Does the Picard Group Vanish for Semilocal Rings?

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In the discussion on whether the Picard group vanishes for dimension 1 regular semi-local rings, it is concluded that the Picard group does indeed vanish. A dimension 1 regular ring is identified as a Dedekind domain, which implies that a semi-local Dedekind domain is a principal ideal domain (PID). Since the Picard group of a PID is known to vanish, this supports the conclusion. The conversation also raises questions about the behavior of the Picard group in non-regular cases and higher dimensions, but the primary focus remains on dimension 1 regular semi-local rings. Overall, the consensus affirms the vanishing of the Picard group in this specific case.
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For a dimension 1 regular semi-local ring, does the Picard group vanish?

What if it is not regular? (and what if I ask for the ideal class group?)
What if it's dimension greater than 1?
 
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Hurkyl said:
For a dimension 1 regular semi-local ring, does the Picard group vanish?

What if it is not regular? (and what if I ask for the ideal class group?)
What if it's dimension greater than 1?

I don't know very much about this. But I think that the Picard group does vanish. A dimension 1 regular ring is a Dedekind domain. And it's known that a semi-local Dedekind domain is a PID. And the Picard group of a PID vanishes.
 
micromass said:
I don't know very much about this. But I think that the Picard group does vanish. A dimension 1 regular ring is a Dedekind domain. And it's known that a semi-local Dedekind domain is a PID. And the Picard group of a PID vanishes.

Excellent. I thought my first question was an easy one, but for the life of me I couldn't find any references for it, and I kept getting tripped up in my attempts to prove it (e.g. that a semi-local Dedekind domain is a PID).
 

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