some changes

2021_Algebra_I.tex

@@ -33,10 +33,8 @@
 \section{Varieties}
 \input{inputs/varieties}
 
-\iffalse
+% \section{Übersicht}
-    \section{Übersicht}
+% \input{inputs/uebersicht}
-    \input{inputs/uebersicht}
-\fi
 
 \cleardoublepage
 \printvocabindex

inputs/finiteness_conditions.tex

@@ -5,9 +5,9 @@
     Then the following sets coincide
     \begin{enumerate}
         \item
-              $\left\{ \sum_{s \in S'} r_{s} \cdot s ~ |~ S \subseteq S' \text{finite}, r_s \in R, \right\}$,
+            \[\left\{ \sum_{s \in S'} r_{s} \cdot s ~ |~ S \subseteq S' \text{ finite}, r_s \in R\right\},\]
         \item
-              $\bigcap_{\substack{S \subseteq N \subseteq M\\N \text{submodule}}} N$,
+            \[\bigcap_{\substack{S \subseteq N \subseteq M\\N \text{submodule}}} N,\]
         \item
               The $\subseteq$-smallest submodule of $M$ containing $S$.
     \end{enumerate}

inputs/nullstellensatz_and_zariski_topology.tex

@@ -366,7 +366,7 @@ For $f \in R$ let $V(f) = V(fR)$.
 \begin{proof}
     Suppose $f$ vanishes on all zeros of $I$.
     Let $R' \coloneqq \mathfrak{k}[X_1,\ldots,X_n,T]$,
-    $g(X_1,\ldots,X_n,T) \coloneqq 1 - T \cdot f(X_1,\ldots,X_n)$
+    \[g(X_1,\ldots,X_n,T) \coloneqq 1 - T \cdot f(X_1,\ldots,X_n)\]
     and $J \subseteq R'$ the ideal generated by $g$ and the elements of $I$
     (viewed as elements of $R'$ which are constant in the $T$-direction).
 
@@ -600,7 +600,7 @@ Let $X$ be a topological space.
     irreducible.
 
     The remaining assertion follows from the fact,
-    that the bijection is $\subseteq$-antimonotonic
+    that the bijection is $\subseteq$-anti\-monotonic
     and thus maximal ideals correspond to minimal irreducible closed subsets,
     which are the one-point subsets as $\mathfrak{k}^n$ is T${}_1$.
 \end{proof}
@@ -609,7 +609,7 @@ Let $X$ be a topological space.
     Let $Z$ be an irreducible subset of the topological space $X$.
     Let $\codim(Z,X)$ be the maximum of the length $n$ of
     strictly increasing chains
-    $Z \subseteq Z_0 \subsetneq Z_1 \subsetneq \ldots \subsetneq Z_n$
+    \[Z \subseteq Z_0 \subsetneq Z_1 \subsetneq \ldots \subsetneq Z_n\]
     of irreducible closed subsets of $X$ containing $Z$ or $\infty$
     if such chains can be found for arbitrary $n$.
     Let
@@ -617,7 +617,7 @@ Let $X$ be a topological space.
         \dim X \coloneqq
         \begin{cases}
             - \infty & \text{if } X = \emptyset,\\
-            \sup_{\substack{Z \subseteq X \\ Z \text{ irreducible}}} \codim(Z,X) & \text{otherwise}.
+            \sup\limits_{\substack{Z \subseteq X \\ Z \text{ irreducible}}} \codim(Z,X) & \text{otherwise}.
         \end{cases}
     \]
 \end{definition}
@@ -730,7 +730,7 @@ In general, these inequalities may be strict.
     it
     is clear that $\codim(\{0\}, \mathfrak{k}^n) \ge n$.
     Translation by $x \in \mathfrak{k}^n$ gives us
-    $\codim(\{x\}, \mathfrak{k}^n) \ge n$.
+    \[\codim(\{x\}, \mathfrak{k}^n) \ge n.\]
 
     The opposite inequality follows from \ref{upperbounddim}
     ($Z = \mathfrak{k}^n$,
@@ -805,13 +805,13 @@ In general, these inequalities may be strict.
     \begin{enumerate}
         \item[M]
             If $m,n \in X$ and $A \subseteq X$ then
-            $m \in \mathcal{H}( \{n\} \cup A) \setminus \mathcal{H}(A) %
+            \[m \in \mathcal{H}( \{n\} \cup A) \setminus \mathcal{H}(A) %
-               \iff n \in \mathcal{H}(\{m\}  \cup A) \setminus \mathcal{H}(A)$,
+            \iff n \in \mathcal{H}(\{m\}  \cup A) \setminus \mathcal{H}(A),\]
         \item[F]
             $\mathcal{H}(A) = %
                 \bigcup_{F \subseteq A \text{ finite}} \mathcal{H}(F)$.
     \end{enumerate}
-    In this case, $S \subseteq X$ is called \vocab{Independent subset},
+    In this case, $S \subseteq X$ is called \vocab{independent subset},
     if $s \not\in \mathcal{H}(S \setminus \{s\})$ for all $s \in S$ and
     \vocab[Generating subset]{generating} if $X = \mathcal{H}(S)$.
     $S$ is called a \vocab{base}, if it is both generating and independent.
@@ -1944,9 +1944,9 @@ extensions:
     let $L$ be a field extension of $Q(A)$.
     By \ref{intclosure} the set of elements of $L$ integral over $A$ is a
     subring of $L$, the \vocab{integral closure}  of $A$ in $L$.
-    $A$ is \vocab{Domain!integrally closed} in $L$
+    $A$ is \vocab[Domain!integrally closed]{integrally closed} in $L$
     if the integral closure of $A$ in $L$ equals $A$.
-    $A$ is \vocab{Domain!normal} if it is integrally closed in $Q(A)$.
+    $A$ is \vocab[Domain!normal]{normal} if it is integrally closed in $Q(A)$.
 \end{definition}
 
 \begin{proposition}
@@ -2029,7 +2029,8 @@ extensions:
     (\ref{characfixnormalfe}),
     there is a positive integer $k$ with $y^k \in K$.
     As $A$ is normal, we have $y^k \in K \cap B = A$.
-    Thus $y^k \in (A \cap \fq) \setminus (A \cap \fr) = \fp \setminus \fp = \emptyset \lightning$.
+    Thus
+    \[y^k \in (A \cap \fq) \setminus (A \cap \fr) = \fp \setminus \fp = \emptyset \lightning.\]
 
     If $L / K$ is not finite, one applies Zorn's lemma to the poset of pairs
     $(M, \sigma)$ where $M$ is an intermediate field
@@ -2116,7 +2117,7 @@ extensions:
     Japanese.
 \end{remark}
 
-\begin{align*}+[Counterexample to going down]
+\begin{example}+[Counterexample to going down]
     Let $R = \mathfrak{k}[X,Y]$ and $A = \mathfrak{k}[X,Y, \frac{X}{Y}]$.
     Then going down does not hold for $A / R$: 
 
@@ -2127,7 +2128,7 @@ extensions:
     and thus proper, we have $(X,Y)_R = \fq \cap R$.
     The prime ideal $(\frac{X}{Y},Y)_A = (\frac{X}{Y}, X,Y)_A$ is lying over
     $(X,Y)_R$, so going down is violated.
-\end{align*}
+\end{example}
 
 
 \subsubsection{Proof of \texorpdfstring{$\codim(\{y\},Y) = \trdeg(\mathfrak{K}(Y) / \mathfrak{k})$}{codim(\{y\},Y) = trdeg(K(Y) /k)}}
@@ -2280,7 +2281,7 @@ We will use the following
     $(a_i)_{i = 1}^m$ for $K / \mathfrak{l}$
     with $a_i \in A$ for $1 \le i \le m$.
 \end{lemma}
-\begin{align*}
+\begin{proof}
     The proof is similar to the proof of \ref{ltrdegresfieldtrbase}.
     There are a natural number $m \ge n$ and elements
     $(a_i)_{i = n+1}^m \in A^{m-n}$ which generate $K$
@@ -2296,7 +2297,7 @@ We will use the following
     But then $K$ is algebraic over its subfield generated by $\mathfrak{l}$
     and the $(a_i)_{i=1}^{m-1}$,
     contradicting the minimality of $m$.
-\end{align*}
+\end{proof}
 
 \begin{theorem}
     \label{htandtrdeg}
@@ -2724,13 +2725,15 @@ this more general theorem.
     defines a homeomorphism between $(A \times B) \cap \Delta$ and $A \cap B$.
     Thus, $C$ is homeomorphic to an irreducible component $C'$ of
     $(A \times B) \cap \Delta$ and
-    \begin{align*}
+    \begin{IEEEeqnarray*}{rCl}
-        \codim(C, \mathfrak{k}^n) = n - \dim(C) = n - \dim(C')
+        \codim(C, \mathfrak{k}^n)
-            = n - \dim(A \times B) + \codim(C', A \times B) \\
+            &=& n - \dim(C)\\
-        \overset{\text{\ref{corpithm}}}{\le }2n - \dim(A \times B)
+            &=& n - \dim(C')\\
-            \overset{\text{\ref{dimprod}}}{=} 2n - \dim(A) - \dim(B)
+            &=& n - \dim(A \times B) + \codim(C', A \times B) \\
-            = \codim(A,\mathfrak{k}^n) + \codim(B, \mathfrak{k}^n)
+            &\overset{\text{\ref{corpithm}}}{\le }&2n - \dim(A \times B)\\
-    \end{align*}
+            &\overset{\text{\ref{dimprod}}}{=}& 2n - \dim(A) - \dim(B)\\
+            &=& \codim(A,\mathfrak{k}^n) + \codim(B, \mathfrak{k}^n)
+    \end{IEEEeqnarray*}
     by the general
     properties of dimension and codimension,
     \ref{corpithm} applied to $(X_i - Y_i)_{i=1}^n$,
@@ -2778,7 +2781,7 @@ this more general theorem.
 \subsection{The Jacobson radical}
 \begin{proposition}
     For a ring $A$,
-    $\bigcap_{\mathfrak{m} \in \MaxSpec A} \mathfrak{m} = \{a \in A | \forall x \in A ~ 1 - ax \in A^{\times }\} \text{\reflectbox{$\coloneqq$}} \rad(A)$,
+    \[\bigcap_{\mathfrak{m} \in \MaxSpec A} \mathfrak{m} = \{a \in A | \forall x \in A ~ 1 - ax \in A^{\times }\} \text{\reflectbox{$\coloneqq$}} \rad(A).\]
     the \vocab{Jacobson radical} of $A$.
 \end{proposition}
 \begin{proof}

inputs/projective_spaces.tex

@@ -65,12 +65,12 @@ Let $\mathfrak{l}$ be any field.
     We call the $r_d$ the \vocab{homogeneous components} of $r$.
 
     An ideal $I \subseteq A$ is called \vocab{homogeneous} if
-    $r \in I \implies \forall d \in \mathbb{I} ~ r_d \in I_d$
+    \[r \in I \implies \forall d \in \mathbb{I} ~ r_d \in I_d\]
     where $I_d \coloneqq I \cap A_d$.
 
     By a \vocab{graded ring}  we understand an $\N$-graded ring.
-    Tin this case,
+    In this case,
-    $A_{+} \coloneqq \bigoplus_{d=1}^{\infty} A_d = \{r \in A | r_0 = 0\} $
+    \[A_{+} \coloneqq \bigoplus_{d=1}^{\infty} A_d = \{r \in A | r_0 = 0\}\]
     is called the \vocab{augmentation ideal}  of $A$.
 \end{definition}
 \begin{remark}[Decomposition of $1$]
@@ -133,16 +133,18 @@ Let $\mathfrak{l}$ be any field.
 \subsubsection{The Zariski topology on $\mathbb{P}^n$}
 \begin{notation}
     Recall that for
-    $\alpha \in \N^{n+1}$ $|\alpha| = \sum_{i=0}^{n} \alpha_i$
+    $\alpha \in \N^{n+1}$
-    and $x^\alpha = x_0^{\alpha_0} \cdot \ldots \cdot x_n^{\alpha_n}$.
+    \[|\alpha| = \sum_{i=0}^{n} \alpha_i  \text{ and } 
+    x^\alpha = x_0^{\alpha_0} \cdot \ldots \cdot x_n^{\alpha_n}.\]
 \end{notation}
 \begin{definition}[Homogeneous polynomials]
     Let $R$ be any ring and
-    $f = \sum_{\alpha \in \N^{n+1}} f_\alpha X^{\alpha}\in R[X_0,\ldots,X_n]$.
+    \[f = \sum_{\alpha \in \N^{n+1}} f_\alpha X^{\alpha}\in R[X_0,\ldots,X_n].\]
     We say that $f$ is \vocab{homogeneous of degree $d$}
-    if $|\alpha| \neq d \implies f_\alpha = 0$ .
+    if
+    \[|\alpha| \neq d \implies f_\alpha = 0.\]
     We denote the subset of homogeneous polynomials of degree $d$  by
-    $R[X_0,\ldots,X_n]_d \subseteq R[X_0,\ldots,X_n]$.
+    \[R[X_0,\ldots,X_n]_d \subseteq R[X_0,\ldots,X_n].\]
 \end{definition}
 \begin{remark}
     This definition gives $R$ the structure of a graded ring.
@@ -151,8 +153,8 @@ Let $\mathfrak{l}$ be any field.
     \label{ztoppn}
     Let $A = \mathfrak{k}[X_0,\ldots,X_n]$.%
     \footnote{As always, $\mathfrak{k}$ is algebraically closed}
-    For $f \in A_d = \mathfrak{k}[X_0,\ldots,
+    For $f \in A_d = \mathfrak{k}[X_0,\ldots,X_n]_d$,
-    _n]_d$, the validity of the equation $f(x_0,\ldots,x_{n}) = 0$
+    the validity of the equation $f(x_0,\ldots,x_{n}) = 0$
     does not depend on the choice of homogeneous coordinates, as
     \[
         f(\lambda x_0,\ldots, \lambda x_n) 0 \lambda^d f(x_0,\ldots,x_n).
@@ -320,8 +322,7 @@ Let $\mathfrak{l}$ be any field.
 \begin{proof}
     $\impliedby$ is clear.
     Let $\Vp(I) \subseteq \Vp(f)$.
-    If $x = (x_0,
+    If $x = (x_0,\ldots,x_n) \in \Va(I)$, then either $x = 0$ in
-    ldots,x_n) \in \Va(I)$, then either $x = 0$ in
     which case $f(x) = 0$ since $d > 0$
     or the point $[x_0,\ldots,x_n] \in \mathbb{P}^n$ is
     well-defined and belongs to $\Vp(I) \subseteq \Vp(f)$,
@@ -330,8 +331,8 @@ Let $\mathfrak{l}$ be any field.
     (\ref{hns3}).
 \end{proof}
 
-\begin{definition}
+\begin{definition}%
-    \footnote{This definition is not too important, the characterization in the following remark suffices.}.
+    \footnote{This definition is not too important, the characterization in the following remark suffices.}
     For a graded ring $R_\bullet$,
     let $\Proj(R_\bullet)$ be the set of $\fp \in \Spec R$
     such that $\fp$ is a homogeneous ideal and $\fp \not\supseteq R_+$.
@@ -432,17 +433,18 @@ Let $\mathfrak{l}$ be any field.
 \end{proof}
 
 \begin{remark}
-    If $R_\bullet$ is $\N$-graded and $\fp \in \Spec R_0$,
+    If $R_\bullet$ is $\N$-graded and $\fp \in \Spec R_0$, then
-    then $\fp \oplus R_+ = \{r \in R | r_0 \in  \fp\} $
+    \[\fp \oplus R_+ = \{r \in R | r_0 \in  \fp\}\]
     is a homogeneous prime ideal of $R$.
-    \[
+    \begin{IEEEeqnarray*}{rCl}
-        \{\fp  \in \Spec R | \fp \text{ is a homogeneous ideal of } R_\bullet\}
+        &&\{\fp  \in \Spec R | \fp \text{ is a homogeneous ideal of } R_\bullet\}\\
-        = \Proj(R_\bullet) \sqcup \{\fp \oplus R_+ | \fp \in \Spec R_0\}.
+        &=&\Proj(R_\bullet) \sqcup \{\fp \oplus R_+ | \fp \in \Spec R_0\}.
-    \]
+    \end{IEEEeqnarray*}
 \end{remark}
 
+\pagebreak
 \subsection{Dimension of $\mathbb{P}^n$}
-\begin{proposition}
+\begin{proposition}\,
     \begin{itemize}
         \item
               $\mathbb{P}^n$ is catenary.
@@ -450,11 +452,11 @@ Let $\mathfrak{l}$ be any field.
               $\dim(\mathbb{P}^n) = n$.
               Moreover, $\codim(\{x\} ,\mathbb{P}^n) = n$ for every $x \in \mathbb{P}^n$.
         \item
-              If  $X \subseteq \mathbb{P}^n$ is irreducible and $x \in X$,
+              If  $X \subseteq \mathbb{P}^n$ is irreducible and $x \in X$, then
-              then $\codim(\{x\}, X) = \dim(X) = n - \codim(X, \mathbb{P}^n)$.
+              \[\codim(\{x\}, X) = \dim(X) = n - \codim(X, \mathbb{P}^n).\]
         \item
               If $X \subseteq Y \subseteq \mathbb{P}^n$ are irreducible subsets,
-              then $\codim(X,Y) = \dim(Y) - \dim(X)$.
+              then \[\codim(X,Y) = \dim(Y) - \dim(X).\]
     \end{itemize}
 \end{proposition}
 \begin{proof}
@@ -472,12 +474,13 @@ Let $\mathfrak{l}$ be any field.
     $\codim(X,Z) = \codim(X \cap \mathbb{A}^n, Z \cap \mathbb{A}^n)$
     and $\codim(Y,Z) = \codim(Y \cap \mathbb{A}^n, Z \cap \mathbb{A}^n)$.
     Thus
-    \begin{align*}
+    \begin{IEEEeqnarray*}{rCl}
         \codim(X,Y) + \codim(Y,Z)
-        & = \codim(X \cap \mathbb{A}^n, Y \cap \mathbb{A}^n) + \codim(Y \cap \mathbb{A}^n, Z \cap \mathbb{A}^n) \\
+        & = &\codim(X \cap \mathbb{A}^n, Y \cap \mathbb{A}^n)\\
-        & = \codim(X \cap \mathbb{A}^n, Z \cap \mathbb{A}^n)\\
+        &   & + \codim(Y \cap \mathbb{A}^n, Z \cap \mathbb{A}^n) \\
-        & = \codim(X, Z)
+        & = &\codim(X \cap \mathbb{A}^n, Z \cap \mathbb{A}^n)\\
-    \end{align*}
+        & = &\codim(X, Z)
+    \end{IEEEeqnarray*}
     because $\mathfrak{k}^n$ is catenary and the first point follows.
     The remaining assertions can easily be derived from the first two.
 \end{proof}
@@ -492,17 +495,15 @@ Let $\mathfrak{l}$ be any field.
     If $X = \Vp(I)$ where $I \subseteq A_+ = \mathfrak{k}[X_0,\ldots,X_n]_+$
     is homogeneous, then $C(X) = \Va(I)$.
 \end{definition}
-\begin{proposition}
+\begin{proposition}\,
     \label{conedim}
     \begin{itemize}
         \item
               $C(X)$ is irreducible iff  $X$ is irreducible or $X = \emptyset$.
         \item
               If $X$ is irreducible, then
-
+              \[\dim(C(X)) = \dim(X) + 1\] and
-              $\dim(C(X)) = \dim(X) + 1$ and
+              \[\codim(C(X), \mathfrak{k}^{n+1}) = \codim(X, \mathbb{P}^n).\]
-
-              $\codim(C(X), \mathfrak{k}^{n+1}) = \codim(X, \mathbb{P}^n)$
     \end{itemize}
 \end{proposition}
 \begin{proof}
@@ -525,7 +526,7 @@ Let $\mathfrak{l}$ be any field.
     Hence $\dim(C(X)) \ge  1 + d$ and
       $\codim(C(X), \mathfrak{k}^{n+1}) \ge n-d$.
     Since
-    $\dim(C(X)) + \codim(C(X), \mathfrak{k}^{n+1}) = \dim(\mathfrak{k}^{n+1}) = n+1$,
+    \[\dim(C(X)) + \codim(C(X), \mathfrak{k}^{n+1}) = \dim(\mathfrak{k}^{n+1}) = n+1,\]
     the two inequalities must be equalities.
 \end{proof}
 \subsubsection{Application to hypersurfaces in $\mathbb{P}^n$}

inputs/varieties.tex

@@ -315,7 +315,7 @@ The following is somewhat harder than in the affine case:
     \end{itemize}
 \end{remark}
 \subsubsection{Examples of categories}
-\begin{example}
+\begin{example}\,
     \begin{itemize}
         \item
               The category of sets.
@@ -357,7 +357,7 @@ The following is somewhat harder than in the affine case:
     addition $\Hom_\mathcal{B}(X,Y) = \Hom_\mathcal{A}(X,Y)$ for
     arbitrary $X,Y \in \Ob(\mathcal{B})$.
 \end{definition}
-\begin{example}
+\begin{example}\,
     \begin{itemize}
         \item
               The category of abelian groups is a full subcategory of the
@@ -407,7 +407,7 @@ The following is somewhat harder than in the affine case:
     It is called an \vocab{equivalence of categories} if it is full,
     faithful and essentially surjective.
 \end{definition}
-\begin{example}
+\begin{example}\,
     \begin{itemize}
         \item
               There are \vocab[Functor!forgetful]{forgetful functors}
@@ -552,7 +552,7 @@ The following is somewhat harder than in the affine case:
 
 
 \end{proof}
-\begin{proposition}[About affine varieties]
+\begin{proposition}[About affine varieties]\,
     \label{propaffvar}
     \begin{itemize}
         \item