# Friday, November 12

:::{.remark}
Continuing from last time: this is equivalent to $\Delta(\PP^n\slice \ZZ)$ being closed.
Since every affine scheme is separated, $\Delta(\AA^n\slice \ZZ)$ is closed for every $D(x_i)$.
Suppose a closed point $(P, P')$ lies in the closure of $\Delta(\PP^n\slice \ZZ)$, then if $P, P'\in D(x_i)$ for some $i$ then $P = P'$ since $D(x_i)$ is separated.
We can ensure this is possible by potentially taking a linear change of coordinates.
Introducing new variables $x_i' = \sum n_i x_i$ with $N \da (n_i)_{i\in I} \in \GL_{n+1}(\ZZ)$.
Then there is an isomorphism of graded rings $\ZZ[x_0, \cdots, x_n] \to \ZZ[x_0', \cdots, x_n']$ inducing an isomorphism $\PP^n\slice\ZZ\selfmap$.
By doing this we can replace $x_0$ with any linear combination of $x_i$s, and we need to show that there exists a linear map $L$ such that $L(x) = \sum n_i x_i$ for which $L(x)\neq P, L(x)\neq P'$, so $P, P' \in D(L(x))$.
Consider the image of $L(x)$ in $\ZZ[x_0, \cdots, x_n]/P$ and similarly for $P'$.
These are (finite) fields since $P, P'$ are maximal.
Any map $\ZZ\cartpower{n+1}\to \FF_q$ is a group morphism, and the kernel is a finite index sublattice.
One can always find an element of $\ZZ\cartpower{n+1}$ which isn't on the union of two strict sublattices, i.e. $1/a + 1/b - 1/ab < 1$.
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:::{.example title="?"}
A projective variety over $k$ is proper over $\spec k$.
These are of the form $\Proj \kxn / I$ for $I$ a homogeneous ideal, and thus come with a closed immersion into $\PP^n\slice k$.
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:::{.example title="The main class of examples"}
If $X \mapsvia{f} Y \in \Proj\Var$ or $\Sch\slice k$.
Then the maps $X\to \spec k$ and $Y\to \spec k$ are proper, and the second is separated.
Peeling off the compositions shows $f$ is proper.
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:::{.example title="?"}
Let $X \mapsvia{f} Y$ be any morphism from a projective scheme to a separated scheme of finite type over $k$.
This is also proper, and thus universally closed, and its image in $Y$ is also proper using that closed subschemes of separated schemes are separated and of finite type, and morphisms factor through their images.
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:::{.corollary title="?"}
Any regular function on a projective (or even proper) variety is locally constant.
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:::{.proof title="?"}
A regular function on projective $X$ is a morphism $X \mapsvia{f} \AA^1$, so consider the open immersion $\AA^1\injects \PP^1$.
The composition $x\circ f: X\to \PP^1$ is projective, thus proper, so $(i \circ f )(X) \subseteq \AA^1 \subseteq \PP^1$ is closed, but the only such closed sets are finite.
Thus $i \circ f$ and thus $f$ is constant on a connected component of $X$.
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:::{.corollary title="?"}
Any morphism from a proper variety to an affine variety is locally constant.
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:::{.proof title="?"}
If $X \mapsvia{f} Y$ with $X$ proper and $Y$ affine, then there is an open immersion $\iota: Y\injects \AA^n$.
The composition of $\iota \circ f$ is locally constant on each coordinate by the previous corollary, making $f$ locally constant.
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:::{.example title="?"}
For $X$ any variety and $Y$ proper, $X\times Y\to X$ is proper because it is the base change along $Y\to \spec k$.
So e.g. $\PP^1\times \AA^1 \to \AA^1$ is proper.
Another example: blow up a projective variety at an ideal.
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:::{.slogan}
Proper means compact fibers.
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