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

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).

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)

### Citations:

Zbl 1470.11282
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\textit{Y. Konomi} and \textit{T. Morisawa}, J. Commut. Algebra 11, No. 1, 69--80 (2019; Zbl 1426.11126)

### References:

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[3] | ——–, Clifford regular domains, J. Algebra 238 (2001), 703–722. · Zbl 1027.13011 |

[4] | S. Bazzoni and L. Salce, Groups in the class semigroups of valuation domains, Israel J. Math. 95 (1996), 135–155. · Zbl 0869.13006 |

[5] | P. Zanardo and U. Zannier, The class semigroup of orders in number fields, Math. Proc. Cambr. Philos. Soc. 115 (1994), 379–391. · Zbl 0828.11068 |

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