Example 26.20.7 (01KG)—The Stacks project

Example 26.20.7. Let $k$ be a field. Consider the structure morphism $p : \mathbf{P}^1_ k \to \mathop{\mathrm{Spec}}(k)$ of the projective line over $k$ , see Example 26.14.4. Let us use the valuative criterion above to prove that $p$ is universally closed. By construction $\mathbf{P}^1_ k$ is covered by two affine opens and hence $p$ is quasi-compact. Let a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_\xi \ar[d] & \mathbf{P}^1_ k \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r]^\varphi & \mathop{\mathrm{Spec}}(k) }$

be given, where $A$ is a valuation ring and $K$ is its field of fractions. Recall that $\mathbf{P}^1_ k$ is gotten by glueing $\mathop{\mathrm{Spec}}(k[x])$ to $\mathop{\mathrm{Spec}}(k[y])$ by glueing $D(x)$ to $D(y)$ via $x = y^{-1}$ (or more symmetrically $xy = 1$ ). To show there is a morphism $\mathop{\mathrm{Spec}}(A) \to \mathbf{P}^1_ k$ fitting diagonally into the diagram above we may assume that $\xi$ maps into the open $\mathop{\mathrm{Spec}}(k[x])$ (by symmetry). This gives the following commutative diagram of rings

$\xymatrix{ K & k[x] \ar[l]^{\xi ^\sharp } \\ A \ar[u] & k \ar[u] \ar[l]_{\varphi ^\sharp } }$

By Algebra, Lemma 10.50.4 we see that either $\xi ^\sharp (x) \in A$ or $\xi ^\sharp (x)^{-1} \in A$ . In the first case we get a ring map

$k[x] \to A, \ \lambda \mapsto \varphi ^\sharp (\lambda ), \ x \mapsto \xi ^\sharp (x)$

fitting into the diagram of rings above, and we win. In the second case we see that we get a ring map

$k[y] \to A, \ \lambda \mapsto \varphi ^\sharp (\lambda ), \ y \mapsto \xi ^\sharp (x)^{-1}.$

This gives a morphism $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(k[y]) \to \mathbf{P}^1_ k$ which fits diagonally into the initial commutative diagram of this example (check omitted).

The Stacks project

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