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22 votes
4 answers
2k views

Two questions about finiteness of ideal classes in abstract number rings

Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite. (I ...
Pete L. Clark's user avatar
34 votes
4 answers
5k views

What is the right definition of the Picard group of a commutative ring?

This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...
Pete L. Clark's user avatar
15 votes
6 answers
2k views

Seeking Noetherian normal domain with vanishing Picard group but not a UFD

Once again, the question says it all. My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
Pete L. Clark's user avatar
10 votes
1 answer
612 views

Bézout ring with non-trivial Picard group?

[I asked this on stackexchange here a few weeks ago to no response] A ring is called Bézout when its finitely generated ideals are principal. Q: Is there a nice example of a Bézout ring $R$ with ...
Badam Baplan's user avatar
4 votes
1 answer
194 views

When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$

Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
user443060's user avatar