The canonical trace of determinantal rings

TL;DR

This paper calculates the canonical trace of generic determinantal rings and provides conditions for its specialization, with applications to Cohen-Macaulay rings of codimension two, revealing explicit generators.

Contribution

It introduces a method to compute the canonical trace for determinantal rings and characterizes the trace in Cohen-Macaulay rings of codimension two.

Findings

Canonical trace of generic determinantal rings computed.

Explicit generators of the trace in Cohen-Macaulay codimension two rings identified.

Trace is generated by minors of the Hilbert-Burch matrix.

Abstract

We compute the canonical trace of generic determinantal rings and provide a sufficient condition for the trace to specialize. As an application we determine the canonical trace $\mbox{tr}(\omega_R)$ of a Cohen-Macaulay ring $R$ of codimension two, which is generically Gorenstein. It is shown that if the defining ideal $I$ of $R$ is generated by $n$ elements, then $\mbox{tr}(\omega_R)$ is generated by the $(n-2)$-minors of the Hilbert-Burch matrix of $I$.