The well-known fundamental identity in number theory expresses the degree of an extension of global fields in terms of local information. In this article we show a generalized fundamental identity for arbitrary Dedekind domains. As an application, we show that any Japanese Dedekind domain is excellent.
The author is partially support by the MoST grant 109-2115-M-001-002-MY3.
The author thanks Raymond Heitmann who kindly informed him an example of a locally Nagata principal ideal domain which is not Nagata . He also thanks the referee for a careful reading and the reference .
"Japanese Dedekind Domains Are Excellent." Taiwanese J. Math. 26 (3) 483 - 500, June, 2022. https://doi.org/10.11650/tjm/220102