Krull dimension of polynomial rings over Noetherian rings via homological methods

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.

• 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? – Sky May 29 at 6:43