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 2 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 = tggH (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 2 G. The compactness of G implies that every open subgroup of G has finite index. Furthermore, GnH = tggH where g runs over the left-coset representatives except for the coset H i.e gH 6= 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 f1g is closed in G. Indeed, suppose G is a topological group for which f1g is closed and suppose g, h 2 G are such that g 6= h. Then − − fg 1hg is closed and thus, Gnfg 1hg 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 − Gnfgh 1g is open and contains U × V where U and V are nonempty neighbourhoods of 1 in G. We see that gU \ hV = Æ, otherwise, we have gu = hv for some u 2 U and v 2 V and − − − − − since f (U × V) = UV 1 ⊆ Gnfg 1hg we see g 1h = uv 1 2 Gnfg 1hg which is impossible. ≤ 9 ⊆ ⊆ [ 3. If H G and U o G (open subset) with U non-empty such that U H, then H = h2HhU where hU is open for all h 2 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 ⊆ [ ≤ a 2 U o G is an open neighbourhood of 1. Hence U = a2I Ka where Ka o G for all I and \ [ \ ≤ thus, L = H U = a2I H Ka. 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.