The Stacks project

Exercise 111.17.3. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $n \geq 1$. Let $\mathfrak m' = (\mathfrak m, x_1, \ldots , x_ n)$ in the polynomial ring $R[x_1, \ldots , x_ n]$. Show that

\[ \dim (R[x_1, \ldots , x_ n]_{\mathfrak m'}) = \dim (R) + n. \]

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View Exercise 111.17.3 as pdf