# Friday, January 14

## Posets, Zariski Topologies

:::{.remark}
Recall the definition of a **poset**.
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:::{.example title="?"}
Given a polytope, one can take its face poset $\FP(P) = \ts{F \leq P}$ where $F_1 \leq F_2$ iff $F_1 \subseteq F_2$ for the faces $F_i$.
More generally, one can take a complex of polytopes, i.e. a collection of polytopes that only intersect at faces.
An example of a complex is the fan of a toric variety.

Similarly, one can take **cones** $\sum c_i \vector v_i \subseteq \RR^d$ for some positive coefficients.
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:::{.remark}
Conversely, given a poset $I$, one can associate a simplicial complex $\realize{I}$, the geometric realization.
Any chain $i_{n_1} < \cdots i_{n_k}$ is sent to a face and glued.
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:::{.example title="?"}
Consider a polytope $P$, taking the face poset $\FP(P)$, and its geometric realization $\realize{\FP(P)}$.
A square has

- $\size P_2 = 1$
- $\size P_1 = 4$
- $\size P_0 = 1$

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![](figures/2022-01-14_10-37-55.png)

Note that one can take the geometric realization of a category by using the nerve to first produce a poset.
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:::{.remark}
With the right choices, there exists a continuous map $\realize{I} \to I$ where $I$ is given the order topology.
Pulling back sheaves on the latter yields constructible sheaves on convex objects, which are locally constant on the interior components.
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:::{.remark}
A first version of the Zariski topology:
let $k = \bar{k} \in \Field$ and let $R\in\Alg^\fg\slice k$ be of the form $R = \kxn/\gens{f_a}$.
We can consider $X\da \mspec R \subseteq \AA^n\slice k$ as the points $\vector x\in k\cartpower{n}$ such that $f_a(\vector x) = 0$.
Recall Noether's theorem -- the $f_a$ can be replaced with a finite collection.
The closed subsets are of the form $V(g_b)$. 
Note that this is $T_1$ since points are closed: given $\vector p = \tv{p_0,\cdots, p_n}$, take $f(\vector p) = \prod_{i\leq n} (x-p_i)$ so that $V(f) = \ts{\vector p}$.
These points biject with maximal ideals in $R$.
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:::{.remark}
An improved version of the Zariski topology: $X = \spec R$, including prime ideals.
The points are as before, and additionally for every irreducible subvariety $Z \subseteq X$, there is a generic point $\eta_Z$.
This adds new points which can't be described in coordinates.
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:::{.remark}
Note that this generalizes to arbitrary (associative, commutative) rings.
For rings that aren't finitely generated, one loses the coordinate interpretation.
These generally won't embed into $\AA^n\slice k$ for any $n$, but can be embedded into (say) $\AA^1\slice R$.
Use that a closed embedding $X\embeds Y$ corresponds precisely to a surjection of associated rings $R_Y \surjects R_X$.
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## Sheaves

:::{.example title="?"}
Let $U \subseteq \Omega \subseteq \CC$ and consider $C^0(\Omega, \CC) \da \Hom_\Top(\Omega, \CC)$ -- this forms a sheaf of abelian groups, $\CC\dash$algebras, rings, sets, etc.

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![](figures/2022-01-14_11-04-15.png)

We'll refer to this as $\OO^\cts_X$.
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:::{.remark}
Some properties:

- For every $\iota: V \subseteq U \implies$ there is a restriction ma
\[
\mcf(\iota): \mcf(U) &\to \mcf(V) \\
f &\mapsto \ro{f}{V}
.\]

- $\mcf(\initial) = \terminal$, so e.g. for rings $\terminal = \ts{0}$ is the zero ring.

- (Sheaf 1, uniqueness): if $\mcu \covers U$ and $s_1, s_2 \in \mcf(\mcu)$, then $\ro{s_1}{U_i} = \ro{s_2}{U_i} \implies s_1 = s_2$.
- (Sheaf 2, existence): if $s_i\in \mcf(U_{ij})$ and $\ro{s_1}{U_{ij}} = \ro{s_2}{U_{ij}}$, then there is a global section $s\in \mcf(U_1 \union U_2)$.

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:::{.example title="?"}
Other examples of sheaves:

- $\OO^\cts_X$. One can check the sheaf properties directly.
- $\OO^\hol_X = \OO^\an_X$ the holomorphic (complex differentiable) and thus analytic (locally equal to a convergent power series) functions on $X$.
- Given a fixed continuous map $f: Y\to X$, setting $\mcf(U) = \ts{s: U\to Y}$ the set of continuous sections of $f$.

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