ON FREE RESOLUTIONS OF IWASAWA MODULES

Let Lambda (isomorphic to Z(p) [[T]]) denote the usual Iwasawa algebra and G denote the Galois group of a finite Galois extension L / K of totally real fields. When the non-primitive Iwasawa module over the cyclotomic Z(p)-extension has a free resolution of length one over the group ring Lambda [G], we prove that the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive p-adic L-function (which is an element of a K-1-group) in a maximal Lambda-order. This integrality result involves a study of the Dicudonne determinant. Using a cohomolgoical criterion of Greenberg, we also deduce the precise conditions under which the non-primitive Iwasawa module has a free resolution of length one. As one application of the last result, we consider an elliptic curve over Q with a cyclic isogeny of degree p(2). We relate the characteristic ideal in the ring Lambda of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over Lambda, associated to two non-primitive classical Iwasawa modules.