Canonical and $n$-canonical modules on a Noetherian algebra

TL;DR

This paper introduces and studies canonical and n-canonical modules on Noetherian algebras, generalizing classical theorems on syzygies and module descent to a non-commutative setting.

Contribution

It defines n-canonical modules and extends key theorems, including a non-commutative version of Aoyama's theorem and a codimension two argument for modules.

Findings

Generalized a theorem on (n,C)-syzygy using n-canonical modules

Proved a non-commutative version of Aoyama's theorem on module descent

Established a codimension two-argument for modules over a coherent sheaf of algebras

Abstract

We define canonical and $n$-canonical modules on a module-finite algebra over a Noether commutative ring and study their basic properties. Using $n$-canonical modules, we generalize a theorem on $(n,C)$-syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a non-commutative version of Aoyama's theorem which states that a canonical module descends with respect to a flat local homomorphism. We also prove the codimension two-argument for modules over a coherent sheaf of algebras with a $2$-canonical module, generalizing a result of the author.