Characteristic ideals and Iwasawa theory
Let \(A\) be a Noetherian Krull domain and \(M\) a finitely generated torsion \(A\)-module. The structure theorem for such modules asserts that \(M\) is pseudo-isomorphic to an elementary module of the form \(\bigoplus_{1\leq i\leq n} A/\mathcal{P}_i^{e_i}\), where the \(\mathcal{P}_i\)'s are height 1 prime ideals of \(A\). This allows to define an important invariant of \(M\), the characteristic ideal \(\mathrm{Ch}_A(M) :=\prod_{1\leq i\leq n} \mathcal{P}_i^{e_i}\), which plays a major role in the (commutative) Iwasawa theory for global fields, where \(A\) is the complete group algebra \(\mathbb{Z}_p\Gamma\) of an abelian \(p\)-adic Lie group \(\Gamma\). A classical example is \(\Gamma \cong \mathbb{Z}_p^d\) when studying multiple \(\mathbb{Z}_p\)-extensions of global fields, in which case \(A\) is isomorphic (non canonically) to the power series ring \(\mathbb{Z}_pt_1,\dots,t_d\). But over a function field \(F\) of characteristic \(p\), extensions \(\mathcal{F}/F\) with \(\mathrm{Gal}(\mathcal{F}/F) \cong \mathbb{Z}_p^\infty\) occur naturally, in which case the Noetherian setting is no longer adapted and the above definition of the characteristic ideal no longer applies. One way out is to consider a filtration of \(\mathbb{Z}_p^d\)-extensions for \(\mathcal{F}\), define the characteristic ideals at the \(\mathbb{Z}_p^d\)-levels and then pass to the inverse limit.NEWLINENEWLINEThe authors' main technical theorem reads: ``Let \(\{\Lambda_d\}_{d\geq 0}\) be an inverse system of Noetherian Krull domains such that \(\Lambda_d \cong \Lambda_{d+1} /\mathcal{P}_{d+1}\) and \(\Lambda_{d+1} \cong \varprojlim \Lambda_{d+1} /\mathcal{P}_{d+1}^n\) for any \(d \geq 0\) (where \(\mathcal{P}_{d+1}\) is a principal prime ideal of \(\Lambda_{d+1}\) of height \(1\)). Let \(\Lambda := \varprojlim \Lambda_d\) and consider a finitely generated \(\Lambda\)-module \(M = \varprojlim M_d\) (where each \(M_d\) is a \(\Lambda_d\)-module). Assume for any \(d \geq 1\) the two following properties: (1) The \(\mathcal{P}_d\)-torsion submodule of \(M_d\) is \(\Lambda_{d-1}\)-pseudo-null , and (2) \(\mathrm{Ch}_{\Lambda_{d-1}}(M_d /\mathcal{P}_d)\subseteq \mathrm{Ch}_{\Lambda_{d-1}}(M_{d-1})\). Then the ideals \(\mathrm{Ch}_{\Lambda_d}(M_d)\) form a projective system (w.r.t. the connecting homomorphisms defining \(\Lambda\)).''NEWLINENEWLINEThis allows to define the \(\Lambda\)-pro-characteristic ideal \(\widetilde{\mathrm{Ch}}_{\Lambda}(M) := \varprojlim \mathrm{Ch}_{\Lambda_d}(M_d)\).NEWLINENEWLINEApply this to the Iwasawa theory for class groups in function fields: \(F\) is a global function field of characteristic \(p\); let \(\mathcal{F}/F\) be a \(\mathbb{Z}_p^\infty\)-extension with Galois group \(\Gamma\) and Iwasawa algebra \(\Lambda\), unramified outside a finite set of places \(S\); choose a \(\mathbb{Z}_p\)-basis \(\{\gamma_i\}_{i\in\mathbb{N}}\) for \(\Gamma\) and for any \(d\geq 0\), let \(\mathcal{F}_d \subset \mathcal{F}\) be the fixed field of \(\{\gamma_i\}_{i>d}\). Assume that no place in \(S\) splits completely in \(\mathcal{F}_1\) (there is no loss of generality). For any finite extension \(L\) of \(F\), let \(\mathcal{M}(L)\) be the pro-\(p\)-completion of the group of divisor classes of \(L\) and \(\mathcal{A}(L)\) the kernel of the degree map \(\mathrm{deg}_L : \mathcal{M}(L) \to \mathbb{Z}_p\). Suppose from now on that the (finitely many) ramified places of \(\mathcal{F}/F\) are totally ramified. Let \(\mathcal{A}(F) := \varprojlim \mathcal{A}(F_d)\).NEWLINENEWLINEThe main technical theorem applies and allows to define the \(\Lambda\)-pro-characteristic ideal \(\widetilde{\mathrm{Ch}}_{\Lambda}(\mathcal{A}(\mathcal{F})) := \varprojlim \mathrm{Ch}_{\Lambda_d}(\mathcal{A}(\mathcal{F}_d))\), which is independent of the filtration \(\{\mathcal{F}_d\}\). Recall that in the appendix of [\textit{D. Burns}, Invent. Math. 184, No. 2, 221--256 (2011; Zbl 1234.11080)], Burns, Lai and Tan have proved the geometric Iwasawa main conjecture (GIMC) at finite level: \(\mathrm{Ch}_{\Lambda_d}(\mathcal{A}(\mathcal{F}_d))=(\theta_{S,\mathcal{F}_d/F})\), where \(\theta_{S,\mathcal{F}_d/F}\) is the classical Stickelberger element.NEWLINENEWLINEThe main technical theorem above yields the following non Noetherian GIMC: \(\widetilde{\mathrm{Ch}}_{\Lambda}(\mathcal{A}(\mathcal{F})) := \varprojlim (\theta_{S,\mathcal{F}_d/F}):= (\theta_{S,\mathcal{F}/F})\).
scientific article; zbMATH DE number 6359494