Does the Picard Group Vanish for Semilocal Rings?

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SUMMARY

The discussion confirms that for a dimension 1 regular semi-local ring, the Picard group does indeed vanish. This conclusion is based on the fact that a dimension 1 regular ring qualifies as a Dedekind domain, which is known to be a Principal Ideal Domain (PID). Consequently, the Picard group of a PID is established to vanish. The conversation also raises questions regarding the behavior of the Picard group in non-regular cases and higher dimensions, indicating areas for further exploration.

PREREQUISITES
  • Understanding of Dedekind domains
  • Knowledge of Principal Ideal Domains (PIDs)
  • Familiarity with the concept of Picard groups
  • Basic principles of semi-local rings
NEXT STEPS
  • Research the properties of Dedekind domains in greater detail
  • Explore the relationship between semi-local rings and Principal Ideal Domains
  • Investigate the structure of Picard groups in non-regular rings
  • Examine the ideal class group in the context of higher-dimensional rings
USEFUL FOR

Mathematicians, algebraists, and graduate students studying ring theory, particularly those interested in the properties of semi-local rings and their implications on the Picard group.

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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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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