorem 3.26 shows that, there exists no two-sided ideal I in K[[G]] such that Cdim(K[[G]]/I) = 2. If G is defined to be torsion free, then K[[G]] is a domain (similar argument as Theorem 2.21); if I is assumed to be a prime ideal then by Theorem A in [3] Cdim(K[[G]]/I) ̸= 1. Therefore either Cdim(K[[G]]/I) = 0 which implies that I is a maximal ideal or Cdim(K[[G]]/I) = 3 which means that I = 0. Thus we have the following result (Theorem C in [7]). Z Theorem 3.31. If G is an open torsion free subgroup of SL2( 2), then every prime ideal in K[[G]] is either maximal or the zero ideal.

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