# Separatedness (Thursday, October 22)

:::{.example}
Consider $\AA^1$, whose polynomial functions are $k[x]$.
Consider now $D(x) \subset \AA^1$, which is isomorphic to the affine variety $V(xy-1)$.
Then the regular functions on $D(x)$ are given by 
\[  
A(D(x)) = 
\frac{ k[x, y] }{ \gens{xy-1} } \cong k[x, x^{-1} ]
.\]
:::

## Products of Prevarieties

Recall that a *prevariety* is a ringed space $(X, \OO_X)$ such that $X$ has a finite open cover by affine varieties $(U_i, \ro{\OO_X}{U_i})$, and a *morphism* of prevarieties is a morphism of ringed spaces.
We saw that one can construct prevarieties by gluing finite collections of prevarieties or affine varieties along open sets, and all prevarieties arise this way.
Similar to varieties, the product $P$ of prevarieties $X, Y$ will satisfy a universal property:

\begin{center}
\begin{tikzcd}
P \arrow[rrd, "\pi_X", bend left] \arrow[rdd, "\pi_Y"', bend right] &                                                     &   \\
                                                                    & Z \arrow[d] \arrow[r] \arrow[lu, "\exists !", dashed] & X \\
                                                                    & Y                                                   &  
\end{tikzcd}
\end{center}

[Link to Diagram](https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAAUQBfU9TXfIRQBGUsKq1GLNgC1uvEBmx4CRAExiJ9Zq0QgAGvL7LBRUWq1TdIAJrcJMKAHN4RUADMAThAC2SUSA4EEgAzDwe3n6IAUFIauEgXr7+1LGIZCAMWGDWUHRwABaOINTa0noAOhUwAB5YcDgICUlRGWkaIABGMGBQoRll1lVoWAD6hs2RSG3BiCHU3b1IALQhA1Zsw2N21Ax03Qzs-CpCIJ5YTgU49lxAA)

:::{.proposition title="?"}
The product is unique up to unique isomorphism, i.e. there is a unique isomorphism between any two products.
:::

:::{.proof}
Standard exercise in category theory.
:::

### Constructing the product of prevarieties 

:::{.example}
Consider $\AA^1 \times \AA^1$, then the product is (and should be) $\AA^2$, but $\AA^2$ does not have the product topology.
For example, one problem is that the Zariski open set $D(x-y)$ is not covered by products of open sets in $\AA^1$.
:::

This happens because the Zariski topology is too weak.
Strategy to fix: use gluing.
Let $X, Y$ be prevarieties and $\ts{U_i}, \ts{V_i}$ be open affine covers of $X$ and $Y$ respectively.
We can construct the product $U_i \cross V_j \subset \AA^{n+m}$, which is an affine variety and satisfies the universal property for products.
We then glue two such products $U_{i_1} \cross V_{j_1}$ and $U_{i_2} \cross V_{j_2}$ along their common open subset in $\qty{U_{i_1}\cap U_{i_2} }\intersect\qty{V_{j_1} \cap V_{j_2}} \subseteq X\cross Y$.

Let 
$\tilde U \da U_{i_1} \intersect U_{i_2} \cross V_{j_1} \intersect V_{j_2}$, we then need that

\[  
\qty{ \tilde U, \ro{ \OO_{U_{i_1} \cross V_{j_1}} }{\tilde U} } \cong
\qty{ \tilde U, \ro{ \OO_{U_{i_2} \cross V_{j_2}} }{\tilde U} }
.\]
This follows from the universal property of products, since the open set $(U\cross V, \ro{ \OO_{X\cross Y} }{U\cross V} )$ is a categorical product of ringed spaces, and the identity provides a unique isomorphism.
By the gluing construction, this produces a ringed space $(X\cross Y, \OO_{X\cross Y})$, we just need to check that this satisfies the universal property.
We have projections $\pi_X, \pi_Y$ set-theoretically, which restrict to morphisms on every $U_i \cross V_j$.
For any prevariety $Z$, we get a unique set map $h:Z\to X\cross Y$ which commutes, so it suffices to check that $h$ is a morphism of ringed spaces.

So consider $h^{-1} (U_i \cross V_j) \subset Z$, which is an open subset of $Z$ given by $f^{-1}(U) \cross f^{-1}(V)$.
Take an open cover and let $W$ be an element in it. 
We can then restrict $f$ and $g$ to get $\ro{f}{W}:W\to U_i$ and $\ro{g}{W}:W\to V_j$ and their product is a morphism of ringed spaces.
So $Z$ is covered by open sets for which $h$ is a morphism of ringed spaces, making $h$ itself a morphism.

What was the point of constructing the product?
We want some notion analogous to being Hausdorff to distinguish spaces like $\PP^1/k$ from the line with the doubled origin.
The issue is that these spaces with the Zariski topology are never Hausdorff.
So we make the following definition:

:::{.definition title="Separated"}
A prevariety is **separated** iff the diagonal morphism
\[  
\Delta_X: X &\to X\cross X \\
x &\mapsto (\id_X \cross \id_X)(x) \da (x, x)
\]
is a closed embedding.
:::

:::{.definition title="Variety"}
A **variety** is a separated prevariety.
:::