Iwasawa algebra

In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology of G into account. More precisely, Λ(G) is the inverse limit of the group rings Zp(G/H) as H  runs through the open normal subgroups of G. Commutative Iwasawa algebras were introduced by Iwasawa (1959) in his study of Zp extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact p-adic analytic groups were introduced by Lazard (1965). In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology of G into account. More precisely, Λ(G) is the inverse limit of the group rings Zp(G/H) as H  runs through the open normal subgroups of G. Commutative Iwasawa algebras were introduced by Iwasawa (1959) in his study of Zp extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact p-adic analytic groups were introduced by Lazard (1965). In the special case when the profinite group G is isomorphic to the additive group of the ring of p-adic integers Zp, the Iwasawa algebra Λ(G) is isomorphic to the ring of the formal power series Zp] in one variable over Zp. The isomorphism is given by identifying 1 + T with a topological generator of G. This ring is a 2-dimensional complete Noetherian regular local ring, and in particular a unique factorization domain. It follows from the Weierstrass preparation theorem for formal power series over a complete local ring that the prime ideals of this ring are as follows: The rank of a finitely generated module is the number of times the module Zp] occurs in it. This is well-defined and is additive for short exact sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module is a torsion module, which happens if and only if the support has dimension at most 1. Many of the modules over this algebra that occur in Iwasawa theory are finitely generated torsion modules. The structure of such modules can be described as follows. A quasi-isomorphism of modules is a homomorphism whose kernel and cokernel are both finite groups, in other words modules with support either empty or the height 2 prime ideal. For any finitely generated torsion module there is a quasi-isomorphism to a finite sum of modules of the form Zp]/(fn) where f is a generator of a height 1 prime ideal. Moreover, the number of times any module Zp]/(f) occurs in the module is well defined and independent of the composition series. The torsion module therefore has a characteristic power series, a formal power series given by the product of the power series fn, that is uniquely defined up to multiplication by a unit. The ideal generated by the characteristic power series is called the characteristic ideal of the Iwasawa module. More generally, any generator of the characteristic ideal is called a characteristic power series. The μ-invariant of a finitely-generated torsion module is the number of times the module Zp]/(p) occurs in it. This invariant is additive on short exact sequences of finitely generated torsion modules (though it is not additive on short exact sequences of finitely generated modules). It vanishes if and only if the finitely generated torsion module is finitely generated as a module over the subring Zp. The λ-invariant is the sum of the degrees of the distinguished polynomials that occur. In other words, if the module is pseudo-isomorphic to where the fj are distinguished polynomials, then