The Stacks project

Exercise 111.38.1. Let $k$ be a field. Let $k[t] \subset k(t)$ be the inclusion of the polynomial ring into its fraction field. Let $X$ be a finite type scheme over $k$. Show that for any morphism

\[ \varphi : \mathop{\mathrm{Spec}}(k(t)) \longrightarrow X \]

over $k$, there exist a nonzero $f \in k[t]$ and a morphism $\psi : \mathop{\mathrm{Spec}}(k[t, 1/f]) \to X$ over $k$ such that $\varphi $ is the composition

\[ \mathop{\mathrm{Spec}}(k(t)) \longrightarrow \mathop{\mathrm{Spec}}(k[t, 1/f]) \longrightarrow X \]

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View Exercise 111.38.1 as pdf