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I ask a question about $\prod k$ in Mathematics about several days.https://math.stackexchange.com/q/2766054/453628. And I have the following question:

1.What is the global dimension about $\prod k$?

2.is there an example of a ring which is absolutely flat and the global dimension is infinite?of course,if this exists,it can't be Noetherian.since finite presented flat module is projective. and the global dimension is the supremum of all projective dimension of finite generated modules.

3.can someone help give an example of a ring which has infinite global dimension and every finite generated module has finite projective dimension.

Thank you in advance!

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  • $\begingroup$ Could you explain how $\operatorname{Spec} k$ is not a counterexample to what you write in 2? $\endgroup$
    – R. van Dobben de Bruyn
    May 7, 2018 at 16:13
  • $\begingroup$ @R.vanDobbendeBruyn thanks.I will correct it. $\endgroup$
    – Jian
    May 7, 2018 at 23:35
  • $\begingroup$ I see an answer to your question 1 already in the MSE link. For more information about this interesting phenomenon, check mathoverflow.net/questions/68436/…. $\endgroup$
    – Fred.Fred
    May 8, 2018 at 8:36

1 Answer 1

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As pointed out in comments, the answer to question 1 has come up here before. But just for completeness, in

Osofsky, B.L., Homological dimension and cardinality, Trans. Am. Math. Soc. 151, 641-649 (1970). ZBL0209.07101,

Osofsky proves that the product of countably many copies of a field has global dimension $k+1$ if $2^{\aleph_0}=\aleph_k$ for some $k<\infty$ and infinite global dimension otherwise (so the global dimension is $2$ if the continuum hypothesis is true).

She also proves that the product of $\aleph_\omega$ copies of a field has infinite global dimension, so that gives an example for question $2$ that doesn't depend on assumptions about $2^{\aleph_0}$.

The famous example of Nagata, in the appendix of his book on Local Rings, of a commutative Noetherian ring with infinite Krull dimension answers question 3.

There's a nice and accessible discussion in Section 5G of

Lam, T.Y., Lectures on modules and rings, Graduate Texts in Mathematics. 189. New York, NY: Springer. xxiii, 557 p. (1999). ZBL0911.16001.

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