On the class semigroup of the cyclotomic \(\mathbb Z_p\)-extension of the rational numbers. (English) Zbl 1426.11126

Konomi, Yutaka; Morisawa, Takayuki

J. Commut. Algebra 11, No. 1, 69-80 (2019).

For a commutative ring \(R\) we let \(F_R\), \(I_R\) and \(P_R\) be the sets of all non-zero fractional, invertible and principal ideals of \(R\), respectively. The quotient \(\mathrm{Cl}_R := I_R / P_R\) is called the class group of \(R\) and is very frequently studied. The quotient \(S_R := F_R / P_R\) is a commutative monoid and is called the class semigroup of \(R\).
Let \(p\) be a prime. In this paper the authors study the case where \(R = \mathcal{O}\) is the ring of integers in the cyclotomic \(\mathbb Z_p\)-extension \(\mathbb Q_{\infty}\) of \(\mathbb Q\). Then \(\mathcal{O}\) is not Dedekind, but it is still a Prüfer domain of finite character. The latter means that every non-zero ideal of \(\mathcal{O}\) is contained in only finitely many maximal ideals. It is shown that the complements \(F_{\mathcal{O}} \setminus I_{\mathcal{O}}\) and \(S_{\mathcal{O}} \setminus \mathrm{Cl}_{\mathcal{O}}\) are in fact groups, where multiplication is induced by the usual multiplication of ideals. The unit element of \(F_{\mathcal{O}} \setminus I_{\mathcal{O}}\) and \(S_{\mathcal{O}} \setminus \mathrm{Cl}_{\mathcal{O}}\) is the unique maximal ideal \(\mathfrak{p}\) of \(\mathcal{O}\) above \(p\) and its class \([\mathfrak p]\) in \(S_{\mathcal{O}}\), respectively. These groups are therefore subsemigroups, but not submonoids of \(F_{\mathcal{O}}\) and \(S_{\mathcal{O}}\), respectively.
Moreover, one has an isomorphism \[ S_{\mathcal{O}} \setminus \mathrm{Cl}_{\mathcal{O}} \simeq \mathrm{Cl}_{\mathcal{O}} \times \mathbb R / \mathbb Z[1/p] \] as \(\Gamma\)-modules, where \(\Gamma := \mathrm{Gal}(\mathbb Q_{\infty} / \mathbb Q)\) acts trivially on the second factor.
The main step in the proof is to show that every non-zero ideal \(I\) of \(\mathcal{O}\) is of the from \(I = \mathfrak{a} \mathfrak{p}^{\alpha}\), where \(\mathfrak{a}\) is an invertible ideal and \(\alpha \in \mathbb R\). Here, the expression \(\mathfrak{p}^{\alpha}\) means the following: For each non-negative integer \(n\) denote the \(n\)-th layer of the \(\mathbb Z_p\)-extension \(\mathbb Q_{\infty}/\mathbb Q\) by \(\mathbb Q_n\) and let \(\mathfrak{p}_n\) be the unique maximal ideal above \(p\) in \(\mathbb Q_n\). Set \(\alpha_n := \lfloor p^n \alpha + 1 \rfloor\). Then \[ \mathfrak{p}^{\alpha} = \bigcup_{n=0}^{\infty} \mathfrak{p}_n^{\alpha_n}. \] Note that \(\mathfrak{p}^{0} = \mathfrak{p}\). One can show that the ideals \(\mathfrak{p}^{\alpha}\) are not invertible, and that one has \(\mathfrak{p}^{\alpha + \beta} = \mathfrak{p}^{\alpha} \mathfrak{p}^{\beta}\). In order to establish the above displayed isomorphism, one finally has to show that \([\mathfrak{p}^{\alpha}] = [\mathfrak{p}]\) if and only if \(\alpha \in \mathbb Z[1/p]\).
We point out that the authors [J. Pure Appl. Algebra 223, No. 9, 3665–3680 (2019; Zbl 1470.11282)] have meanwhile generalized this to arbitrary number fields (albeit with some mild concessions as the final step requires the ideals \(\mathfrak{p}_n\) to be principal).

Reviewer: Andreas Nickel (Essen)

MSC:

11R23	Iwasawa theory
11R29	Class numbers, class groups, discriminants

Keywords:

class semigroup; class group; cyclotomic extension

Citations:

Zbl 1470.11282

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