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Inaugural-Dissertation InauguRal-DisseRtation zur Erlangung der Doktorwürde der Naturwissenschaftlich-Mathematischen Gesamtfakultät der RupRecht-KaRls-UniveRsitÄt HeidelbeRg vorgelegt von Dipl.-Math. Oliver Thomas aus Lüneburg Tag der mündlichen Prüfung: Thema: On Analytic and Iwasawa Cohomology Betreuer: Prof. Dr. Otmar Venjakob Abstract We generalise spectral sequences for Iwasawa adjoints of Jannsen to higher dimensional coefficient rings by systematically employing Matlis, local, Koszul and Tate duality. With the same strategy we achieve a generalisation of Venjakob’s local duality theorem for Iwasawa algebras and compute the Λ-torsion of the first Iwasawa cohomology group, both locally and globally. Furthermore, we develop a flexible framework to prove standard results of group coho- mology for topologised monoids with coefficients in topologised modules, using explicit methods dating back to Hochschild and Serre. This closes a few argumentative gaps in the literature. We also prove a form of Poincaré duality for Lie groups over arbitrary complete non- archimedean fields of characteristic zero. Finally, we take tentative steps towards applying these results to ¹ϕ; Γº-modules. Zusammenfassung Wir verallgemeinern Spektralfolgen für Iwasawa-Adjungierte von Jannsen auf höher- dimensionale Koeffizientenringe, indem wir systematisch Matlis-, lokale, Koszul- und Tate-Dualität verwenden. So erreichen wir auch eine Verallgemeinergung von Venjakobs Theorem über lokale Dualität für Iwasawa-Algebren und bestimmen die Λ-Torsion der er- sten Iwasawa-Kohomologie-Gruppen, sowohl lokal als auch global. Ferner entwickeln wir ein flexibles Framework, um Standard-Resultate der Gruppenko- homologie für topologisierte Monoide mit Koeffizienten in topologisierten Moduln zu zeigen. Hier nutzen wir explizite Methoden, die auf Hochschild und Serre zurückgehen. Dies schließt einige argumentative Lücken in der Literatur. Wir zeigen eine Form von Poincaré-Dualität für Lie-Gruppen über beliebigen vollständi- gen nicht-archimedischen Körpern der Charakteristik null. Schlussendlich gehen wir erste Schritte in Richtung einer Anwendung dieser Resultate auf ¹ϕ; Γº-Moduln. Diese Arbeit wäre nicht entstanden ohne die vielen interessanten und hilfreichen mathe- matischen Diskussionen mit meinem Betreuer Otmar Venjakob. Für seine Geduld, sein offenes Ohr und sein Vertrauen in meine Fähigkeiten bedanke ich mich herzlich. Auch gebührt Kay Wingberg besonderer Dank, der nicht nur oft das rechte Wort zur rechten Zeit wusste, sondern auch die Entwicklung dieser Arbeit mit Interesse verfolgt hat. Beiden möchte ich ferner für ihr Vertrauen in mich und für die mir gewährten Freiheiten in meiner Zeit als ihr jeweiliger Assistent danken. Meinen Bürokollegen, der Arbeitsgruppe, dem Mathematischen Institut und dem Frisbee- Seminar danke ich für die vielen Ablenkungen, die meine Zeit als Doktorand so abwechs- lungsreich und spannend gemacht haben. Ohne euch hätte es weniger Spaß gemacht! Schließlich danke ich meiner Mutter für, letztlich, eigentlich alles. Contents Introduction ix Chapter 1. Spectral Sequences for Iwasawa Adjoints 1 1.1. Overview 1 1.2. Conventions 2 1.3. A Few Facts on R-Modules 3 1.4. (Avoiding) Matlis Duality 9 1.5. Tate Duality and Local Cohomology 12 1.6. Iwasawa Adjoints 16 1.7. Local Duality for Iwasawa Algebras 20 1.8. Torsion in Iwasawa Cohomology 21 Chapter 2. A Framework for Topologised Groups 23 2.1. Topological Categories 23 2.2. Topologised Groups 25 2.3. Rigidified G-Modules 27 2.4. The Induced Module 30 Chapter 3. Cohomology of Topologised Monoids 31 3.1. Abstract Monoid Cohomology 31 3.2. Setup 32 3.3. Cohomology of Topologised Monoids 35 3.4. Cohomology and Extensions 36 3.5. Shapiro’s Lemma for Topologised Groups 38 3.6. A Hochschild-Serre Spectral Sequence 44 3.7. A Double Complex 58 3.8. Shapiro’s Lemma for Topologised Monoids 66 Chapter 4. Duality for Analytic Cohomology 67 4.1. Some Functional Analysis 67 4.2. Analytic Actions of Lie Groups 73 4.3. Duality for Lie Algebras 75 4.4. Tamme’s Comparison Results 78 4.5. The Duality Theorem 78 Chapter 5. Two Applications to ¹ϕ; Γº-modules 83 5.1. An Exact Sequence of Berger and Fourquaux 83 5.2. So Many Rings 85 5.3. Lubin-Tate ¹ϕ; Γº-modules 86 5.4. Towards Duality in the Herr Complex 88 Bibliography 93 vii Introduction Cohomology has jokingly been referred to as the fifth operation of elementary arithmetic after addition, substraction, multiplication and division.¹ The development of the cohomo- logical machinery started in the early twentieth century in the area of algebraic topology. Since then it has been an invaluable tool in many areas of mathematics, algebraic num- ber theory chief among them. Since at least the publication of Artin’s and Tate’s notes on class field theory (cf. [AT68]) it is universally accepted that the cohomological point of view is the most conceptual and insightful one for proving the results of class field theory – a major achievement in algebraic number theory in the twentieth century. It is therefore no wonder that many modern results in algebraic number theory are phrased in a cohomological language, where a rich machinery is available. One example of this is Iwasawa theory. In his seminal paper [Iwa73], Iwasawa uses an ad hoc construction for something that is nowadays called an Iwasawa adjoint, and which can be described as an Ext-group. Norm compatible elements in field extensions can via class field theory also be expressed as corestriction compatible systems of Galois cohomology classes, which are closely linked to Euler systems and play a key role in the construction of p-adic L-functions, a major area of research in algebraic number theory in general and Iwasawa theory in particular. The first chapter of this thesis is devoted to studying Iwasawa adjoints and Iwasawa coho- mology. We systematically employ Matlis duality, local duality, Koszul duality and Tate duality to derive more general versions of important spectral sequences due to Jannsen (cf. [Jan89; Jan14]): Where he only considered p -coefficients, we allow higher dimen- sional local rings. This also allows us to prove a quite general local duality theorem for Iwasawa algebras, generalising the results of [Ven02]. We also show that the first Iwa- sawa cohomology groups are torsion-free over their Iwasawa algebras. This chapter has been accepted for publication as [TV19]. • ¹ ; º The first chapter mainly handles variants of two cohomology theories: ExtR M R where R is a commutative ring, and group cohomology H •¹G; Aº, where G is a locally compact group and A a discrete G-module. By cleverly combining these two types of cohomology • ¹ ; º theories, we also gain access to the cohomology groups ExtΛ M Λ , where Λ is the (non- commutative) Iwasawa algebra. The arguments are not purely algebraic in nature: We often employ continuous duals and have to take care about how we topologise certain modules. However, as the modules in question are finitely generated, this does not pose an actual obstacle. Everything becomes significantly more complicated if instead of considering discrete G- modules A, one wants to look at coefficients with more complicated topological data. It gets even worse when G and A are not only topological groups, but also have the struc- ture of e. g. manifolds. We will call these groups and coefficients topologised to denote a potentially finer structure than that of topological groups. Developing cohomology the- ories which make use of this finer structure, one is confronted with a fundamental issue: ¹We would like to point out that even the addition of natural numbers can benefit from a cohomological point of view, cf. [Grü17]. ix x INTRODUCTION The classical machinery of homological algebra refuses to function and results aremuch more difficult to prove. There seem to be three possibilities to circumvent these problems. The most conceptional one, due to Flach (cf. [Fla08]), embeds topological modules in a strictly larger category. He is able to elegantly recover standard results – however, many of the objects appearing are not representable by actual cohomology groups and only exist in this larger category, which makes explicit determinations rather difficult. A middle ground between explicit calculations and a more abstract point of view can be found for example in the work of Casselman and Wigner (cf. [CW74]), comprehensively in the book of Borel and Wal- lach (cf. [BW00]) and with applications to locally analytic representations in the work of Kohlhaase (cf. [Koh11]). Here, injective resolutions are replaced with certain acyclic reso- lutions with “strictly injective” differentials or similarly, whose existence in applications is somewhat unclear. The third possibility is the one originally used by Hochschild and Serre (cf. [HS53]), where everything is explicitly done on the level of cochains. One result one would like to recover is a version of the Hochschild-Serre spectral sequence for topologised groups. The discrete version of the theorem reads as follows: THeoRem. Let G be a group and N a normal subgroup. Let A be a G-module. Then there is a convergent first-quadrant spectral sequence + Hp ¹G/N; H q ¹N; Aºº =) Hp q ¹G; Aº: Another important result is known as Shapiro’s lemma, which in the discrete case can be stated as such: THeoRem. Let G be a group and H ≤ G a subgroup. Let A be an H-module. Then there is H ¹ º; an easily defined induced module IndG A on which G operates, such that there are natural isomorphisms for all n: n¹ ; H ¹ ºº n¹ ; º: H G IndG A H H A There are numerous instances in the literature where variants of these two results areused for topologised groups and coefficients – sometimes in a generality we find implausible, often in ways that turn out to be correct, but where we find the arguments insufficient, cf. e. g. [Lec12, lemma 5.26, theorem 5.27], [GR18, proposition 3.3], [Ked16, lemma 3.3], [Pot13, theorem 2.8], [BF17, remarks after definition 2.1.2].
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