Exercise 111.17.1. Construct a ring A with finitely many prime ideals having dimension > 1.
111.17 Dimension
Exercise 111.17.2. Let f \in \mathbf{C}[x, y] be a nonconstant polynomial. Show that \mathbf{C}[x, y]/(f) has dimension 1.
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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