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@ -1457,10 +1457,10 @@ Recall the definition of a normal field extension in the case of finite field ex
It follows from \ref{cintclosure} and \ref{locandquot} that the integral closure of $A$ in some field extension $L$ of $Q(A)$ is always normal.
\end{remark}
\begin{remark}
A finite field extension of $\Q$ is called an \vocab{algebraic number field} (ANF). If $K$ is an ANF, let $\cO_K$ (the \vocab[Ring of integers in an ANF]{ring of integers in $K$}) be the integral closure of $\Z$ in $K$.
A finite field extension of $\Q$ is called an \vocab{algebraic number field} (ANF). If $K$ is an ANF, let $\mathcal{O}_K$ (the \vocab[Ring of integers in an ANF]{ring of integers in $K$}) be the integral closure of $\Z$ in $K$.
One can show that this is a finitely generated (hence free, by results of EInführung in die Algebra % EINFALG
) abelian group.
We have $\cO_{\Q} = \Z$ by the proposiiton.
We have $\mathcal{O}_{\Q} = \Z$ by the proposiiton.
\end{remark}
\subsubsection{Action of \texorpdfstring{$\Aut(L / K)$}{Aut(L / K)} on prime ideals of a normal ring extension}
@ -1486,7 +1486,7 @@ Recall the definition of a normal field extension in the case of finite field ex
\end{proof}
\begin{remark}
The theorem is very important for its own sake. For instance, if $K$ is an ANF which is a Galois extension of $\Q$ it shows that $\Gal(K / \Q)$ transitively acts on the set of prime ideals of $\cO_K$ over a given prime number $p$. More generally, if $L / K$ is a Galois extension of ANF then $\Gal(L / K)$ transitively acts on the set of $\fq \in \Spec \cO_L$ for which $\fq \cap K$ is a given $\fp \in \Spec \cO_K$.
The theorem is very important for its own sake. For instance, if $K$ is an ANF which is a Galois extension of $\Q$ it shows that $\Gal(K / \Q)$ transitively acts on the set of prime ideals of $\mathcal{O}_K$ over a given prime number $p$. More generally, if $L / K$ is a Galois extension of ANF then $\Gal(L / K)$ transitively acts on the set of $\fq \in \Spec \mathcal{O}_L$ for which $\fq \cap K$ is a given $\fp \in \Spec \mathcal{O}_K$.
\end{remark}
\subsubsection{A going-down theorem}
@ -2306,7 +2306,7 @@ If $P_e $ with $e < d$ was $\neq 0$, it could not be a multiple of $P$ contradic
The bijectivity of the $\phi_{U, (U_i)_{i \in I}}$ is called the \vocab{sheaf axiom}.
\end{definition}
\begin{dtrivial}
A presheaf is a contravariant functor $\cG : \cO(X) \to C$ where $\cO(X)$ denotes the category of open subsets of $X$ with inclusions as morphisms and $C$ is the category of sets, rings or (abelian) groups.
A presheaf is a contravariant functor $\cG : \mathcal{O}(X) \to C$ 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.
\end{dtrivial}
\begin{definition}
A subsheaf $\cG'$ is defined by subsets (resp. subrings or subgroups) $\cG'(U) \subseteq \cG(U)$ for all open $U \subseteq X$ such that the sheaf axioms still hold.
@ -2339,21 +2339,21 @@ If $P_e $ with $e < d$ was $\neq 0$, it could not be a multiple of $P$ contradic
\end{example}
\begin{example}
If $X = \R^n$, $\bK \in \{\R, \C\}$ and $\cO(U)$ is the sheaf of $\bK$-valued $C^{\infty}$-functions on $U$, then $\cO$ is a subsheaf of the sheaf (of rings) of $\bK$-valued continuous functions on $X$.
If $X = \R^n$, $\bK \in \{\R, \C\}$ and $\mathcal{O}(U)$ is the sheaf of $\bK$-valued $C^{\infty}$-functions on $U$, then $\mathcal{O}$ is a subsheaf of the sheaf (of rings) of $\bK$-valued continuous functions on $X$.
\end{example}
\begin{example}
If $X = \C^n$ and $\cO(U)$ the set of holomorphic functions on $X$, then $\cO$ is a subsheaf of the sheaf of $\C$-valued $C^{\infty}$-functions on $X$.
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 $X$.
\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 $\cO_X(U)$ 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 $\phi(y) = \frac{f(y)}{g(y)}$.
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 $\phi(y) = \frac{f(y)}{g(y)}$.
\end{definition}
\begin{remark}\label{structuresheafcontinuous}
$\cO_X$ is a subsheaf (of rings) of the sheaf of $\mathfrak{k}$-valued functions on $X$.
The elements of $\cO_X(U)$ are continuous:
$\mathcal{O}_X$ 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 $U$.
\end{remark}
@ -2361,18 +2361,18 @@ Let $X \subseteq \mathfrak{k}^n$ be open. Let $R = \mathfrak{k}[X_1,\ldots,X_n]$
\begin{proposition}\label{structuresheafri}
Let $X = V(I)$ where $I = \sqrt{I} \subseteq R$ is an ideal. Let $A = R / I$. Then
\begin{align}
\phi: A &\longrightarrow \cO_X(X) \\
\phi: A &\longrightarrow \mathcal{O}_X(X) \\
f \mod I &\longmapsto f\defon{X}
\end{align}
is an isomorphism.
\end{proposition}
\begin{proof}
It is easy to see that the map $A \to \cO_X(X)$ is well-defined and a ring homomorphism.
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 $\phi \in \cO_X(X)$. 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 $U_x$.
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 $U_x$.
\begin{claim}
\Wlog we can assume $U_x = X \setminus V(g_x)$.
\end{claim}
@ -2412,24 +2412,24 @@ is an isomorphism.
Let $X \subseteq \bP^n$ be closed and $R_\bullet = \mathfrak{k}[X_0,\ldots,X_n]$ with its usual grading.
\begin{definition}\label{structuresheafpn}
For open $U \subseteq X$, let $\cO_X(U)$ 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 $y = [y_0,\ldots,y_n] \in W$.
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 $y = [y_0,\ldots,y_n] \in W$.
\end{definition}
\begin{remark}
This is a subsheaf of rings of the sheaf of $\mathfrak{k}$-valued functions on $X$.
Under the identification $\bA^n =\mathfrak{k}^n$ with $\bP^n \setminus \Vp(X_0)$, one has $\cO_X \defon{X \setminus \Vp(X_0)} = \cO_{X \cap \bA^n}$ as subsheaves of the sheaf of $\mathfrak{k}$-valued functions, where the second sheaf is a sheaf on a closed subset of $\mathfrak{k}^n$:
Under the identification $\bA^n =\mathfrak{k}^n$ with $\bP^n \setminus \Vp(X_0)$, one has $\mathcal{O}_X \defon{X \setminus \Vp(X_0)} = \mathcal{O}_{X \cap \bA^n}$ 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 $X \cap \bA^n$ 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 $d \in \N$.
\end{remark}
\begin{remark}
It follows from the previous remark and the similar result in the affine case that the elements of $\cO_X(U)$ are continuous on $U \setminus V(X_0)$.
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 $U$.
\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 $\cO_X\left( X \right) $ are constant functions on $X$.
If $X$ is connected (e.g. irreducible), then the elements of $\mathcal{O}_X\left( X \right) $ are constant functions on $X$.
\end{proposition}
@ -2512,18 +2512,18 @@ The following is somewhat harder than in the affine case:
\subsection{The category of varieties}
\begin{definition}[Algebraic variety]\label{defvariety}
An \vocab{algebraic variety} or \vocab{prevariety} over $\mathfrak{k}$ is a pair $(X, \cO_X)$, where $X$ is a topological space and $\cO_X$ 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 $f \in \cO_X(V) \iff \iota\st_x(f) \in \cO_{Y_x}(\iota_x^{-1}(V))$,
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 $f \in \mathcal{O}_X(V) \iff \iota\st_x(f) \in \mathcal{O}_{Y_x}(\iota_x^{-1}(V))$,
In this, the \vocab{pull-back} $\iota_x\st(f)$ of $f$ is defined by $(\iota_x\st(f))(\xi) \coloneqq f(\iota_x(\xi))$.
A morphism $(X, \cO_X) \to (Y, \cO_Y)$ of varieties is a continuous map $X \xrightarrow{\phi} Y$ such that for all open $U \subseteq Y$ and $f \in \cO_Y(U)$, $\phi\st(f) \in \cO_X(\phi^{-1}(U))$.
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 $f \in \mathcal{O}_Y(U)$, $\phi\st(f) \in \mathcal{O}_X(\phi^{-1}(U))$.
An isomorphism is a morphism such that $\phi$ is bijective and $\phi^{-1}$ also is a morphism of varieties.
\end{definition}
\begin{example}
\begin{itemize}
\item If $(X, \cO_X)$ is a variety and $U \subseteq X$ open, then $(U, \cO_X\defon{U})$ 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 $\bP^n$, then $(X, \cO_X)$ is a variety, where $\cO_X$ is the structure sheaf on $X$ (\ref{structuresheafkn}, reps. \ref{structuresheafpn}).
\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 $\bP^n$, 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. $\bP^n$).
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.
@ -2538,9 +2538,9 @@ The following is somewhat harder than in the affine case:
\begin{lemma}\label{localinverse}
Let $X$ be any $\mathfrak{k}$-variety and $U \subseteq X$ open.
\begin{enumerate}[i)]
\item All elements of $\cO_X(U)$ 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 $\lambda \defon{V_x} \in \cO_X(V_x)$, then $\lambda \in \cO_X(U)$.
\item If $\vartheta \in \cO_X(U)$ and $\vartheta(x) \neq 0$ for all $x \in U$, then $\vartheta \in \cO_X(U)^{\times }$.
\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 $\vartheta \in \mathcal{O}_X(U)^{\times }$.
\end{enumerate}
\end{lemma}
\begin{proof}
@ -2548,12 +2548,12 @@ The following is somewhat harder than in the affine case:
\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 $\cO_X$ there is $\ell \in \cO_X(U)$ with $\ell\defon{V_x} =\lambda_x$. It follows that $\ell=\lambda$.
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 $\lambda \in \cO_X(U)$.
We have $\lambda \cdot \vartheta = 1$, thus $\vartheta \in \cO_X(U)^{\times}$.
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 $\vartheta \in \mathcal{O}_X(U)^{\times}$.
\end{enumerate}
@ -2563,15 +2563,15 @@ The following is somewhat harder than in the affine case:
\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_{\foralllg_\mathfrak{k}}(\cO_Y(Y), \cO_X(X)) \\
(X \xrightarrow{f} Y) &\longmapsto (\cO_Y(Y) \xrightarrow{f\st} \cO_X(X))
\phi: \Hom_{\Var_\mathfrak{k}}(X,Y) &\longrightarrow \Hom_{\foralllg_\mathfrak{k}}(\mathcal{O}_Y(Y), \mathcal{O}_X(X)) \\
(X \xrightarrow{f} Y) &\longmapsto (\mathcal{O}_Y(Y) \xrightarrow{f\st} \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 \foralllg_\mathfrak{k} \\
X &\longmapsto \cO_X(X)\\
(X\xrightarrow{f} Y) &\longmapsto (\cO_X(X) \xrightarrow{f\st} \cO_Y(Y))
X &\longmapsto \mathcal{O}_X(X)\\
(X\xrightarrow{f} Y) &\longmapsto (\mathcal{O}_X(X) \xrightarrow{f\st} \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 $\cA$ of $\foralllg_\mathfrak{k}$,
having the $\mathfrak{k}$-algebras $A$ of finite type with $\nil A = \{0\} $ as objects.
@ -2579,11 +2579,11 @@ The following is somewhat harder than in the affine case:
\end{proposition}
\begin{remark}
It is clear that $\nil(\cO_X(X)) = \{0\}$ for arbitrary varieties. For general varieties it is however not true that $\cO_X(X)$ is a $\mathfrak{k}$-algebra of finite type.
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 $\cO_X(X) \cong R / I$ (see \ref{structuresheafri}) is a $\mathfrak{k}$-algebra of finite type.
Then $\mathcal{O}_X(X) \cong R / I$ (see \ref{structuresheafri}) is a $\mathfrak{k}$-algebra of finite type.
\end{remark}
\begin{proof}
@ -2593,8 +2593,8 @@ The following is somewhat harder than in the affine case:
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\st(\xi_i) $. Thus $f$ is uniquely determined by $\cO_Y(Y) \xrightarrow{f\st} \cO_X(X)$.
Conversely, let $Y = V(I)$ and $\cO_Y(Y) \xrightarrow{\phi} \cO_X(X)$ be a morphism of $\mathfrak{k}$-algebras. Define $f_i \coloneqq \phi(\xi_i)$ and consider $X \xrightarrow{f = (f_1,\ldots,f_n)} Y\subseteq \mathfrak{k}^n$.
By definition $f_i = f\st(\xi_i) $. Thus $f$ is uniquely determined by $\mathcal{O}_Y(Y) \xrightarrow{f\st} \mathcal{O}_X(X)$.
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 $X \xrightarrow{f = (f_1,\ldots,f_n)} Y\subseteq \mathfrak{k}^n$.
\begin{claim}
$f$ has image contained in $Y$.
\end{claim}
@ -2607,10 +2607,10 @@ The following is somewhat harder than in the affine case:
\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 $\lambda \in \cO_Y(\Omega)$ 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 $\alpha \coloneqq \phi(a)\defon{W} \in \cO_X(W)$, $\beta \coloneqq \phi(b)\defon{W} \in \cO_X(W)$.
By the second part of \ref{localinverse} $\beta \in \cO_X(W)^{\times}$ and $f\st(\lambda)\defon{W} = \frac{\alpha}{\beta} \in \cO_X(W)$.
The first part of \ref{localinverse} shows that $f\st(\lambda) \in \cO_X(U)$.
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 $f\st(\lambda)\defon{W} = \frac{\alpha}{\beta} \in \mathcal{O}_X(W)$.
The first part of \ref{localinverse} shows that $f\st(\lambda) \in \mathcal{O}_X(U)$.
\end{subproof}
By definition of $f$, we have $f\st = \phi$. This finished the proof of the first point.
@ -2624,7 +2624,7 @@ The following is somewhat harder than in the affine case:
If $A \in \Ob(\cA)$ 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 $x \in \sqrt{I}$, $x \mod I \in \nil(R / I) \cong \nil(A) = \{0\}$.
Thus $A \cong\cO_X(X)$ where $X = V(I)$.
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}
@ -2646,30 +2646,30 @@ If $X$ is a set, then $\cB \subseteq \mathcal{P}(X)$ is a base for some topology
\end{definition}
\begin{proposition}\label{oxulocaf}
Let $X$ be an affine variety over $\mathfrak{k}$, $\lambda \in \cO_X(X)$ and $U = X \setminus V(\lambda)$.
Then $U$ is an affine variety and the morphism $\phi: \cO_X(X)_\lambda \to \cO_X(U)$ defined by the restriction $\cO_X(X) \xrightarrow{\cdot |_U } \cO_X(U)$ and the universal property of the localization is an isomorphism.
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.
\end{proposition}
\begin{proof}
Let $X$ be an affine variety over $\mathfrak{k}, \lambda \in \cO_X(X)$ and $U = X \setminus V(\lambda)$. The fact that $\lambda\defon{U} \in \cO_x(U)^{\times}$ follows from \ref{localinverse}.
Thus the universal property of the localization $\cO_X(X)_\lambda$ can be applied to $\cO_X(X) \xrightarrow{\cdot |_U} \cO_X(U)$.
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 $\mathcal{O}_X(X) \xrightarrow{\cdot |_U} \mathcal{O}_X(U)$.
\[
\begin{tikzcd}
\cO_X(X) \arrow{d}{\cdot |_U}\arrow{r}{x \mapsto \frac{x}{1}} & \cO_X(X)_\lambda \arrow[dotted, bend left]{dl}{\existsone \phi} \\
\cO_X(U) &
\mathcal{O}_X(X) \arrow{d}{\cdot |_U}\arrow{r}{x \mapsto \frac{x}{1}} & \mathcal{O}_X(X)_\lambda \arrow[dotted, bend left]{dl}{\existsone \phi} \\
\mathcal{O}_X(U) &
\end{tikzcd}
\]
\[
\begin{tikzcd}
&Y \arrow[bend right, swap]{ld}{\pi_0} \arrow[bend right, swap]{d}{\pi}&\cO_Y(Y) \cong A_\lambda \arrow{d}{\mathfrak{s}}& \\
X \arrow[hookrightarrow]{r}{}& U \arrow[swap]{u}{\sigma} & \cO_X(U)
&Y \arrow[bend right, swap]{ld}{\pi_0} \arrow[bend right, swap]{d}{\pi}&\mathcal{O}_Y(Y) \cong A_\lambda \arrow{d}{\mathfrak{s}}& \\
X \arrow[hookrightarrow]{r}{}& U \arrow[swap]{u}{\sigma} & \mathcal{O}_X(U)
\end{tikzcd}
\]
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 $A \coloneqq \cO_X(X) \cong R / I$ and there is $\ell \in R$ such that $\ell\defon{X} = \lambda$.
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 $\cO_Y(Y) \cong \mathfrak{k}[Z,X_1,\ldots,X_n] / J \cong A[Z] / (1 -\lambda Z) \cong A_\lambda$.
By the proposition about affine varieties (\ref{propaffvar}), the morphism $\mathfrak{s}: \cO_Y(Y) \cong A_\lambda \to \cO_X(U)$ corresponds to a morphism $U \xrightarrow{\sigma} Y$.
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 $\mathfrak{s}(X_i \mod J) = X_i \mod I$.
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
@ -2743,38 +2743,38 @@ If $X$ is a set, then $\cB \subseteq \mathcal{P}(X)$ is a base for some topology
\end{remark}
\subsubsection{The local ring of an affine variety}
\begin{definition}
If $X$ is a variety, the stalk $\cO_{X,x}$ 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 $\mathfrak{m}_x = \{f \in \cO_{X,x} | f(x) = 0\}$.
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 $\mathfrak{m}_x = \{f \in \mathcal{O}_{X,x} | f(x) = 0\}$.
\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 $\cO_{X,x} \setminus \mathfrak{m}_x$ are units in $\cO_{X,x}$.
Let $g = (U, \gamma)/\sim \in \cO_{X,x}$ and $g(x) \neq 0$.
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 $\mathcal{O}_{X,x}$.
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 $\gamma \in \cO_X(U)^{\times}$.
By the third point of \ref{localinverse} we have $\gamma \in \mathcal{O}_X(U)^{\times}$.
$(\gamma^{-1})_x$ 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 $A = \cO_X(X) \cong R / I$.
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 $\lambda_x \in \cO_{X,x}^{\times}$, where $\lambda_x$ denotes the image under $A \cong \cO_X(X) \to \cO_{X,x}$.
By the universal property of the localization, there exists a unique ring homomorphism $A_{\mathfrak{m}_x} \xrightarrow{\iota} \cO_{X,x}$
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
\[
\begin{tikzcd}
A \arrow{r}{} \arrow{d}{\lambda \mapsto \lambda_x} & A_{\mathfrak{m}_x} \arrow[dotted, bend left]{ld}{\existsone \iota} \\
\cO_{X,x}
\mathcal{O}_{X,x}
\end{tikzcd}
\]
commutes.
The morphism $A_{\mathfrak{m}_x}\xrightarrow{\iota} \cO_{X,x}$ is an isomorphism.
The morphism $A_{\mathfrak{m}_x}\xrightarrow{\iota} \mathcal{O}_{X,x}$ is an isomorphism.
\end{proposition}
\begin{proof}
To show surjectivity, let $\ell = (U, \lambda) / \sim \in \cO_{X,x}$, where $U$ is an open neighbourhood of $x$ in $X$.
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}, $\cO_X(U) \cong A_f$, and $\lambda = f^{-n}\vartheta$ for some $n \in \N$ and $\vartheta \in A$.
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 $A_{\mathfrak{m}_x}$.
@ -2782,20 +2782,20 @@ If $X$ is a set, then $\cB \subseteq \mathcal{P}(X)$ is a base for some topology
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 $\cO_X(W) \cong A_h$, 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 $\lambda = 0$.
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 $\lambda = 0$.
\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 $x \in \bP^{2}$.
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 $x \in U = \bP^2 \setminus V(\ell)$ and the rational functions $\gamma = \ell^{-g}G, \eta = \ell^{-h}H$ are elements of $\cO_{\bP^2}(U)$.
Let $I_x(G,H) \subseteq \cO_{\bP^2,x}$ denote the ideal generated by $\gamma_x$ and $\eta_x$.
Let $\ell\in R_1$ such that $\ell(x) \neq 0$. Then $x \in U = \bP^2 \setminus V(\ell)$ and the rational functions $\gamma = \ell^{-g}G, \eta = \ell^{-h}H$ are elements of $\mathcal{O}_{\bP^2}(U)$.
Let $I_x(G,H) \subseteq \mathcal{O}_{\bP^2,x}$ denote the ideal generated by $\gamma_x$ and $\eta_x$.
\noindent The dimension $\dim_{\mathfrak{k}}(\cO_{X,x} / I_x(G,H)) \text{\reflectbox{$\coloneqq$}} i_x(G,H)$ is called the \vocab{intersection multiplicity} of $G$ and $H$ at $x$.
\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 $x$.
\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 $\cO_{\bP^2}(U) \to \cO_{\bP^2,x}$ is a unit, showing that the image of $\tilde \gamma = \tilde \ell^{-g} G$ in $\cO_{\bP^2,x}$ is multiplicatively equivalent to $\gamma_x$, and similarly for $\eta_x$.
If $\tilde \ell \in R_1$ also satisfies $\tilde \ell(x) \neq 0$, then the image of $\tilde \ell / \ell$ under $\mathcal{O}_{\bP^2}(U) \to \mathcal{O}_{\bP^2,x}$ is a unit, showing that the image of $\tilde \gamma = \tilde \ell^{-g} G$ in $\mathcal{O}_{\bP^2,x}$ 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 $\ell(x) \neq 0$.
\end{remark}
\begin{theorem}[Bezout's theorem]