# Friday, October 15

:::{.remark}
Last time: open and closed subschemes, where openness was easy since we required $f:X\to Y$ to be a homeomorphism onto a closed subset of $Y$ with $f^\#$ surjective.
Any example of a closed subscheme is locally of the following form:
$V(I) \da \spec A/I \to \spec A$ induced by some $A\surjects A/I$ in rings.
Here $A\localize{g} \surjects (A/I)\localize{g}$ implies that $f^(D(g))$ surjective for every distinguished open, so $f^\#$ is a surjective sheaf map.
However, this need not be surjective on global sections.
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:::{.example title="?"}
Recall that $\PP^1\slice k = \AA^1 \Disjoint_{x\mapsto x\inv}\AA^1$ and $\OO_{\PP^1}(\PP^1) = k$ where we glued $k[t] \intersect k[s] = k$ along $s=1/t$.
Consider the closed subscheme of $\AA^1$ given by $X\da \spec \CC[t]/ t^2$ and the global restriction map
\[
f^\#(\PP^1): \OO_{\PP^1}(\PP^1) &\to \OO_X(X) \\
\CC &\to\CC[t]/t^2
,\]
which is not surjective.
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:::{.example title="?"}
Consider $\AA^2\slice k$ for $k=\kbar$, how many closed subschemes are homeomorphic onto the origin $0$ corresponding to $\gens{x, y}\in \spec k[x, y]$.
Since they're all locally of the form $V(I)$, these correspond to ideals $I$ where $V(I) = 0$.
These are ideals $I$ where $\gens{x, y}$ is the only ideal containing $I$, so we can write this as $\ts{I \st \sqrt I = \gens{x, y}}$, i.e. the primary decomposition of $I$ has only 1 prime, namely $\gens{x, y}$.
Some ideals of this form:

- $\gens{x, y}^k$ for any $k\geq 0$,
- $\gens{x^ay^b, x^cy^d}$ where $\det \matt a b c d \neq 0$, e.g. $\gens{x^2, y}$.
- $\gens{(x+y)^2, y}$
- $\gens{f, g}$ where $V(f) \intersect V(g) = 0$ as a set, e.g. two curves only intersecting at the origin.

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:::{.remark}
What kinds of schemes are these?
For example, considering $V(x-y^2)$ and $V(y)$, we have $\gens{y-x^2, y} = \gens{x^2, y}$, yielding a non-reduced scheme.
We have $k[x,y]/\gens{x^2, y}= k \bigoplus kx\in\mods{k}$, thought of as functions as the tangent vector at 0 pointing horizontally.
Similarly, $k[x,y] = \gens{x^2, xy, y^2} = k \oplus kx \oplus ky$, which can be thought of as functions on $\T_p \AA^2$ for $p=0$.
The rough idea: we want $\T_0 \AA^2 \cong \spec k[x,y]/\gens{x, y}^2$.
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:::{.definition title="Reduced subscheme structures"}
Let $Z \subseteq \realize{Y}$ be closed, then there exists a unique scheme structure $X$ on $Z$ such that $\realize{X} = Z$, the **reduced subscheme structure on $Z$**.
Affine locally, for $Z \subseteq \realize{\spec A}$ given by $V(I)$ for some ideal $I$, and we define this as $\spec A/\sqrt{I}$.
This will glue because passing to reduction will commute with localization, i.e. $(A_\red)\localize{f_\red} = (A\localize{f})_\red$ where $A_\red = A/\sqrt{0}$.
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:::{.example title="?"}
Take $0\in \abs{\AA^2\slice k}$, then its reduced subscheme structure is $\spec k[x, y] / \gens{x, y}$.
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:::{.remark}
Any closed subscheme structure along $Z$ is locally given by $\spec A/I$ with $V(I) = Z$, and there's always a map $\spec A/\sqrt{I} \to \spec A/I$ dual to the reduction map $A/I \to (A/I)_\red$.
For any closed subscheme $X \subseteq Y$, we define $X_\red$ as the reduced subscheme associated to $\abs{X}$, and there is a morphism $X_\red \to X$.

> Idea: this is a space such that all of its functions kill nilpotents.

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:::{.proposition title="?"}
$X_\red$ is well-defined
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:::{.proof title="?"}
Let $Y\in \Sch$ and $Z \subset \realize{Y}\in \Top$ closed.
Take a cover $\mcu \to Y$ with $U_i = \spec A_i$, then $Z \intersect \realize{U_i}$ is closed and thus equal to some $V(I_i)$.
Define a reduced scheme $X_i \da \spec (A_i / \sqrt{I_i})$, which we'll try to glue to define $X_\red$.
Note that we can write 
\[
\sqrt{I_i} = \Intersect _{\mfp\in Z \intersect \realize{U_i}} \mfp
,\]
which generalizes $\nilrad{R} = \Intersect_{\mfp\in \spec R} \mfp$.

To give a gluing amounts to defining isomorphisms:
\[
f_{ij}: X_i \intersect U_{j} \to X_j \intersect U_i
.\]

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So pass to an open affine cover.
We'll have $(A_i)\localize{f} = (A_j)\localize{g}$ for some $f, g$, and this will induce isomorphisms 
\[
(A_i)\localize{f} / \sqrt{I_i} \iso \OO_{X_j \intersect U_i}(Z \intersect D(f))
.\]

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