# Monday, September 13

## Affine Schemes

:::{.definition title="Restricted sheaves"}
Let $(X, \OO_X) \in \RingedSpace$ and $U \subseteq X$ be open, then for $V \subseteq U$ open, define the restricted sheaf $\OO_{X}\ro{}{V}(V) \da \OO_X(V)$.
:::

:::{.warnings}
\[
\Sh\slice X \ni\ro{\OO_X}{U}\neq \OO_X(U) \in \Ring
.\]
:::

:::{.remark}
Recall the definition of a ringed space $(X, \OO_X)$.
The quintessential example: $X$ a smooth manifold and $\OO_X \da C^{\infty}(\wait; \RR)$ the sheaf of smooth functions.
For defining morphisms, consider a map $f:X\to Y$, then an alternative way of defining $f$ to be smooth is that there is a pullback 
\[
f^*: C^0(V, \RR) &\to C^0(U, \RR) \\
g &\mapsto g \circ f
\]
for $U \subseteq X, V \subseteq Y$, and that $f^*$ in fact restricts to $f^*: C^\infty(V; \RR) \to C^\infty(U; \RR)$, i.e. preserving smooth functions.
:::

:::{.definition title="Morphisms of ringed spaces"}
A morphism of ringed spaces is a pair
\[
(M, \OO_M) \mapsvia{(\varphi, \varphi^\#)} (N, \OO_N)
.\]
where $\varphi \in C^0(M, N)$ and $\varphi^\# \in \Mor_{\Sh\slice N}(\OO_N, \varphi_* \OO_M)$.

This is an **isomorphism** of ringed spaces if

1. $\varphi$ is a homeomorphism, and
2. $\varphi^\#$ is an isomorphism of sheaves.
:::

:::{.remark}
In the running example, 
\[
\varphi^\#(U): \OO_N(U) \to \varphi_* \OO_M(M) = \OO_M(\varphi\inv(U))
.\]
This implies that maps of ringed spaced here induce smooth maps, and so there is an embedding $\smooth\Mfd\slice{\RR} \injects \Ringedspace$.
:::

:::{.remark}
We'll try to set up schemes the same way one sets up smooth manifolds.
A topological manifold is a space locally homeomorphic to $\RR^n$, and a smooth manifold is one in which it's locally isomorphic as a ringed space to $(\RR^n, C^\infty(\wait; \RR))$ with its sheaf of smooth functions.
:::

:::{.definition title="Smooth manifolds, alternative definition"}
A **smooth manifold** is a ringed space $(M, \OO_M)$ that is locally isomorphic to $(\RR^d, C^\infty(\wait; \RR))$, i.e. there is an open cover $\mcu \covers M$ such that 
\[
(U_i, \ro{\OO_M}{U_i}) \cong (\RR^n, C^{\infty}(\wait; \RR))
.\]
:::

:::{.example title="?"}
An example of a morphism of ringed spaces that is not an isomorphism:
take $(\RR, C^0) \to (\RR, C^\infty)$ given by $(\id, \id^\#)$ where $\id^\#: C^\infty \to \id_* C^0$ is given by $\id^\#(U): C^\infty(U) \to C^0(U)$ is the inclusion of continuous functions into smooth functions.
:::

:::{.remark}
We'll define schemes similarly: build from simpler pieces, namely an open cover with isomorphisms to affine schemes. 
A major difference is that there may not exist a *unique* isomorphism type among all of the local charts, i.e. the affine scheme can vary across the cover.
:::

:::{.remark}
Recall that for $A$ a ring we defined $(\spec A, \OO_{\spec A})$, where $\spec A \da \ts{\text{Prime ideals } \mfp \normal A}$, equipped with the Zariski topology generated by closed sets $V(I) \da \ts{\mfp \normal A \st I\contains \mfp}$.
We then defined 
\[
\OO_{\spec A}(U) \da \ts{s: U\to \Disjoint_{\mfp \in U} A\localize{\mfp}
\st
s(\mfp) \in A\localize{\mfp}, \, s\text{ locally a fraction}
}
.\]
We saw that

1. We can identify stalks: $\OO_{\spec A, \mfp} = A\localize{\mfp}$
2. We can identify sections on distinguished opens: 
\[
\OO_{\spec A}(D_f) = A\localize{f} = \ts{a/f^k \st a\in A, k\in \ZZ_{\geq 0}}
,\]
where $D_f \da V(f)^c = \ts{\mfp \in \spec A \st f\not\in \mfp}$.

As a corollary, we get $\OO_{\spec A}(\spec A) = A$, noting $\spec A = d_1$ and $A\localize{1} = A$.
Thus we can recover the ring $A$ from the ringed space $(X, \OO_X) \da (\spec A, \OO_{\spec A})$ by taking global sections, i.e. $\Globsec{\spec A; \OO_{\spec A}} = A$.
:::

## Affine Varieties

:::{.remark}
Let $k = \bar k$ and set $\AA^n\slice k = k^n$ whose regular functions are given by $\kxn$, regarded as maps to $k$.
:::

:::{.definition title="Affine variety"}
An affine variety is any set of the form 
\[
X \da V(f_1,\cdots, f_n) = \ts{p\in \AA^n\slice k \st f_1(p) = \cdots = f_m(p) = 0}
\]
for $f_i \in \kxn$,
:::

:::{.remark}
Writing $I = \gens{f_1,\cdots, f_m}$, we have $X = V(\sqrt I)$.
Letting $I(S) = \ts{f\in\kxn \st \ro{f}{S} = 0}$, then by the Nullstellensatz, $IV(I) = \sqrt{I}$.
This gives a bijection between affine varieties in $\AA^n\slice k$ and radical ideals $I \normal \kxn$.
:::

:::{.definition title="Coordinate rings of affine varieties"}
The **coordinate ring** of an affine variety $X$ is $k[X] \da \kxn/I(X)$, regarded as polynomial functions on $X$.
:::

:::{.remark}
We quotient here because if the difference of functions is in $I(X)$, these functions are equal when restricted to $X$.
For example, $y=x$ in $k[x, y]/ \gens{x-y}$, which are different functions where for $X\da \Delta$, we have $\ro{x}{\Delta} = \ro{y}{\Delta}$.
:::

:::{.remark}
As an application of the Nullstellensatz, there is a correspondence

\[
\adjunction{I(\wait)}{V(\wait)}{\ts{\text{Points } p\in X}}{\mspec k[X]}
\]
:::

:::{.remark}
Why is an affine variety $X$ an example of an affine scheme $\spec k[X]$?
These don't have the same underlying topological space:
\[
\tau(X) &\da \ts{V(I) \da\ts{p\in X \st f_i(p) = 0 \,\, \forall f_i \in I} \st I\normal k[X]} \\
\tau(\mspec k[X]) &\da \ts{ V(I) \da \ts{ \mfm \in \mspec k[X] \st \mfm \contains I} \st I\normal k[X] }
.\]

However, they are closely related: 
\[
\ro{\tau(\mspec k[X])}{\spec k[X]} = \tau(X_\zar)
,\]
i.e. the space $\spec k[X]$ with the restricted topology from $\mspec k[X]$ is homeomorphic to $X$ with the Zariski topology.
I.e. restricting to *closed points* recovers regular functions on $X$.

:::

:::{.warnings}
Defining things that are locally isomorphic to schemes would work for objects but not morphisms!
:::