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