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\subsection{Sheaves}
\begin{definition}[Sheaf]
Let $X$ be any topological space.
A \vocab{presheaf} $\mathcal{G}$ of sets (or rings,
(abelian) groups) on $X$
associates a set (or rings, or (abelian) group) $\mathcal{G}(U)$ to
every open
subset $U$ of $X$, and a map (or ring or group homomorphism)
$\mathcal{G}(U)
\xrightarrow{r_{U,V}} \mathcal{G}(V)$ to every inclusion $V
\subseteq U$ of
open subsets of $X$ such that $r_{U,W} =
r_{V,W} r_{U,V}$ for inclusions
$U
\subseteq V \subseteq W$ of open subsets.
Elements of $\mathcal{G}(U)$ are often called \vocab{sections}
of
$\mathcal{G}$ on $U$ or \vocab{global sections} when $U = X$.
Let $U \subseteq X$ be open and $U = \bigcup_{i \in I} U_i$ an open
covering.
A family $(f_i)_{i \in I} \in
\prod_{i \in I} \mathcal{G}(U_i)$ is called
\vocab[Sections!
compatible]{compatible} if $r_{U_i, U_i \cap U_j}(f_i) =
r_{U_j, U_i \cap U_j}(f_j)$ for all $i,j \in I$.
Consider the map
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\begin{align}
\phi_{U, (U_i)_{i \in I}}: \mathcal{G}(U)
& \longrightarrow \{(f_i)_{i \in I} \in \prod_{i \in I} \mathcal{G}(U_i) |
r_{U_i, U_i \cap U_j}(f_i) = r_{U_j, U_i \cap U_j}(f_j) \text{ for } i,j \in I
\} \\ f & \longmapsto (r_{U, U_i}( f))_{i \in I}
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\end{align}
A presheaf is called \vocab[Presheaf!
separated]{separated} if $\phi_{U, (U_i)_{i \in I}}$ is
injective for all such $U$ and
$(U_i)_{i \in I}$.\footnote{This also called ``locality''.}
It satisfies \vocab{gluing} if $\phi_{U, (U_i)_{i \in I}}$ is
surjective.
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A presheaf is called a \vocab{sheaf} if it is separated and
satisfies gluing.
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The bijectivity of the $\phi_{U, (U_i)_{i \in I}}$ is called the
\vocab{sheaf
axiom}.
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\end{definition}
\begin{trivial}
+
A presheaf is a contravariant functor $\mathcal{G} :
\mathcal{O}(X) \to C$ where $\mathcal{O}(X)$ denotes the category
of open subsets of $X$ with inclusions as morphisms and $C$ is the category of
sets, rings or (abelian) groups.
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\end{trivial}
\begin{definition}
A subsheaf $\mathcal{G}'$ is defined by subsets (resp.
subrings or subgroups) $\mathcal{G}'(U) \subseteq
\mathcal{G}(U)$ for all open $U \subseteq X$ such that the sheaf axioms
still hold.
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\end{definition}
\begin{remark}
If $\mathcal{G}$ is a sheaf on $X$ and $\Omega \subseteq X$ open,
then
$\mathcal{G}\defon{\Omega}(U) \coloneqq \mathcal{G}(U)$
for open $U \subseteq
\Omega$ and $r_{U,V}^{(\mathcal{G}\defon{\Omega})}(f) \coloneqq
r_{U,V}^{(\mathcal{G})}(f)$ is a sheaf of the same kind as
$\mathcal{G}$ on
$\Omega$.
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\end{remark}
\begin{remark}
The notion of restriction of a sheaf to a closed subset, or of preimages under
general continuous maps, can be defined but this is a bit harder.
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\end{remark}
\begin{notation}
It is often convenient to write $f \defon{V}$ instead of
$r_{U,V}(f)$.
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\end{notation}
\begin{remark}
Applying the \vocab{sheaf axiom} to the empty covering of $U = \emptyset$,
one
finds that $\mathcal{G}(\emptyset) = \{0\} $.
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\end{remark}
\subsubsection{Examples of sheaves}
\begin{example}
Let $G$ be a set and let $\mathfrak{G}(U)$ be the set of arbitrary maps
$U
\xrightarrow{f} G$.
We put $r_{U,V}(f) = f\defon{V}$.
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It is easy to see that this defines a sheaf.
If $\cdot $ is a group operation on $G$, then $(f\cdot g)(x) \coloneqq
f(x)\cdot g(x)$ defines the structure of a sheaf of group on
$\mathfrak{G}$.
Similarly, a ring structure on $G$ can be used to define the structure of a
sheaf of rings on $\mathfrak{G}$.
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\end{example}
\begin{example}
If in the previous example $G$ carries a topology and $\mathcal{G}(U)
\subseteq
\mathfrak{G}(U)$ is the subset (subring, subgroup) of continuous
functions $U
\xrightarrow{f} G$, then $\mathcal{G}$ is a subsheaf of
$\mathfrak{G}$, called
the sheaf of continuous $G$-valued functions on (open subsets of) $X$.
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\end{example}
\begin{example}
If $X = \R^n$, $\mathbb{K} \in \{\R, \C\}$ and
$\mathcal{O}(U)$ is the sheaf
of $\mathbb{K}$-valued $C^{\infty}$-functions on
$U$, then $\mathcal{O}$ is a
subsheaf of the sheaf (of rings) of $\mathbb{K}$-valued continuous
functions on
$X$.
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\end{example}
\begin{example}
If $X = \C^n$ and $\mathcal{O}(U)$ the set of holomorphic functions
on $X$,
then $\mathcal{O}$ is a subsheaf of the sheaf of $\C$-valued
$C^{\infty}$-functions on $X$.
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\end{example}
\subsubsection{The structure sheaf on a closed subset of $\mathfrak{k}^n$}
Let $X \subseteq \mathfrak{k}^n$ be open.
Let $R = \mathfrak{k}[X_1,\ldots,X_n]$.
\begin{definition}
\label{structuresheafkn}
For open subsets $U \subseteq X$, let $\mathcal{O}_X(U)$ be the set
of
functions $U \xrightarrow{\phi} \mathfrak{k}$ such that every $x
\in U$ has a
neighbourhood $V$ such that there are $f,g \in R$ such that for $y \in V$ we
have $g(y) \neq 0$ and $\phi(y) = \frac{f(y)}{g(y)}$.
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\end{definition}
\begin{remark}
\label{structuresheafcontinuous}
$\mathcal{O}_X$ is a subsheaf (of rings) of the sheaf of
$\mathfrak{k}$-valued functions on $X$.
The elements of $\mathcal{O}_X(U)$ are continuous: Let $M \subseteq
\mathfrak{k}$ be closed.
We must show the closedness of $N \coloneqq \phi^{-1}(M)$ in $U$.
For $M = \mathfrak{k}$ this is trivial.
Otherwise $M$ is finite and we may assume $M = \{t\} $ for some $t \in
\mathfrak{k}$.
For $x \in U$, there are open $V_x \subseteq U$ and $f_x, g_x \in R$ such that
$\phi = \frac{f_x}{g_x}$ on $V_x$.
Then $N \cap V_x = V(f_x - t\cdot g_x) \cap V_x)$ is closed in $V_x$.
As the $V_x$ cover $U$ and $U$ is quasi-compact, $N$ is closed in $U$.
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\end{remark}
\begin{proposition}
\label{structuresheafri}
Let $X = V(I)$ where $I = \sqrt{I} \subseteq R$ is an ideal.
Let $A = R / I$.
Then
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\begin{align}
\phi: A & \longrightarrow \mathcal{O}_X(X) \\ f \mod I
& \longmapsto f\defon{X}
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\end{align}
is an isomorphism.
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\end{proposition}
\begin{proof}
It is easy to see that the map $A \to \mathcal{O}_X(X)$ is
well-defined and a
ring homomorphism.
Its injectivity follows from the Nullstellensatz and $I =
\sqrt{I}$
(
\ref{hns3}).
Let $\phi \in \mathcal{O}_X(X)$.
for $x \in X$, there are an open subset $U_x \subseteq X$ and $f_x, g_x \in R$
such that $\phi = \frac{f_x}{g_x}$ on $U_x$.
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\begin{claim}
Without loss of generality loss of generality we can assume $U_x = X \setminus
V(g_x)$.
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\end{claim}
\begin{subproof}
The closed subsets $(X \setminus U_x) \subseteq \mathfrak{k}^n$ has
the form
$X\setminus U_x = V(J_x)$ for some ideal $J_x \subseteq R$.
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As $x \not\in X \setminus V_x$ there is $h_x \in J_x$ with $h_x(x) \neq 0$.
Replacing $U_x$ by $X \setminus V(h_x)$, $f_x$ by $f_xh_x$ and $g_x$ by
$g_xh_x$, we may assume $U_x = X \setminus V(g_x)$.
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\end{subproof}
\begin{claim}
Without loss of generality loss of generality we can assume $V(g_x) \subseteq
V(f_x)$.
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\end{claim}
\begin{subproof}
Replace $f_x$ by $f_xg_x$ and $g_x$ by $g_x^2$.
\end{subproof}
As $X$ is quasi-compact, there are finitely many points
$(x_i)_{i=1}^m$ such
that the $U_{x_i}$ cover $X$.
Let $U_i \coloneqq U_{x_i}, f_i \coloneqq
f_{x_i}, g_i \coloneqq g_{x_i}$.
As the $U_i = X \setminus V(g_i)$ cover $X$, $V(I) \cap
\bigcap_{i=1}^m V(g_i)
= X \cap \bigcap_{i=1}^m V(g_i) = \emptyset$.
By the Nullstellensatz (
\ref{hns1}) the ideal of $R$ generated by
$I$ and the
$a_i$ equals $R$.
There are thus $n \ge m \in \N$ and elements
$(g_i)_{i = m+1}^n$ of $I$ and
$(a_i)_{i=1}^n \in R^n$ such that $1 =
\sum_{i=1}^{n} a_ig_i$.
Let for $i > m$ $f_i \coloneqq 0$, $F = \sum_{i=1}^{n} a_if_i =
\sum_{i=1}^{m}
a_if_i \in R$.
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\begin{claim}
For all $x \in X $ ~ $f_i(x) = \phi(x) g_i(x)$.
\end{claim}
\begin{subproof}
If $x \in V_i$ this follows by our choice of $f_i$ and $g_i$.
If $x \in X \setminus V_i$ or $i > m$ both sides are zero.
\end{subproof}
It follows that
\[
\phi(x) = \phi(x) \cdot 1 =
\phi(x) \cdot \sum_{i=1}^{n}
a_i(x) g_i(x) = \sum_{i=1}^{n} a_i(x) f_i(x) = F(x)
\]
Hence $\phi =
F\defon{X}$.
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\end{proof}
\subsubsection{The structure sheaf on closed subsets of $\mathbb{P}^n$}
Let $X \subseteq \mathbb{P}^n$ be closed and $R_\bullet =
\mathfrak{k}[X_0,\ldots,X_n]$ with its usual grading.
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\begin{definition}
\label{structuresheafpn}
For open $U \subseteq X$, let $\mathcal{O}_X(U)$ be the set of
functions $U
\xrightarrow{\phi} \mathfrak{k}$ such that for every $x \in U$,
there are an
open subset $W \subseteq U$, a natural number $d$ and $f,g \in R_d$ such that
$W \cap \Vp(g) = \emptyset$ and $\phi(y) =
\frac{f(y_0,\ldots,y_n)}{g(y_0,\ldots,y_n)}$ for $y = [y_0,\ldots,y_n] \in W$.
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\end{definition}
\begin{remark}
This is a subsheaf of rings of the sheaf of $\mathfrak{k}$-valued
functions on
$X$.
Under the identification $\mathbb{A}^n =\mathfrak{k}^n$
with $\mathbb{P}^n
\setminus \Vp(X_0)$, one has $\mathcal{O}_X
\defon{X \setminus \Vp(X_0)} =
\mathcal{O}_{X \cap \mathbb{A}^n}$ as subsheaves of the
sheaf of
$\mathfrak{k}$-valued functions, where the second sheaf is a sheaf on
a closed
subset of $\mathfrak{k}^n$:
Indeed, if $W$ is as in the definition then $\phi([1,y_1,\ldots,y_n]) =
\frac{f(1,y_1,\ldots,y_n)}{g(1,y_1,\ldots,y_n)}$ for $[1,y_1,\ldots,y_n] \in
W$.
Conversely if $\phi([1,y_1,\ldots,y_n]) =
\frac{f(y_1,\ldots,y_n)}{g(y_1,\ldots,y_n)}$ on an open subset $W $ of $X \cap
\mathbb{A}^n$ then $\phi([y_0,\ldots,y_n]) =
\frac{F(y_0,\ldots,y_n)}{G(y_0,\ldots,y_n)}$ on $W$ where $F(X_0,\ldots,X_n)
\coloneqq X_0^d f(\frac{X_1}{X_0}, \ldots, \frac{X_n}{X_0})$ and
$G(X_0,\ldots,X_n) = X_0^d g(\frac{X_1}{X_0},\ldots,
\frac{X_n}{X_0})$ with a
sufficiently large $d \in \N$.
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\end{remark}
\begin{remark}
It follows from the previous remark and the similar result in the affine case
that the elements of $\mathcal{O}_X(U)$ are continuous on $U
\setminus V(X_0)$.
Since the situation is symmetric in the homogeneous coordinates, they are
continuous on all of $U$.
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\end{remark}
The following is somewhat harder than in the affine case:
\begin{proposition}
If $X$ is connected (e.g. irreducible), then the elements of
$\mathcal{O}_X\left( X \right) $ are constant functions on
$X$.
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\end{proposition}
% Lecture 14
\subsection{The notion of a category}
\begin{definition}
A \vocab{category} $\mathcal{A}$ consists of:
\begin{itemize}
\item
A class
$\Ob \mathcal{A}$ of \vocab[Objects]{objects of $\mathcal{A}$}.
\item
For two arbitrary objects $A, B \in \Ob \mathcal{A}$, a
\textbf{set} $\Hom_\mathcal{A}(A,B)$ of
\vocab[Morphism]{morphisms for $A$ to $B$ in $\mathcal{A}$}.
\item
A map $\Hom_\mathcal{A}(B,C) \times \Hom_\mathcal{A}(A,B)
\xrightarrow{\circ} \Hom_\mathcal{A}(A,C)$, the composition of
morphisms, for arbitrary triples $(A,B,C)$ of objects of
$\mathcal{A}$.
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\end{itemize}
The following conditions must be satisfied:
\begin{enumerate}[A]
\item
For
morphisms $A \xrightarrow{f} B\xrightarrow{g} C
\xrightarrow{h} D$, we have $h
\circ (g \circ f) = (h \circ g) \circ f$.
\item
For every $A \in \Ob(\mathcal{A})$, there is an $\Id_A \in
\Hom_{\mathcal{A}}(A,A)$ such that $\Id_A \circ f = f$ (reps. $g \circ \Id_A =
g$) for arbitrary morphisms $B \xrightarrow{f}
A$ (reps.
$A \xrightarrow{g}
C).
$
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\end{enumerate}
A morphism $X \xrightarrow{f} Y$ is called an \vocab[Isomorphism]{isomorphism
(in $\mathcal{A} $)}
if there is a morphism $Y \xrightarrow{g} X$ (called the
\vocab[Inverse morphism]{inverse $f^{-1}$ of $f$)} such that $g \circ f =
\Id_X$ and $f \circ g = \Id_Y$.
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\end{definition}
\begin{remark}
\begin{itemize}
\item
The distinction between classes and sets is important here.
\item
We will usually omit the composition sign $\circ$.
\item
It is easy to see that $\Id_A$ is uniquely determined by the above condition
$B$, and that the inverse $f^{-1}$ of an isomorphism
$f$ is uniquely determined.
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\end{itemize}
\end{remark}
\subsubsection{Examples of categories}
\begin{example}
\begin{itemize}
\item
The category of sets.
\item
The category of groups.
\item
The category of rings.
\item
If $R$ is a ring, the category of $R$-modules and the category $\Alg_R$ of
$R$-algebras
\item
The category of topological spaces
\item
The category $\Var_\mathfrak{k}$ of varieties over
$\mathfrak{k}$ (see
\ref{defvariety})
\item
If $\mathcal{A}$ is a category, then the \vocab{opposite category}
or \vocab{dual category} is defined by $\Ob(\mathcal{A}\op) =
\Ob(\mathcal{A})$ and $\Hom_{\mathcal{A}\op}(X,Y) =
\Hom_\mathcal{A}(Y,X)$.
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\end{itemize}
In most of these cases, isomorphisms in the category were just called
`isomorphism'.
The isomorphisms in the category of topological spaces are the homeomophisms.
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\end{example}
\subsubsection{Subcategories}
\begin{definition}[Subcategories]
A \vocab{subcategory} of
$\mathcal{A}$ is a category $\mathcal{B}$ such that
$\Ob(\mathcal{B}) \subseteq \Ob(\mathcal{A})$, such that
$\Hom_\mathcal{B}(X,Y) \subseteq \Hom_\mathcal{A}(X,Y)$ for
objects $X$ and $Y$ of $\mathcal{B}$, such that for every object $X
\in \Ob(\mathcal{B})$, the identity $\Id_X$ of $X$ is the same in
$\mathcal{B}$ as in $\mathcal{A}$, and such that for
composable morphisms in $\mathcal{B}$, their compositions in
$\mathcal{A}$ and $\mathcal{B}$ coincide.
We call $\mathcal{B}$ a \vocab{full subcategory} of
$\mathcal{A}$ if in
addition $\Hom_\mathcal{B}(X,Y) = \Hom_\mathcal{A}(X,Y)$ for
arbitrary $X,Y \in
\Ob(\mathcal{B})$.
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\end{definition}
\begin{example}
\begin{itemize}
\item
The category of abelian groups is a full subcategory of the category of groups.
It can be identified with the category of $\Z$-modules.
\item
The category of finitely generated $R$-modules as a full subcategory of the
category of $R$-modules.
\item
The category of $R$-algebras of finite type as a full subcategory of $\Alg_R$.
\item
The category of affine varieties over $\mathfrak{k}$ as a full
subcategory of the category of varieties over $\mathfrak{k}$.
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\end{itemize}
\end{example}
\subsubsection{Functors and equivalences of categories}
\begin{definition}
A \vocab[Functor!
covariant]{(covariant) functor} (resp. \vocab[Functor!contravariant]{contravariant functor}) between categories $\mathcal{A}
\xrightarrow{F} \mathcal{B}$ is a map
$\Ob(\mathcal{A}) \xrightarrow{F} \Ob(\mathcal{B})$ with
a family of maps $\Hom_\mathcal{A}(X,Y) \xrightarrow{F}
\Hom_\mathcal{B}(F(X),F(Y))$ (resp. $\Hom_\mathcal{A}(X,Y)
\xrightarrow{F} \Hom_\mathcal{B}(F(Y),F(X))$ in the case of
contravariant functors), where $X$ and $Y$ are arbitrary objects of
$\mathcal{A}$, such that the following conditions hold:
\begin{itemize}
\item
$F(\Id_X) = \Id_{F(X)}$
\item
For morphisms $X \xrightarrow{f}
Y \xrightarrow{g} Z$ in $\mathcal{A}$, we have $F(gf) =
F(g)F(f)$ ( resp.
$F(gf) = F(f)
F(g)$)
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\end{itemize}
A functor is called \vocab[Functor!
essentially surjective]{essentially surjective} if every object of
$\mathcal{B}$ is isomorphic to an element of the image of
$\Ob(\mathcal{A}) \xrightarrow{F} \Ob(\mathcal{B})$.
A functor is called \vocab[Functor!
full]{full} (resp. \vocab[Functor!faithful]{faithful}) if it induces surjective (resp.
injective) maps between sets of morphisms.
It is called an \vocab{equivalence of categories} if it is full, faithful and
essentially surjective.
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\end{definition}
\begin{example}
\begin{itemize}
\item
There are \vocab[Functor!forgetful]{forgetful functors} from rings to abelian groups or from abelian
groups to sets which drop the multiplicative structure of a ring or the group
structure of a group.
\item
If $\mathfrak{k}$ is any vector space there is a contravariant
functor from $\mathfrak{k}$-vector spaces to itself sending $V$ to
its dual vector space $V\subseteq$ and $V \xrightarrow{f}
W$ to the dual linear map $W^{\ast}
\xrightarrow{f^{\ast}} V^{\ast}$.
When restricted to the full subcategory of finite-dimensional vector spaces it
becomes a contravariant self-equivalence of that category.
\item
The embedding of a subcategory is a faithful functor.
In the case of a full subcategory it is also full.
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\end{itemize}
\end{example}
\subsection{The category of varieties}
\begin{definition}[Algebraic variety]
\label{defvariety}
An \vocab{algebraic variety} or \vocab{prevariety} over
$\mathfrak{k}$ is a
pair $(X, \mathcal{O}_X)$, where $X$ is a topological space and
$\mathcal{O}_X$
a subsheaf of the sheaf of $\mathfrak{k}$-valued functions on $X$
such that
for every $x \in X$, there are a neighbourhood $U_x$ of $x$ in $X$, an open
subset $V_x$ of a closed subset $Y_x$ of
$\mathfrak{k}^{n_x}$\footnote{By the
result of
\ref{affopensubtopbase} it can be assumed that $V_x = Y_x$ without
altering the definition.
} and a homeomorphism $V_x
\xrightarrow{\iota_x}
U_x$ such that for every open subset $V \subseteq U_x$ and every function
$V\xrightarrow{f} \mathfrak{k}$, we have $f \in
\mathcal{O}_X(V) \iff
\iota^{\ast}_x(f) \in
\mathcal{O}_{Y_x}(\iota_x^{-1}(V))$,
In this, the \vocab{pull-back} $\iota_x^{\ast}(f)$ of $f$ is
defined by
$(\iota_x^{\ast}(f))(\xi) \coloneqq f(\iota_x(\xi))$.
A morphism $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ of
varieties is a
continuous map $X \xrightarrow{\phi} Y$ such that for all open $U
\subseteq Y$
and $f \in \mathcal{O}_Y(U)$, $\phi^{\ast}(f) \in
\mathcal{O}_X(\phi^{-1}(U))$.
An isomorphism is a morphism such that $\phi$ is bijective and
$\phi^{-1}$ also
is a morphism of varieties.
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\end{definition}
\begin{example}
\begin{itemize}
\item
If $(X, \mathcal{O}_X)$ is a variety and $U \subseteq X$ open, then
$(U, \mathcal{O}_X\defon{U})$ is a variety (called an
\vocab{open subvariety} of $X$), and the embedding $U \to X$ is a morphism of
varieties.
\item
If $X$ is a closed subset of $\mathfrak{k}^n$ or
$\mathbb{P}^n$, then $(X, \mathcal{O}_X)$ is a
variety, where $\mathcal{O}_X$ is the structure sheaf on $X$
(
\ref{structuresheafkn}, reps.
\ref{structuresheafpn}).
A variety is called \vocab[Variety!
affine]{affine} (resp. \vocab[Variety!projective]{projective}) if it is isomorphic to a variety of
this form, with $X $ closed in $\mathfrak{k}^n$ (resp.
$\mathbb{P}^n$).
A variety which is isomorphic to and open subvariety of $X$ is called
\vocab[Variety!
quasi-affine]{quasi-affine} (resp. \vocab[Variety!quasi-projective]{quasi-projective}).
\item
If $X = V(X^2 - Y^3) \subseteq \mathfrak{k}^2$ then
$\mathfrak{k} \xrightarrow{t \mapsto (t^3,t^2)}
X$ is a morphism which is a homeomorphism of topological spaces but not an
isomorphism of varieties.
% TODO
\item
The composition of two morphisms $X \to Y \to Z$ of varieties is a morphism of
varieties.
\item
$X\xrightarrow{\Id_X}
X$ is a morphism of varieties.
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\end{itemize}
\end{example}
\subsubsection{The category of affine varieties}
\begin{lemma}
\label{localinverse}
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Let $X$ be any $\mathfrak{k}$-variety and $U \subseteq X$ open.
\begin{enumerate}[i)]
\item
All elements of $\mathcal{O}_X(U)$ are continuous.
\item
If $U \subseteq X$ is open, $U \xrightarrow{\lambda} \mathfrak{k}$
any function and every $x \in U$ has a neighbourhood $V_x \subseteq U$ such
that $\lambda \defon{V_x} \in \mathcal{O}_X(V_x)$,
then $\lambda \in \mathcal{O}_X(U)$.
\item
If $\vartheta \in \mathcal{O}_X(U)$ and $\vartheta(x)
\neq 0$ for all $x \in U$, then $\vartheta \in
\mathcal{O}_X(U)^{\times }$.
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\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[i)]
\item
The property is local on $U$, hence it is sufficient to show it in the
quasi-affine case.
This was done in
\ref{structuresheafcontinuous}.
\item
For the second part, let $\lambda_x \coloneqq \lambda \defon{V_x}
$.
We have $\lambda_x\defon{V_x \cap V_y} = \lambda \defon{V_x \cap V_y} =
\lambda_y \defon{V_x \cap V_y} $.
The $V_x$ cover $U$.
By the sheaf axiom for $\mathcal{O}_X$ there is $\ell \in
\mathcal{O}_X(U)$
with $\ell\defon{V_x} =\lambda_x$.
It follows that $\ell=\lambda$.
\item
By the definition of variety, every $x \in U$ has a quasi-affine neighbourhood
$V \subseteq U$.
We can assume $U$ to be quasi-affine and $X = V(I) \subseteq
\mathfrak{k}^n$,
as the general assertion follows by an application of ii).
If $x \in U$ there are a neighbourhood $x \in W \subseteq U$ and $a,b \in R =
\mathfrak{k}[X_1,\ldots,X_n]$ such that $\vartheta(y) =
\frac{a(y)}{b(y)}$ for
$y \in W$, with $b(y) \neq 0$.
Then $a(x) \neq 0$ as $\vartheta(x) \neq 0$.
Replacing $W$ by $W \setminus V(a)$, we may assume that $a$ has no zeroes on
$W$.
Then $\lambda(y) = \frac{b(y)}{a(y)}$ for $y \in W$ has a
non-vanishing
denominator and $\lambda \in \mathcal{O}_X(U)$.
We have $\lambda \cdot \vartheta = 1$, thus $\vartheta \in
\mathcal{O}_X(U)^{\times}$.
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\end{enumerate}
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\end{proof}
\begin{proposition}[About affine varieties]
\label{propaffvar}
\begin{itemize}
\item
Let $X,Y$ be varieties over $\mathfrak{k}$.
Then the map
\begin{align}
\phi: \Hom_{\Var_\mathfrak{k}}(X,Y) & \longrightarrow
\Hom_{\Alg_\mathfrak{k}}(\mathcal{O}_Y(Y), \mathcal{O}_X(X)) \\ (X
\xrightarrow{f} Y) & \longmapsto (\mathcal{O}_Y(Y) \xrightarrow{f^{\ast}}
\mathcal{O}_X(X))
\end{align}
is injective when $Y$ is quasi-affine and
bijective when $Y$ is affine.
\item
The contravariant functor
\begin{align}
F: \Var_\mathfrak{k} & \longrightarrow \Alg_\mathfrak{k} \\ X & \longmapsto
\mathcal{O}_X(X) \\ (X\xrightarrow{f} Y) & \longmapsto (\mathcal{O}_X(X)
\xrightarrow{f^{\ast}} \mathcal{O}_Y(Y))
\end{align}
restricts to an
equivalence of categories between the category of affine varieties over
$\mathfrak{k}$ and the full subcategory $\mathcal{A}$ of
$\Alg_\mathfrak{k}$,
having the $\mathfrak{k}$-algebras $A$ of finite type with $\nil A =
\{0\} $ as
objects.
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\end{itemize}
\end{proposition}
\begin{remark}
It is clear that $\nil(\mathcal{O}_X(X)) = \{0\}$ for arbitrary
varieties.
For general varieties it is however not true that
$\mathcal{O}_X(X)$ is a
$\mathfrak{k}$-algebra of finite type.
There are counterexamples even for quasi-affine $X$.
%TODO
If, however, $X$ is affine, we may assume w.l.o.g.
that $X = V(I)$ where $I = \sqrt{I} \subseteq R$ is an ideal
with $R = \mathfrak{k}[X_1,\ldots,X_n]$.
Then $\mathcal{O}_X(X) \cong R / I$ (see
\ref{structuresheafri})
is a
$\mathfrak{k}$-algebra of finite type.
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\end{remark}
\begin{proof}
It suffices to investigate $\phi$ when $Y$ is an open subset of $V(I) \subseteq
\mathfrak{k}^n$, where $I = \sqrt{I} \subseteq R$ is
an ideal and $Y = V(I)$
when $Y$ is affine.
Let $(f_1,\ldots,f_n)$ be the components of $X \xrightarrow{f} Y
\subseteq
\mathfrak{k}^n$.
Let $Y \xrightarrow{\xi_i} \mathfrak{k}$ be the $i$-th
coordinate.
By definition $f_i = f^{\ast}(\xi_i) $.
Thus $f$ is uniquely determined by $\mathcal{O}_Y(Y)
\xrightarrow{f^{\ast}}
\mathcal{O}_X(X)$.
Conversely, let $Y = V(I)$ and $\mathcal{O}_Y(Y)
\xrightarrow{\phi}
\mathcal{O}_X(X)$ be a morphism of
$\mathfrak{k}$-algebras.
Define $f_i \coloneqq \phi(\xi_i)$ and consider $X
\xrightarrow{f =
(f_1,\ldots,f_n)} Y\subseteq \mathfrak{k}^n$.
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\begin{claim}
$f$ has image contained in $Y$.
\end{claim}
\begin{subproof}
For $x \in X, \lambda \in I$ we have $\lambda(f(x)) =
(\phi(\lambda \mod I))(x)
= 0$ as $\phi$ is a morphism of $\mathfrak{k}$-algebras.
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Thus $f(x) \in V(I) = Y$.
\end{subproof}
\begin{claim}
$f$ is a morphism in $\Var_\mathfrak{k}$
\end{claim}
\begin{subproof}
For open $\Omega \subseteq Y, U = f^{-1}(\Omega) = \{x
\in X | \forall \lambda
\in J ~ (\phi(\lambda))(x) \neq 0\}$ is open in $X$, where $Y
\setminus \Omega
= V(J)$.
If $\lambda \in \mathcal{O}_Y(\Omega)$ and $x \in U$, then $f(x)$
has a
neighbourhood $V$ such that there are $a,b \in R$ with $\lambda(v)
=
\frac{a(v)}{b(v)}$ and $b(v) \neq 0$ for all $v \in V$.
Let $W \coloneqq f^{-1}(V)$.
Then $\alpha \coloneqq \phi(a)\defon{W} \in
\mathcal{O}_X(W)$, $\beta
\coloneqq \phi(b)\defon{W} \in
\mathcal{O}_X(W)$.
By the second part of
\ref{localinverse} $\beta \in
\mathcal{O}_X(W)^{\times}$
and $f^{\ast}(\lambda)\defon{W} =
\frac{\alpha}{\beta} \in \mathcal{O}_X(W)$.
The first part of
\ref{localinverse} shows that
$f^{\ast}(\lambda) \in
\mathcal{O}_X(U)$.
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\end{subproof}
By definition of $f$, we have $f^{\ast} = \phi$.
This finished the proof of the first point.
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\begin{claim}
The functor in the second part maps affine varieties to objects of
$\mathcal{A}$ and is essentially surjective.
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\end{claim}
\begin{subproof}
It follows from the remark that the functor maps affine varieties to objects of
$\mathcal{A}$.
If $A \in \Ob(\mathcal{A})$ then $ A /\mathfrak{k}$ is of
finite type, thus $A
\cong R / I$ for some $n$.
Since $\nil(A) = \{0\}$ we have $I = \sqrt{I}$,
as for $x \in \sqrt{I}$, $x
\mod I \in \nil(R / I) \cong \nil(A) = \{0\}$.
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Thus $A \cong\mathcal{O}_X(X)$ where $X = V(I)$.
\end{subproof}
Fullness and faithfulness of the functor follow from the first point.
\end{proof}
\begin{remark}
Note that giving a contravariant functor $\mathcal{C} \to
\mathcal{D}$ is
equivalent to giving a functor $\mathcal{C} \to
\mathcal{D}\op$.
We have thus shown that the category of affine varieties is equivalent to
$\mathcal{A}\op$, where $\mathcal{A} \subsetneq
\Alg_\mathfrak{k}$ is the full
subcategory of $\mathfrak{k}$-algebras $A$ of finite type with
$\nil(A) =
\{0\}$.
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\end{remark}
\subsubsection{Affine open subsets are a topology base}
\begin{definition}
A set $\mathcal{B}$ of open subsets of a topological space $X$ is
called a
\vocab{topology base} for $X$ if every open subset of $X$ can be written as a
(possibly empty) union of elements of $\mathcal{B}$.
\end{definition}
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\begin{fact}
If $X$ is a set, then $\mathcal{B} \subseteq
\mathcal{P}(X)$ is a base for some
topology on $X$ iff $X = \bigcup_{U \in \mathcal{B}} U$ and for arbitrary $U, V
\in \mathcal{B}, U \cap V$ is a union of elements of
$\mathcal{B}$.
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\end{fact}
\begin{definition}
Let $X$ be a variety.
An \vocab{affine open subset} of $X$ is a subset which is an affine variety.
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\end{definition}
\begin{proposition}
\label{oxulocaf}
Let $X$ be an affine variety over $\mathfrak{k}$, $\lambda \in
\mathcal{O}_X(X)$ and $U = X \setminus V(\lambda)$.
Then $U$ is an affine variety and the morphism $\phi:
\mathcal{O}_X(X)_\lambda
\to \mathcal{O}_X(U)$ defined by the restriction
$\mathcal{O}_X(X)
\xrightarrow{\cdot |_U } \mathcal{O}_X(U)$ and the universal
property of the
localization is an isomorphism.
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\end{proposition}
\begin{proof}
Let $X$ be an affine variety over $\mathfrak{k}, \lambda \in
\mathcal{O}_X(X)$
and $U = X \setminus V(\lambda)$.
The fact that $\lambda\defon{U} \in
\mathcal{O}_x(U)^{\times}$ follows
from
\ref{localinverse}.
Thus the universal property of the localization
$\mathcal{O}_X(X)_\lambda$ can
be applied to $\mathcal{O}_X(X) \xrightarrow{\cdot |_U}
\mathcal{O}_X(U)$.
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\[
\begin{tikzcd}
\mathcal{O}_X(X) \arrow{d}{\cdot |_U}\arrow{r}{x \mapsto \frac{x}{1}} & \mathcal{O}_X(X)_\lambda \arrow[dotted, bend left]{dl}{\existsone \phi} \\
\mathcal{O}_X(U) &
\end{tikzcd}
\]
\[
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\begin{tikzcd}
&Y \arrow[bend right, swap]{ld}{\pi_0} \arrow[bend right, swap]{d}{\pi}&\mathcal{O}_Y(Y) \cong A_\lambda \arrow{d}{\mathfrak{s}}& \\
X \arrow[hookrightarrow]{r}{}& U \arrow[swap]{u}{\sigma} & \mathcal{O}_X(U)
\end{tikzcd}
\]
For the rest of the proof, we may assume $X = V(I) \subseteq
\mathfrak{k}^n$ where $I = \sqrt{I} \subseteq R
\coloneqq\mathfrak{k}[X_1,\ldots,X_n]$ is an ideal.
Then $A \coloneqq \mathcal{O}_X(X) \cong R / I$ and there is $\ell
\in R$ such
that $\ell\defon{X} = \lambda$.
Let $Y = V(J) \subseteq \mathfrak{k}^{n+1}$ where $J \subseteq
\mathfrak{k}[Z,X_1,\ldots,X_n]$ is generated by the elements of $I$ and $1 -
Z\ell(X_1,\ldots,X_n)$.
Then $\mathcal{O}_Y(Y) \cong \mathfrak{k}[Z,X_1,\ldots,X_n] / J \cong
A[Z] / (1
-\lambda Z) \cong A_\lambda$.
By the proposition about affine varieties (
\ref{propaffvar}), the
morphism
$\mathfrak{s}: \mathcal{O}_Y(Y) \cong A_\lambda \to
\mathcal{O}_X(U)$
corresponds to a morphism $U \xrightarrow{\sigma} Y$.
We have $\mathfrak{s}(Z \mod J) = \lambda^{-1}$ and
$\mathfrak{s}(X_i \mod J) =
X_i \mod I$.
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Thus $\sigma(x) = (\lambda(x)^{-1}, x)$ for $x \in U$.
Moreover, the projection $Y \xrightarrow{\pi_0} X$ dropping the
$Z$-coordinate
has image contained in $U$, as for $(z,x) \in Y$ the equation
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\[
1 =
z\lambda(x)
\]
implies $\lambda(x) \neq 0$.
It thus defines a morphism $Y \xrightarrow{\pi} U$ and by the description
of
$\sigma$ it follows that $\sigma \pi = \Id_U$.
Similarly it follows that $\sigma \pi = \Id_Y$.
Thus, $\sigma$ and $\pi$ are inverse to each other.
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\end{proof}
\begin{corollary}
\label{affopensubtopbase}
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The affine open subsets of a variety $X$ are a topology base on $X$.
\end{corollary}
\begin{proof}
Let $X = V(I) \subseteq \mathfrak{k}^n$ with $I =
\sqrt{I}$.
If $U \subseteq X$ is open then $X \setminus U = V(J)$ with $J \supseteq I$ and
$U = \bigcup_{f \in J} (X \setminus V(f))$.
Thus $U$ is a union of affine open subsets.
The same then holds for arbitrary quasi-affine varieties.
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Let $X$ be any variety, $U \subseteq X$ open and $x \in U$.
By the definition of variety, $x$ has a neighbourhood $V_x$ which is
quasi-affine, and replacing $V_x$ by $U \cap V_x$ which is also quasi-affine we
may assume $V_x \subseteq U$.
$V_x$ is a union of its affine open subsets.
Because $U$ is the union of the $V_x$, $U$ as well is a union of affine open
subsets.
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\end{proof}
% Lecture 14A TODO?
% Lecture 15
% CRTPROG
\subsection{Stalks of sheaves}
\begin{definition}[Stalk]
Let $\mathcal{G}$ be a presheaf of sets on
the topological space $X$, and let $x \in X$.
The \vocab{stalk} (\vocab[Stalk]{Halm}) of $\mathcal{G}$ at $x$ is the set of
equivalence classes of pairs $(U, \gamma)$, where $U$ is an open neighbourhood
of $x$ and $\gamma \in \mathcal{G}(U)$ and the equivalence relation
$\sim $ is
defined as follows: $( U , \gamma) \sim (V, \delta)$ iff there exists
an open
neighbourhood $W \subseteq U \cap V$ of $x$ such that $\gamma
\defon{W} =
\delta \defon{W}$.
If $\mathcal{G}$ is a presheaf of groups, one can define a groups
structure on
$\mathcal{G}_x$ by
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\[
((U, \gamma) / \sim ) \cdot \left( (V,\delta) / \sim
\right) = (U \cap V, \gamma \defon{U \cap V} \cdot
\delta\defon{U \cap V}) /
\sim
\]
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If $\mathcal{G}$ is a presheaf of rings, one can similarly define a
ring
structure on $\mathcal{G}_x$.
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If $U$ is an open neighbourhood of $x \in X$, then we have a map (resp.
homomorphism)
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\begin{align}
\cdot_x : \mathcal{G}(U) & \longrightarrow \mathcal{G}_x \\
\gamma & \longmapsto \gamma_x \coloneqq (U, \gamma) / \sim
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\end{align}
\end{definition}
\begin{fact}
Let $\gamma,\delta \in \mathcal{G}(U)$.
If $\mathcal{G}$ is a sheaf\footnote{or, more generally, a separated presheaf}
and if for all $x \in U$, we have $\gamma_x = \delta_x$, then $\gamma =
\delta$.
In the case of a sheaf, the image of the injective map $\mathcal{G}(U)
\xrightarrow{\gamma \mapsto (\gamma_x)_{x \in U}} \prod_{x \in U}
\mathcal{G}_x$ is the set of all
$(g_x)_{x \in U} \in \prod_{x \in U}
\mathcal{G}_x $ satisfying the following \vocab{coherence condition}:
For
every $x \in U$, there are an open neighbourhood $W_x \subseteq U$ of $x$ and
$g^{(x)} \in \mathcal{G}(W_x)$ with $g_y^{(x)} =
g_y$ for all $y \in W_x$.
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\end{fact}
\begin{proof}
Because of $\gamma_x = \delta_x$, there is $x \in W_x \subseteq U$ open such
that $\gamma\defon{W_x} = \delta\defon{W_x}$.
As the $W_x$ cover $U$, $\gamma = \delta$ by the sheaf axiom.
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\end{proof}
\begin{definition}
Let $\mathcal{G}$ be a sheaf of functions.
Then $\gamma_x$ is called the \vocab{germ} of the function $\gamma$
at $x$.
The \vocab[Germ!
value at $x$]{value at $x$ } of $g = (U, \gamma) / \sim \in
\mathcal{G}_x$ defined as $g(x) \coloneqq \gamma(x)$,
which is independent of the choice of the representative $\gamma$.
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\end{definition}
\begin{remark}
If $\mathcal{G}$ is a sheaf of
$C^{\infty}$-functions (resp.
holomorphic functions), then $\mathcal{G}_x$ is called the ring of
germs of $C^\infty$-functions (resp. of holomorphic functions) at $x$.
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\end{remark}
\subsubsection{The local ring of an affine variety}
\begin{definition}
If $X$ is a variety, the stalk $\mathcal{O}_{X,x}$ of the structure sheaf
at
$x$ is called the \vocab{local ring} of $X$ at $x$.
This is indeed a local ring, with maximal ideal $\mathfrak{m}_x =
\{f \in
\mathcal{O}_{X,x} | f(x) = 0\}$.
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\end{definition}
\begin{proof}
By
\ref{localring} it suffices to show that $\mathfrak{m}_x$
is a proper ideal,
which is trivial, and that the elements of $\mathcal{O}_{X,x} \setminus
\mathfrak{m}_x$ are units in $\mathcal{O}_{X,x}$.
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Let $g = (U, \gamma)/\sim \in \mathcal{O}_{X,x}$ and $g(x) \neq 0$.
$\gamma$ is Zariski continuous (first point of
\ref{localinverse}).
Thus $V(\gamma)$ is closed.
By replacing $U$ by $U \setminus V(\gamma)$ we may assume that $\gamma$
vanishes nowhere on $U$.
By the third point of
\ref{localinverse} we have $\gamma \in
\mathcal{O}_X(U)^{\times}$.
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$(\gamma^{-1})_x$ is an inverse to $g$.
\end{proof}
\begin{proposition}
\label{proplocalring}
Let $X = \Va(I) \subseteq \mathfrak{k}^n$ be
equipped with its usual structure
sheaf, where $I = \sqrt{I} \subseteq R =
\mathfrak{k}[X_1,\ldots,X_n]$ .
Let $x \in X$ and $A = \mathcal{O}_X(X) \cong R / I$.
$\{P \in R | P(x) = 0\} \text{\reflectbox{$\coloneqq$}} \fn_x \subseteq R$ is maximal,
$I \subseteq \fn_x$ and $\mathfrak{m}_x \coloneqq \fn_x / I$ is the
maximal ideal of elements of $A$ vanishing at $x$.
If $\lambda \in A \setminus \mathfrak{m}_x$, we have $\lambda_x \in
\mathcal{O}_{X,x}^{\times}$, where $\lambda_x$ denotes the image under $A \cong
\mathcal{O}_X(X) \to \mathcal{O}_{X,x}$.
By the universal property of the localization, there exists a unique ring
homomorphism $A_{\mathfrak{m}_x} \xrightarrow{\iota}
\mathcal{O}_{X,x}$ such
that
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\[
\begin{tikzcd}
A \arrow{r}{} \arrow{d}{\lambda \mapsto \lambda_x} &
A_{\mathfrak{m}_x} \arrow[dotted, bend left]{ld}{\existsone \iota} \\
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\mathcal{O}_{X,x}
\end{tikzcd}
\]
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commutes.
The morphism $A_{\mathfrak{m}_x}\xrightarrow{\iota}
\mathcal{O}_{X,x}$ is an
isomorphism.
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\end{proposition}
\begin{proof}
To show surjectivity, let $\ell = (U, \lambda) / \sim \in
\mathcal{O}_{X,x}$,
where $U$ is an open neighbourhood of $x$ in $X$.
We have $X \setminus U = V(J)$ where $J \subseteq A$ is an ideal.
As $x \in U$ there is $f \in J$ with $f(x) \neq 0$.
Replacing $U $ by $X \setminus V(f)$ we may assume $U = X \setminus V(f)$.
By
\ref{oxulocaf}, $\mathcal{O}_X(U) \cong A_f$, and
$\lambda =
f^{-n}\vartheta$ for some $n \in \N$ and $\vartheta \in
A$.
Then $\ell = \iota(f^{-n} \vartheta)$ where the last fraction is taken in
$A_{\mathfrak{m}_x}$.
Let $\lambda = \frac{\vartheta}{g} \in A_{\mathfrak{m}_x}$
with
$\iota(\lambda) = 0$.
It is easy to see that $\iota(\lambda) = (X \setminus V(g),
\frac{\vartheta}{g}) / \sim $.
Thus there is an open neighbourhood $U$ of $x$ in $X \setminus V(g)$ such that
$\vartheta$ vanishes on $U$.
Similar as before there is $h \in A$ with $h(x) \neq 0$ and $W = X \setminus
V(h) \subseteq U$.
By the isomorphism $\mathcal{O}_X(W) \cong A_h$, there is $n \in
\N$ with
$h^{n}\vartheta = 0$ in $A$.
Since $h \not\in \mathfrak{m}_x$, $h$ is a unit and the image of
$\vartheta$ in
$A_{\mathfrak{m}_x}$ vanishes, implying $\lambda = 0$.
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\end{proof}
\subsubsection{Intersection multiplicities and Bezout's theorem}
\begin{definition}
Let $R = \mathfrak{k}[X_0,X_1,X_2]$ equipped with its usual grading and let $x
\in \mathbb{P}^{2}$.
Let $G \in R_g, H \in R_h$ be homogeneous polynomials with $x \in V(G) \cap
V(h)$.
Let $\ell\in R_1$ such that $\ell(x) \neq 0$.
Then $x \in U = \mathbb{P}^2 \setminus V(\ell)$ and the rational
functions
$\gamma = \ell^{-g}G, \eta = \ell^{-h}H$ are
elements of
$\mathcal{O}_{\mathbb{P}^2}(U)$.
Let $I_x(G,H) \subseteq \mathcal{O}_{\mathbb{P}^2,x}$
denote the ideal
generated by $\gamma_x$ and $\eta_x$.
\noindent The dimension $\dim_{\mathfrak{k}}(\mathcal{O}_{X,x}
/ I_x(G,H)) \text{\reflectbox{$\coloneqq$}} i_x(G,H)$ is called the
\vocab{intersection multiplicity} of $G$ and $H$ at $x$.
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\end{definition}
\begin{remark}
If $\tilde \ell \in R_1$ also satisfies $\tilde \ell(x) \neq
0$, then the image
of $\tilde \ell / \ell$ under
$\mathcal{O}_{\mathbb{P}^2}(U) \to
\mathcal{O}_{\mathbb{P}^2,x}$ is a unit, showing that
the image of $\tilde
\gamma = \tilde \ell^{-g} G$ in
$\mathcal{O}_{\mathbb{P}^2,x}$ is
multiplicatively equivalent to $\gamma_x$, and similarly for $\eta_x$.
Thus $I_x(G,H)$ does not depend on the choice of $\ell \in R_1$ with
$\ell(x)
\neq 0$.
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\end{remark}
\begin{theorem}[Bezout's theorem]
In the above situation, assume that $V(H)$ and $V(G)$
intersect properly in the sense that $V(G) \cap V(H) \subseteq
\mathbb{P}^2$ has no irreducible component of dimension $\ge 1$.
Then
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\[
\sum_{x \in V(G) \cap V(H)} i_x(G,H) = gh
\]
Thus, $V(G) \cap V(H)$ has
$gh$ elements counted by multiplicity.
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\end{theorem}
\printvocabindex
\end{document}