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Maximilian Keßler 2022-02-16 02:10:21 +01:00
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2021_Algebra_I.tex
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@ -2339,7 +2339,7 @@ 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 $\mathcal{O}(U)$ 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 $X$.
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 $X$.
\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 $X$.
@ -2438,19 +2438,19 @@ The following is somewhat harder than in the affine case:
\subsection{The notion of a category}
\begin{definition}
A \vocab{category} $\cA$ consists of:
A \vocab{category} $\mathcal{A}$ consists of:
\begin{itemize}
\item A class $\Ob \cA$ of \vocab[Objects]{objects of $\cA$}.
\item For two arbitrary objects $A, B \in \Ob \cA$, a \textbf{set} $\Hom_\cA(A,B)$ of \vocab[Morphism]{morphisms for $A$ to $B$ in $\cA$}.
\item A map $\Hom_\cA(B,C) \times \Hom_\cA(A,B) \xrightarrow{\circ} \Hom_\cA(A,C)$, the composition of morphisms, for arbitrary triples $(A,B,C)$ of objects of $\cA$.
\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 $\mathcal{A}$.
\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 $A \in \Ob(\cA)$, there is an $\Id_A \in \Hom_{\cA}(A,A)$ such that $\Id_A \circ f = f$ (reps. $g \circ \Id_A = g$) for arbitrary morphisms $B \xrightarrow{f} A$ (reps. $A \xrightarrow{g} C).$
\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. $A \xrightarrow{g} C).$
\end{enumerate}
A morphism $X \xrightarrow{f} Y$ is called an \vocab[Isomorphism]{isomorphism (in $\cA $)} 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 $f \circ g = \Id_Y$.
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 $f \circ g = \Id_Y$.
\end{definition}
\begin{remark}
\begin{itemize}
@ -2468,14 +2468,14 @@ The following is somewhat harder than in the affine case:
\item If $R$ is a ring, the category of $R$-modules and the category $\foralllg_R$ 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 $\cA$ is a category, then the \vocab{opposite category} or \vocab{dual category} is defined by $\Ob(\cA\op) = \Ob(\cA)$ and $\Hom_{\cA\op}(X,Y) = \Hom_\cA(Y,X)$.
\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 $\Hom_{\mathcal{A}\op}(X,Y) = \Hom_\mathcal{A}(Y,X)$.
\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.
\end{example}
\subsubsection{Subcategories}
\begin{definition}[Subcategories]
A \vocab{subcategory} of $\cA$ is a category $\cB$ such that $\Ob(\cB) \subseteq \Ob(\cA)$, such that $\Hom_\cB(X,Y) \subseteq \Hom_\cA(X,Y)$ for objects $X$ and $Y$ of $\cB$, such that for every object $X \in \Ob(\cB)$, the identity $\Id_X$ of $X$ is the same in $\cB$ as in $\cA$, and such that for composable morphisms in $\cB$, their compositions in $\cA$ and $\cB$ coincide.
We call $\cB$ a \vocab{full subcategory} of $\cA$ if in addition $\Hom_\cB(X,Y) = \Hom_\cA(X,Y)$ for arbitrary $X,Y \in \Ob(\cB)$.
A \vocab{subcategory} of $\mathcal{A}$ is a category $\cB$ such that $\Ob(\cB) \subseteq \Ob(\mathcal{A})$, such that $\Hom_\cB(X,Y) \subseteq \Hom_\mathcal{A}(X,Y)$ for objects $X$ and $Y$ of $\cB$, such that for every object $X \in \Ob(\cB)$, the identity $\Id_X$ of $X$ is the same in $\cB$ as in $\mathcal{A}$, and such that for composable morphisms in $\cB$, their compositions in $\mathcal{A}$ and $\cB$ coincide.
We call $\cB$ a \vocab{full subcategory} of $\mathcal{A}$ if in addition $\Hom_\cB(X,Y) = \Hom_\mathcal{A}(X,Y)$ for arbitrary $X,Y \in \Ob(\cB)$.
\end{definition}
\begin{example}
\begin{itemize}
@ -2489,12 +2489,12 @@ The following is somewhat harder than in the affine case:
\subsubsection{Functors and equivalences of categories}
\begin{definition}
A \vocab[Functor!covariant]{(covariant) functor} (resp. \vocab[Functor!contravariant]{contravariant functor}) between categories $\cA \xrightarrow{F} \cB$ is a map $\Ob(\cA) \xrightarrow{F} \Ob(\cB)$ with a family of maps $\Hom_\cA(X,Y) \xrightarrow{F} \Hom_\cB(F(X),F(Y))$ (resp. $\Hom_\cA(X,Y) \xrightarrow{F} \Hom_\cB(F(Y),F(X))$ in the case of contravariant functors), where $X$ and $Y$ are arbitrary objects of $\cA$, such that the following conditions hold:
A \vocab[Functor!covariant]{(covariant) functor} (resp. \vocab[Functor!contravariant]{contravariant functor}) between categories $\mathcal{A} \xrightarrow{F} \cB$ is a map $\Ob(\mathcal{A}) \xrightarrow{F} \Ob(\cB)$ with a family of maps $\Hom_\mathcal{A}(X,Y) \xrightarrow{F} \Hom_\cB(F(X),F(Y))$ (resp. $\Hom_\mathcal{A}(X,Y) \xrightarrow{F} \Hom_\cB(F(Y),F(X))$ 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 $\cA$, we have $F(gf) = F(g)F(f)$ ( resp. $F(gf) = F(f)F(g)$)
\item For morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ in $\mathcal{A}$, we have $F(gf) = F(g)F(f)$ ( resp. $F(gf) = F(f)F(g)$)
\end{itemize}
A functor is called \vocab[Functor!essentially surjective]{essentially surjective} if every object of $\cB$ is isomorphic to an element of the image of $\Ob(\cA) \xrightarrow{F} \Ob(\cB)$.
A functor is called \vocab[Functor!essentially surjective]{essentially surjective} if every object of $\cB$ is isomorphic to an element of the image of $\Ob(\mathcal{A}) \xrightarrow{F} \Ob(\cB)$.
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.
\end{definition}
@ -2573,7 +2573,7 @@ The following is somewhat harder than in the affine case:
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 $\foralllg_\mathfrak{k}$,
restricts to an equivalence of categories between the category of affine varieties over $\mathfrak{k}$ and the full subcategory $\mathcal{A}$ of $\foralllg_\mathfrak{k}$,
having the $\mathfrak{k}$-algebras $A$ of finite type with $\nil A = \{0\} $ as objects.
\end{itemize}
\end{proposition}
@ -2617,12 +2617,12 @@ The following is somewhat harder than in the affine case:
\begin{claim}
The functor in the second part maps affine varieties to objects of $\cA$ and is essentially surjective.
The functor in the second part maps affine varieties to objects of $\mathcal{A}$ and is essentially surjective.
\end{claim}
\begin{subproof}
It follows from the remark that the functor maps affine varieties to objects of $\cA$.
It follows from the remark that the functor maps affine varieties to objects of $\mathcal{A}$.
If $A \in \Ob(\cA)$ then $ A /\mathfrak{k}$ is of finite type, thus $A \cong R / I$ for some $n$.
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 $x \in \sqrt{I}$, $x \mod I \in \nil(R / I) \cong \nil(A) = \{0\}$.
Thus $A \cong\mathcal{O}_X(X)$ where $X = V(I)$.
\end{subproof}
@ -2630,7 +2630,7 @@ The following is somewhat harder than in the affine case:
\end{proof}
\begin{remark}
Note that giving a contravariant functor $\cC \to \cD$ is equivalent to giving a functor $\cC \to \cD\op$. We have thus shown that the category of affine varieties is equivalent to $\cA\op$, where $\cA \subsetneq \foralllg_\mathfrak{k}$ is the full subcategory of $\mathfrak{k}$-algebras $A$ of finite type with $\nil(A) = \{0\}$.
Note that giving a contravariant functor $\cC \to \cD$ is equivalent to giving a functor $\cC \to \cD\op$. We have thus shown that the category of affine varieties is equivalent to $\mathcal{A}\op$, where $\mathcal{A} \subsetneq \foralllg_\mathfrak{k}$ is the full subcategory of $\mathfrak{k}$-algebras $A$ of finite type with $\nil(A) = \{0\}$.
\end{remark}
\subsubsection{Affine open subsets are a topology base}