Exercise 111.38.2 (0DJ2)—The Stacks project

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Table of contents
Part 9: Miscellany
Chapter 111: Exercises
Section 111.38: Morphisms from the projective line
Exercise 111.38.2 (cite)

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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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