arXiv:1009.3729v1 [math.NT] 20 Sep 2010 SNOQIT I: THE Λ[G]-MODULES OF IWASAWA THEORY PREDA MIH ˘ AILESCU Abstract. For Λ = Z p [[T ]], the ring of formal power series in one variable, the structure of the finitely generated Λ - torsion modules is a main concern of Iwasawa theory. In this first part of Snoqit 1 we investigate some topics related to the representation of these Λ-modules as direct sums of certain elementary Λ - submodules, and the growth of the intermediate levels of modules which are projective limits. Contents 1. Introduction 1 1.1. Classical theory of Λ-modules reviewed 3 2. Property F and IW modules 6 2.1. Definition of Property F 7 2.2. Growth of IW-modules 10 2.3. Classification of PIW-modules 16 3. Coalescence and decompositions of Λ modules 17 4. The decomposition of Weierstrass modules 19 5. Appendix: Property F for class groups 21 References 27 1. Introduction Let p be an odd prime and Λ = Z p [[T ]] be the ring of formal power se- ries in the independent variable T . Then Λ is a local ring with maximal ideal M =(p, T ). The maximal ideal verifies the topological condition ∩ n M n = {0}, which allows one to induce M - adic completion on Λ - modules. Therefore all the Λ-modules considered here will be assumed to be complete in the M - adic topology [5], Chapter 5, §11. In number theory, one considers extensions K ∞ /K of finite extensions K/Q, such that Gal(K ∞ /K) ∼ = Z p and K ∞ = ∪ n≥0 K n , [K n : K n−1 ]= Date : Version 1.00 September 21, 2010. Key words and phrases. 11R23 Iwasawa Theory, 11R27 Units. 1