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

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