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.