An Example!
Posted: Wed Jun 08, 2016 1:45 am
Consider the ring homomorphism
\[ \phi \ \colon \mathbb{C}[x,y] \longrightarrow \mathbb{C}[s,t] \\
x \mapsto s \\
y \mapsto st \]and the induced morphism of affine schemes \[ f \ \colon \mathbb{A}^{2}_{\mathbb{C}} = \text{Spec}\ \mathbb{C}[s,t] \longrightarrow \mathbb{A}^{2}_{\mathbb{C}} = \text{Spec} \ \mathbb{C}[x,y] \] Describe the pre-image $ f^{-1}(\mathfrak{m}) $ of the maximal ideal $ \mathfrak{m} = (x-a,y-b) \subset \mathbb{C}[x,y] $, where $ a,b \in \mathbb{C} $. View this result from a geometrical approach (with points in $\mathbb{C}^{2}$).
\[ \phi \ \colon \mathbb{C}[x,y] \longrightarrow \mathbb{C}[s,t] \\
x \mapsto s \\
y \mapsto st \]and the induced morphism of affine schemes \[ f \ \colon \mathbb{A}^{2}_{\mathbb{C}} = \text{Spec}\ \mathbb{C}[s,t] \longrightarrow \mathbb{A}^{2}_{\mathbb{C}} = \text{Spec} \ \mathbb{C}[x,y] \] Describe the pre-image $ f^{-1}(\mathfrak{m}) $ of the maximal ideal $ \mathfrak{m} = (x-a,y-b) \subset \mathbb{C}[x,y] $, where $ a,b \in \mathbb{C} $. View this result from a geometrical approach (with points in $\mathbb{C}^{2}$).