Index-modules and applications

Abstract

Let K be a commutative field, ${A\subseteq K}$ be a Dedekind ring and V be a K-vector space. For any pair of A-lattices R ≠ 0 and S of V, we define an A-submodule ${\left[R : S\right]^{\prime}_{A}}$ of K, their A-index-module. Once the basic properties of these modules are stated, we show that this notion can be used to recover more usual ones: the group-index, the relative invariant, the Fitting ideal of R/S when ${S\subseteq R}$, and the generalized index of Sinnott. As an example, we consider the following situation. Let F/k be a finite abelian extension of global function fields, with Galois group G, and degree g. Let ∞ be a place of k which splits completely in F/k. Let ${{\mathcal O}_{F}}$ be the ring of functions of F, which are regular outside the places of F sitting over ∞. Then one may use Stark units to define a subgroup ${\mathcal E_F}$ of ${{\mathcal O}_{F}^{\times}}$, the group of units of ${\mathcal O_F}$. We use the notion of index-module to prove that for every nontrivial irreducible rational character ψ of G, the ψ-part of ${\mathbb Z\left[g^{-1}\right]\otimes_\mathbb Z\left(\mathcal O_F^\times/\mathcal E_F\right)}$ and the ψ-part of ${\mathbb Z\left[g^{-1}\right]\otimes_\mathbb ZCl(\mathcal O_F)}$ have the same order.