The Stacks project

109.17 Dimension

Exercise 109.17.1. Construct a ring $A$ with finitely many prime ideals having dimension $> 1$.

Exercise 109.17.2. Let $f \in \mathbf{C}[x, y]$ be a nonconstant polynomial. Show that $\mathbf{C}[x, y]/(f)$ has dimension $1$.

Exercise 109.17.3. 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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