[Math] finite morphism is proper

algebraic-geometryschemes

I've got stuck in the exercise II.4.1 in Hartshorne.

Note that it is "a finite morphism is proper".

Most of the solutions starts with "Proper morphism is local on the base…" and uses valuative criterion without noetherian hypothesis.

But it is on the corollary 4.8 which needs noetherian hypothesis.

Hartshorne give me a notice for noetherian hypothesis but I cannot read french (EGA).

So,

  1. Can you give me tight condition for valuative criterion?

  2. Do corollary 4.6 and 4.8 (Properties of separated or proper morphisms) need any additional condition (like noetherian) or need only separatedness/properness?

Best Answer

For Question 1, the most general versions of the various valuative criteria I know of are the following.

Consider a morphism $f\colon X \to S$, and the following commutative diagram: $$\require{AMScd} \begin{CD} \operatorname{Spec}(K) @>>> X\\ @VVV @VVfV\\ \operatorname{Spec}(A) @>>> S \end{CD}\label{eq:star}\tag{$*$}$$ where $A$ is a valuation ring with field of fractions $K$.

Valuative criterion for universal closedness [Stacks, Tag01KA]. Assume $f$ is quasi-compact. Then, $f$ is universally closed if and only if there exists a morphism $\operatorname{Spec}(A) \to X$ making the diagram \eqref{eq:star} commute.

Valuative criterion for separatedness [Stacks, Tag 01KY].

  1. If $f$ is separated, then there is at most one morphism $\operatorname{Spec}(A) \to X$ making the diagram \eqref{eq:star} commute.
  2. If there is at most one morphism $\operatorname{Spec}(A) \to X$ making the diagram \eqref{eq:star} commute, and $f$ is quasi-separated, then $f$ is separated.

Valuative criterion for properness [Stacks, Tag 0BX4]. Assume $f$ is quasi-separated and of finite type. Then, $f$ is proper if and only if there exists a unique morphism $\operatorname{Spec}(A) \to X$ making the diagram \eqref{eq:star} commute.

For Question 2, no, you do not need noetherianity for these statements. For example, see [Görtz/Wedhorn, Prop. 9.13] for the analogue of Corollary 4.6, and [Görtz/Wedhorn, Prop. 12.58] for the analogue of Corollary 4.8. In essence, what you need to do (for separatedness) is to determine directly whether the diagonal in question is a closed immersion.

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