Does the Picard Group Vanish for Semilocal Rings?

[Hurkyl]

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?

 

[micromass]

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.

 

[Hurkyl]

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).