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The only examples of commutative rings of finite global dimension I know are either:

  • Dedekind domains (and fields as a degenerate special case)
  • Regular local rings
  • Rings constructed from the previous examples by taking direct sums, or forming the rings of polynomials over a ring of finite global dimension.

Are there other examples? In particular, are there other examples that are finite-dimensional over a field $k$?

(Examples of rings of finite global dimension are easier to come by in the noncommutative case, but I'm specifically curious about the commutative case.)

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What about regular rings with finite Krull dimension? Not all such rings are contained in your list above. –  Donu Arapura Jan 24 '11 at 22:25
In case you aren't sure about how to get such examples as mentioned in my last comment, take the coordinate ring of any nonsingular affine variety. –  Donu Arapura Jan 24 '11 at 22:40
That's the answer I was looking for. (I was only familiar with the noncommutative case, and didn't know the term "regular ring" or Serre's characterization until about ten seconds ago.) So they're actually incredibly common. Sorry, I didn't realize the question was so easy or well known. –  arsmath Jan 24 '11 at 22:40
Donu if you put your answer as an answer, I'll accept it so that this question doesn't linger on the unanswered question list. –  arsmath Jan 24 '11 at 22:42
No problem, that's what we're here for. –  Donu Arapura Jan 24 '11 at 22:43

2 Answers 2

up vote 6 down vote accepted

At arsmath's request, I'm making this official. (This is pretty standard commutative algebra, but I realize not everyone has gone through it.)

A commutative ring $R$ is regular if it's noetherian and its local rings are regular. Using Serre's theorem e.g. Matsumura Commutative Ring Theory p 156, and the fact that $Ext$ commutes with localization, we can see that any regular ring with finite Krull dimension has finite global dimension.

To an algebraic geometer regular = nonsingular. So in particular, so there is a large supply of basic examples arising as coordinate rings of nonsingular affine varieties. This is a bit circular the way I'm saying it, but of course, you can test the condition using the Jacobian criterion...

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An artinian ring $R$ is a finite direct product of local artinian rings. If $R$ is of finite global dimension, so are the factors, and then they are regular local by Serre's theorem. As regular local rings are domains, the Jacobson radical of the factors has to be trivial (for its elements are nilpotent) and then $R$ is a product of fields.

You will thus not get interesting examples of finite dimension.

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