Cohen-Macaulay type of orders, generators and ideal classes

TL;DR

This paper investigates the Cohen-Macaulay type of orders over Dedekind domains in étale algebras, providing bounds, formulas, and classifications with applications to matrices and abelian varieties.

Contribution

It introduces new bounds and formulas for the Cohen-Macaulay type of orders and relates it to minimal generating sets, extending known results to broader classes.

Findings

Derived bounds for the Cohen-Macaulay type of orders.

Formulas for computing the type of overorders.

Classification of ideal classes with multiplicator ring of type 2.

Abstract

In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in \'etale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of minimal generating sets of its fractional ideals, generalizing known results for Gorenstein and Bass orders. Finally, we give a classification of the ideal classes with multiplicator ring of type $2$, with applications to the computations of the conjugacy classes of integral matrices and the isomorphism classes of abelian varieties over finite fields.