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How do I use the valuative criterion for properness:

"Let $f: X \rightarrow Y$ be a morphism of finite type of noetherian schemes. Then $f$ is proper if and only if for all discrete valuation rings $R$ with fields of fractions $K$ and for any $K$-valued point $x \in X(K)$ that maps to a point $f(x)$ that is defined over $R$, there is a unique lift of $x$ to $\overline{x} \in X(R)$"

to show that if $A$ is a Noetherian ring, then the structure morphism $\mathbb{P}^n_A\rightarrow \mathrm{Spec}\, A$ is proper?

Gazerun
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2 Answers2

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In the valuative criterion, you need a morphism $\mathrm{Spec}(R)\to Y$. In the affine $Y$ case, this means $R$ must be an $A$-algebra.

Write $\mathbb P^n_A=\mathrm{Proj}A[T_0, \dots, T_n]$. A rational point $x\in \mathbb P^n_A(K)$ can be represented by homogeneous coordinates $(t_0, t_1, \dots, t_n)$ with $t_i\in K$ and at least one of them is non-zero. Multiplying them by a suitable element of $R$, which does not change the point $x$, one can suppose that all $t_i\in R$ and at least one of them, say $t_0$, belongs to $R^*$ (units of $R$).

Consider the morphism $\bar{x} : \mathrm{Spec}(R)\to D_+(T_0)\subset \mathbb P^n_A$ defined by $$ O(D_+(T_0)) = A\left[\frac{T_1}{T_0}, \dots, \frac{T_n}{T_0}\right]\to R, \quad \frac{T_i}{T_0}\mapsto \frac{t_i}{t_0}.$$ Then it is easy to see that $\bar{x}$ extends $x$.

The uniqueness of the extension comes from the fact that the projective space is separated. One can also check directly as above using coordinates (if $\bar{x}'$ is another extension, suppose for instance that it maps the closed point to some point of $D_+(T_0)$, then show that $t_i/t_0\in R$ and $\bar{x}'=\bar{x}$.)

Cantlog
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Hartshorne proves this at the end of II.4; he shows that a projective morphism is proper which is the case you have. He actually reduces the proof to showing that $\mathbb{P}^n_A\to\mbox{Spec}(A)$ is proper.

rfauffar
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