When Hermite and Smith normal form are discussed in the textbooks (e.g. the one of Adkins and Weintraub) they assume the ring to be a PID. What goes wrong over UFDs? Can one still do something similar putting extra assumptions, like, for example assuming one dimensional UFD? I'd be interested, e.g., in the local ring of holomorphic germs in one variable. Are there some expositions of the subject?
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+1, interesting question. But isn't the local ring of holomorphic germs in one variable a principal ideal domain? Every ideal is generated by some power $z^n$. – joriki Feb 04 '23 at 17:40
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You are correct. The ring of germs of holomorphic functions in one variable is a PID. This is, for example, explained in the first volume of the book of Remmert and Schumacher. I got confused by this discussion: https://math.stackexchange.com/questions/153877/why-is-the-ring-of-holomorphic-functions-not-a-ufd. Thank you, joriki. – HCH Feb 05 '23 at 21:29
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Yes, I thought you might have been thinking of the ring of holomorphic functions. Are you still interested in the rest of the question, though? – joriki Feb 05 '23 at 22:00
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Of course. It bothers me for quite a while. – HCH Feb 05 '23 at 23:35