Let [Formula: see text] be a number field and let [Formula: see text] be an odd rational prime. Let [Formula: see text] be a [Formula: see text]-extension of [Formula: see text] and let [Formula: see text] be a finite extension of [Formula: see text], abelian over [Formula: see text]. In this paper we extend the classical results, e.g. [16], relating characteristic ideal of the [Formula: see text]-quotient of the projective limit of the ideal class groups to the [Formula: see text]-quotient of the projective limit of units modulo Stark units, in the non-semi-simple case, for some [Formula: see text]-irreductible characters [Formula: see text] of [Formula: see text]. The proof essentially uses the theory of Euler systems.

Let K be a totally real number field of degree r = [K : Q] and let p be an odd rational prime. Let K∞ denote the cyclotomic Zp-extension of K and let L∞ be a finite extension of K∞, abelian over K. In this article, we extend results of [Bü 09] relating characteristic ideal of the χ-quotient of the projective limit of the ideal class groups to the χ-quotient of the projective limit of the r-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some Qp-irreductible characters χ of Gal(L∞/K∞). × p be a non-trivial Q p -irreducible character of ∆. Let O := Z p [χ] be the ring generated by the values of χ over Z p and let O(χ) denote the ring O on which ∆ acts via χ. For any Z p [∆]-module M , we define the χ-quotient M χ by M χ := M ⊗ Zp[∆] O(χ).

Let K be a totally real number field of degree r. Let K ∞ denote the cyclotomic Z 2 -extension of K and let L ∞ be a finite extension of K ∞ , abelian over K. The goal of this paper is to compare the characteristic ideal of the χ-quotient of the projective limit of the narrow class groups to the χ-quotient of the projective limit of the r-th exterior power of totally positive units modulo a subgroup of Rubin-Stark units, for some Q 2 -irreducible characters χ of Gal(L ∞ /K ∞ ).

The aim of this paper is to compare the orders of the class groups and the quotients of the r-th exterior power of units modulo Rubin-Stark units.

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