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Semilocal rings vs Picard group

  1. Jul 18, 2011 #1

    Hurkyl

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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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  3. Jul 19, 2011 #2

    micromass

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    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.
     
  4. Jul 19, 2011 #3

    Hurkyl

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    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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