# Friday, October 01

:::{.remark}
Recall the Proj construction: for $S = \bigoplus_{d\geq 0} S_d \in \gr\CRing$ we define the irrelevant ideal $S_+ \da \bigoplus _{d\geq 1} S_d$ and
\[
\Proj S &\da \ts{p\in \spec S \text{ homog } \st p\not\contains S_+} \\
\OO_{\Proj S} &\da \ts{s: U\to \Disjoint_{p\in U} S\primelocalize{p} \st s(p) \in S\primelocalize{p}, s \text{ locally a fraction}}
,\]
recalling that $S\primelocalize{p} = \ts{a /f \st \deg a = \deg f, a,f\in S, f\not\in p}$.
We showed this was a locally ringed space using 
\[
( D(f), \ro{\OO_{\Proj S}}{D(f)} \iso (\spec S\localize{f}, \OO_{\spec S\localize{f}})
,\]
where $D(f) \da \ts{p\in \proj S \st f\not\in p}$, and thus $\Proj S \in \Sch$.
:::

:::{.exercise title="?"}
Check that there is a natural map of schemes $\Proj S\to \spec S_0$.
:::

:::{.remark}
Consider 
\[
\PP^n\slice R \da \Proj R[x_0,\cdots, x_n] && R = k = \bar{k} \in \Field
.\]
Then the closed points of $\PP^n\slice k$ are of the form $\gens{a_i x_j - a_j x_i} \in \mspec \kxn$ for points $[a_0: \cdots : a_n] \in k^n/\sim$ where $\vector a \sim \lambda \vector a$ for $\lambda \in k\units$.
Note that $D(x_i) = \ts{p\in \PP^n\slice k \st x_i \not\in p}$ -- what are the closed points? 
We discard the hyperplane $a_i = 0$ in $\PP^n$ to obtain

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-10-01_11-43.pdf_tex} };
\end{tikzpicture}

Then $x_i \in \mfm_q$ for $q \da [a_0: \cdots : a_n]$ iff $a_i=0$, and
\[
D(x_i) 
&= \spec \kxn\localize{x_i} \\
&= \ts{f(x_0, \cdots, x_n)/x_i^d \st \deg f = d} \\
&= \ts{f\qty{ {x_0\over x_i }, \cdots, 1, \cdots, {x_n \over x_i }}} \\
&= k\adjoin{{x_0\over x_i}, \cdots, {x_n \over x_i} } \\
&\cong \AA^n\slice k
.\]
We claim that $\Union_{i\geq 0} D(x_i) = \PP^n\slice k$, or equivalently $\emptyset = \Intersect_{i \geq 0} V(x_i) = V(\gens{x_0,\cdots, x_n})$.
But this is true since $\gens{x_0, \cdots, x_n} = S^+$ is the irrelevant ideal.
:::

:::{.proposition title="?"}
Let $k = \bar k \in \Field$.
Then there is a fully faithful embedding of categories
\[
F: \Var\slice k \embeds \Sch\slice k
.\]
Here $\Var\slice k$ are defined as topological spaces with sheaves of rings of *regular functions* which admitted an affine cover of the form $V(I) \subseteq \AA^n\slice k$ (viewed as a variety).
:::

:::{.example title="Going from a variety to a scheme"}
Consider $X\da \AA^2\slice k$ as a variety and separately as a scheme $X'$.
As a variety, $X \da k\cartpower{2}$ with the Zariski topology, while as a scheme $X' = \spec k[x, y]$ with the Zariski topology.
Then there is an inclusion $X \injects X'$ which is a bijection on closed points.

More generally, for $X\in \Top$ any space, define $t(X)$ to be the set of irreducible closed subsets.
Some facts:

- For $Y \subseteq X$ closed, $t(Y) \subseteq t(X)$,
- $t(Y_1 \union Y_2) = t(Y_1) \union t(Y_2)$,
- $t( \Intersect_i Y_i) = \Intersect_i t(Y_i)$.

Define a topology on $t(X)$ by declaring closed sets to be any of the form $t(Y)$ for $Y \subseteq X$ closed.
Note that the scheme $X'$ has non-closed points, i.e. irreducible subvarieties, i.e. irreducible closed subsets of $X$ as a variety:

\begin{tikzpicture}
\fontsize{35pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-10-01_12-02.pdf_tex} };
\end{tikzpicture}

Then consider the map
\[
\alpha: X &\to t(X) \\
p &\mapsto \ts{p}
,\]
noting that this is only well-defined if points are closed in $X$.

Now let $V\in \Var\slice k$ with its sheaf of regular functions $\OO_V$ (i.e. restrictions of polynomials).
Define a sheaf of rings on $t(V)$ as $\alpha_* \OO_V$, using that $\alpha$ is continuous, and noting that $\alpha\inv(U) = U \intersect \alpha(X)$.

To see this is a scheme, it suffices to check for $V$ affine since this entire construction is compatible with restriction and we can take an affine cover.
Letting $I = I(V)$ for $V\in \Aff\Var\slice k$, then $(t(V), \alpha_* \OO_V)\iso \spec k[V] \da \spec \kxn/I$.
There is a bijection
\[
t(V) &\mapstofrom \spec k[V] \\
Y &\mapsto I(Y) \\
V(p) &\mapsfrom p
.\]
One can check that the topology on $t(V)$ bijects with the Zariski topology on $\spec k[V]$, and
\[
\alpha_* \OO_V(t(V)) = \OO_V(V) = \OO_{\spec k[V]}(\spec k[V]) = k[V]
.\]
:::

:::{.exercise title="?"}
Check this on open subsets of $t(V)$.
:::

:::{.remark}
$\OO_X \in \Sh(X, \kalg)$ being a sheaf of $k\dash$algebras means the following diagram commutes:
\begin{tikzcd}
	k && {\OO_X(U)} \\
	\\
	&& {\OO_X(V)}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-3]
	\arrow["{\res_{UV}}", from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJrIl0sWzIsMCwiXFxPT19YKFUpIl0sWzIsMiwiXFxPT19YKFYpIl0sWzAsMV0sWzAsMl0sWzEsMiwiXFxyZXNfe1VWfSJdXQ==)

This is the data of a morphism $(X, \OO_X) \to \spec k$.
:::

:::{.remark}
What's the point of the theorem?
Not everything of geometric interest is in the essential image of $F$.
Consider $V(y-x^2) \subseteq \AA^2\slice k$, and consider intersecting it with lines $y=t$:

\begin{tikzpicture}
\fontsize{42pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-10-01_12-19.pdf_tex} };
\end{tikzpicture}

Letting $I_1 = I_1(\gens{y-x^2})$ and $I_2 = I(\gens{y-t})$,
intersecting in varieties yields
\[
V_1 \intersect V_2 = V(I_1 + I_2) = V(\sqrt{I_1 + I_2})
.\]
One can check $I_1 + I_2 = (x-\sqrt t, y-t)\cdot (x+\sqrt t, y-t)$, and $\spec k[x, y] / \gens{y-x^2, y} = \spec k[x]/\gens{x^2}$ when $t=0$ (i.e. when there's a tangency with multiplicity), since the scheme intersection is $\spec k[x, y] / \gens{I_1 + I_2}$.
Note that the regular functions on a point are just constant, so the sheaf of regular functions on a point is $k$ itself and thus doesn't pick up the multiplicity of the intersection.
:::

:::{.remark}
There can be issues for $\spec R$ when $R$ is finitely generated but not reduced!
:::