Flat morphism

Remarks

For a ring $A$, the morphism $\mathbb{A}^n_A \to \operatorname{Spec} A$ is flat. Each local ring of $\mathbb{A}^n_A$ is a localization of $A[x_1, \dots, x_n]$, which is a free $A$-module, and localizations of flat modules are flat. For a field $k$, the closed immersion $\operatorname{Spec} k \to \operatorname{Spec} k[t]$ given by $t \mapsto 0$ is not flat. Its stalk map is $k[t]_{(t)} \to k$, and $k$ is not flat over $k[t]_{(t)}$, since multiplication by $t$ is injective on $k[t]_{(t)}$ but becomes zero on $k$. For a morphism of affine schemes $\operatorname{Spec} B \to \operatorname{Spec} A$, flatness is equivalent to flatness of $B$ as an $A$-module.