arXiv:math/9907215v1 [math.NT] 28 Jul 1999 Euler Characteristics as Invariants of Iwasawa Modules ∗ Susan Howson ∗∗ Introduction Let G be a pro-p, p-adic, Lie group with no element of order p and let Λ(G) denote the Iwasawa algebra of G, defined in the usual way by Λ(G) = lim ←− Z p [G/H ], (1) where H runs over the open, normal subgroups of G, and the inverse limit is taken with respect to the canonical projection maps. Then many situations arise in which one is interested in determining information about the structure of modules which are finitely generated over Λ(G), such as the rank (defined in (2) below). In this paper we describe a number of invariants associated to a finitely generated Λ(G) module, M , and calculated via an Euler characteristic formula. For example, we give a simple formula for the rank of M in terms of an Euler Characteristics formula. These ideas are well known to Algebraists, see for example the book by K. Brown, [2], chapter IX in particular, but do not appear to have been exploited to their full potential in Iwasawa Theory yet. This formula gives the natural generalisation of the strongest form of Nakayama’s lemma (for the case of G isomorphic to Z p ) to other pro-p, p-adic, Lie groups. Thus the first subsection of this paper should be seen as a continuation of the earlier note [1] which explained situations where that result can fail to generalise directly. We then apply these ideas to modules finitely generated over the F p –linear completed group algebra, and consider an invariant of Iwasawa modules which gives the classical Iwasawa µ-invariant in the case G ∼ = Z p , where the idea of expressing this invariant in terms of an Euler characteristic is well known. If one instead starts with the Euler characteristic formula as the definition of a ’homological Λ(G)–rank’ then we can extend it to some situations where the na¨ ıve definition of Λ(G)–rank is not appropriate. For example, we can consider removing the restriction that G is a pro-p group. In the second section we give some discussion of the information this can tell us about a Λ(G)–module. In particular, in the classical case of G ∼ = Z × p this relates to the decomposition of a finitely generated * Mathematics Subject Classification:16E10(11R23) ** School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K. sh@maths.nott.ac.uk