Cohen-Macaulay type of orders, generators and ideal classes
Stefano Marseglia
TL;DR
The paper develops a comprehensive framework for the Cohen-Macaulay type of orders in étale algebras over Dedekind domains, providing explicit local formulas $type_{ \mathfrak p}(S)=\dim_{S/\mathfrak p}(S^{t}/\mathfrak pS^{t})$ and a global bound $type(S)\le\dim_Q(K)-1$, with sharp realizations. It connects CM type to the structure of overorders via $g(S)$, establishes a complete CM type 2 classification, and derives a counting formula for isomorphism classes of fractional ideals and their multiplicator rings, yielding practical consequences for conjugacy classes of integral matrices and abelian varieties over finite fields. The work also situates CM type within broader notions of near-Gorensteinness, relating local completions, trace duals, and conductor-based length criteria to the existing Gorenstein/Bass landscape. Collectively, these results enable algorithmic calculations of ideal classes (via the ideal class monoid and weak equivalence) and illuminate how far an order is from Gorenstein or Bass, with significant applications to computational number theory and arithmetic geometry.
Abstract
In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in étale 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.