Exercise 111.38.2. Let $k$ be a field. Let $k[t] \subset k(t)$ be the inclusion of the polynomial ring into its fraction field. Show that for any morphism
\[ \varphi : \mathop{\mathrm{Spec}}(k(t)) \longrightarrow \mathbf{P}^ n_ k \]
over $k$, there exists a morphism $\psi : \mathop{\mathrm{Spec}}(k[t]) \to \mathbf{P}^ n_ k$ over $k$ such that $\varphi $ is the composition
\[ \mathop{\mathrm{Spec}}(k(t)) \longrightarrow \mathop{\mathrm{Spec}}(k[t]) \longrightarrow \mathbf{P}^ n_ k \]
Hint: the image of $\varphi $ is in a standard open $D_+(T_ i)$ for some $i$; then show that you can “clear denominators”.
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