Ideal class semigroups of overrings

Let $p$ be an odd prime. For a number field $K$ , we let $K_{\infty }$ be the maximal unramified pro- $p$ extension of $K$ ; we call the group $\text{Gal}(K_{\infty }/K)$ the $p$ -class tower group of $K$ . In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$ -group occurs as the $p$ -class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\text{Gal}(K/\mathbb{Q})$ on the $p$ -class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for the groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In the appendix, we show how the probabilities introduced for finite $p$ -groups can be extended in a consistent way to the infinite pro- $p$ groups which can arise in both the real and imaginary quadratic settings.

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Factorizations in Bounded Hereditary Noetherian Prime Rings

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000305 ◽

2018 ◽

Vol 62 (2) ◽

pp. 395-442 ◽

Cited By ~ 4

Author(s):

Daniel Smertnig

Keyword(s):

Class Group ◽

Structure Theory ◽

Projective Modules ◽

Ideal Class ◽

Ideal Class Group ◽

Prime Rings ◽

Sets Of Lengths ◽

Irreducible Elements ◽

Transfer Homomorphism ◽

Zero Sum

AbstractIf H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

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