The following problem is an exercise at $11^{th}$ chapter in Atiyah's book on commutative algebra:

For any Noetherian ring $R$ we have $\dim R[x] = 1 + \dim R$ where $\dim$ stands for Krull dimension.

It is ok if one follows the hint after the exercise but I wonder if there exists another way to prove it via homological methods?

For example, when $R$ is a Noetherian regular ring (not requiring $R$ to be local) then the Krull dimension of $R$ coincides with its global homological dimension, hence the problem is solved by Hilbert's syzygy.

  • $\begingroup$ With the hint, I can only prove $dimR[x]$ is not less than $1+dimR$.And I found a method to prove that it is equal in Eisenbud’s book.Can we prove the exercise with the hint in Atiyah’s book? $\endgroup$ – Sky May 29 at 6:43

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