Generalization of bi-canonical degrees

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $ω$. Attached to each such ring R, when $ω$ is an ideal, there are integers--the type of R, the reduction number of $ω$--that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In arXiv:1701.05592 and arXiv:1711.09480 we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where $ω$ is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.

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