Dimension theorem for Noetherian local rings

For a Noetherian local ring $(A,\mathfrak{m})$, the Krull dimension of $A$, the Krull dimension of the associated graded ring ${\mathrm{gr}}_{\mathfrak{m}}(A)$, and the order of the pole at $t=1$ of its Hilbert series are equal.

If $x$ is a non-zero-divisor in a finite-dimensional Noetherian ring $A$, thenEvery prime chain above $(x)$ can be extended downward by a minimal prime of $A$, because a non-zero-divisor belongs to no minimal prime.