# Monday, November 08

:::{.remark}
Let $f:X\to Y \in \Sch$, then $f$ is **proper** iff

- $f$ is separated,
- $f$ is of finite type,
- $f$ is universally closed, so for all $Y'\to Y$, the base change morphisms $X\fiberprod{Y} Y' \to Y'$ is closed.

The valuative criterion of properness stated that if $R$ is a valuation ring and $K \da \ff(R)$, so $\spec K\to \spec R$, there are unique lifts of the following form:

\begin{tikzcd}
	{\spec K} && X \\
	\\
	{\spec R} && Y
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow[from=3-1, to=3-3]
	\arrow["\Theta"{description}, dashed, from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXHNwZWMgSyJdLFswLDIsIlxcc3BlYyBSIl0sWzIsMiwiWSJdLFsyLDAsIlgiXSxbMywyXSxbMCwxXSxbMCwzXSxbMSwyXSxbMSwzLCJcXFRoZXRhIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)

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:::{.definition title="Projective space over a scheme"}
Let $Y \in \Sch$, then define **projective space over $Y$** as $\PP^n\slice Y \da \PP^n\slice \ZZ \times Y \da \PP^n\slice \ZZ \fiberprod{\spec \ZZ} Y$.
:::

:::{.remark}
This is analogous to $\PP^n\slice R \da \Proj R[x_0,\cdots, x_n]$ for $R\in \CRing$, and these two constructions turn out to be the same.
Note that $R[x_0, \cdots, x_n] \cong \ZZ[x_0,\cdots, x_n] \tensor_\ZZ R$.
:::

:::{.definition title="Projective morphisms"}
A morphism $X \mapsvia{f} Y\in \Sch$ is **projective** iff $f$ factors as $X \injects \PP^n\slice Y \surjects Y$, a closed immersion into projective space followed by projection onto $Y$.
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:::{.example title="?"}
Let $S\in \gr_\ZZ\CRing$ be a graded ring, then $S_0 \leq S$ is a subring.
Suppose $S$ is finitely generated over $S_0$ by $S_1$, so there exists a finite generating set $\ts{x_i}_{i\leq n}$ and a surjective map $S_0[x_0, \cdots, x_n] \surjects S$ sending $x_i$ to elements of $S_1$.
Note that this preserves the grading, since $S_0$ elements have degree zero and $x_i$ have degree 1 on both sides, so we get a map $\psi: \Proj S \to \Proj S_0[x_0, \cdots, x_n] \da \PP^n\slice {S_0}$.
So the ring maps on affine opens will be surjective, since they are localizations of $\psi$, so this yields a closed immersion and thus a projective morphism $\Proj S \to \spec S_0$.
:::


:::{.example title="?"}
If $S = k \in \Field$ with $k = \bar{k}$, then if $R \in \gr_\ZZ \CRing$ is finitely generated in degree 1, then $\proj R\to \spec k$ is projective, since these are exactly quotients of $\kxn$ by a homogeneous ideal.
If this homogeneous ideal is radical, then these correspond to projective varieties over $k$.
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:::{.remark}
We'll show that projective implies proper, which will furnish many examples of proper maps.
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:::{.theorem title="?"}
Any projective morphism $f:X\to Y$ is proper.
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:::{.exercise title="Hartshorne 3.13, checking when a morphism is finite type"}
See Hartshorne, try it!

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:::{.proof title="?"}
We know that base changes of proper morphisms are proper, using the valuative criterion.
With the above example and exercise, it suffices to show $\PP^n\slice \ZZ$ is proper.
Why?
We have a closed immersion $X\to \PP^n\slice Y\da \PP^n\slice{\ZZ} \times Y$ by the definition of $f:X\to Y$ being proper.
Then the projection $\PP^n\slice{\ZZ}\times Y\to Y$ is proper, and is a base change of $\PP^n\slice \ZZ\to \spec \ZZ$.
So if we know the latter is proper, compositions of proper maps are proper and thus $f$ is proper.

\todo[inline]{todo, try to form a diagram here.}

Idea: clear denominators in a minimal way.
We can cover $\PP^n\slice \ZZ$ by affine opens:
\[
D(x_i) \da \ts{ \mfp \in \ZZ[x_0,\cdots, x_n]_{\homog} \st z_i\not\in \mfp}
\cong \spec \ZZ\adjoin{ {x_0 \over x_1}, \cdots, {x_n \over x_1}}
.\]
Then $\ts{D(x_i)}_{1\leq i \leq n+1} \covers \PP^n\slice \ZZ$, making $\PP^n\slice \ZZ$ finite type since it admits a finite cover by $\spec R_i$ for $R_i\in\alg\slice{\ZZ}^\fg$.
It thus suffices to verify the valuative criterion of properness, since this will imply separatedness.
So we'll show any morphism $g: \spec K\to \PP^n\slice \ZZ$ lifts uniquely to a morphism $\spec K\to \PP^n\slice \ZZ$.
Given $g$, $g(0)$ is a single point, and by a linear change of coordinates we can ensure $g(0) \in \Intersect_i D(x_i)$.
So we have
\[
g(0) \in D(x_0, \cdots, x_n) \cong \spec T \da \spec \ZZ\adjoin{\ts{x_i\over x_j}_{0\leq j\leq n}} &&\forall i
.\]
So $g$ factors through the open immersion $\spec T \embeds \PP^n\slice \ZZ$, and is thus a map of affine schemes and equivalently the data of a ring map $\phi: \spec T\to K \in \CRing$.
Let $\phi_{ij} = \phi\qty{x_i\over x_j}$, and note that $\phi_{ij} \in K\units$ for every $i, j$.
These satisfy a cocycle condition $\phi_{ij} \phi_{jk} = \phi_{ik}$, so letting $v_i\da v(\phi_{i, 0})$ for $v$ the valuation, there is some minimal $v_i$.

> Continued next time.

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