From dfd49d15713686f642c396c976fd5dba3f4b5d4e Mon Sep 17 00:00:00 2001 From: arnaucube Date: Fri, 2 Jan 2026 19:33:26 +0100 Subject: [PATCH] port notes on A-algebras & Noether normalization & integral closures (#3) * port notes on A-algebras & integral elems * add R.4.3's proof & 4.4 * add Noether normalization proof --- .github/workflows/typos.yml | 2 +- commutative-algebra-notes.pdf | Bin 333808 -> 351227 bytes commutative-algebra-notes.tex | 274 ++++++++++++++++++++++++++++++---- galois-theory-notes.pdf | Bin 346493 -> 346500 bytes galois-theory-notes.tex | 4 +- 5 files changed, 251 insertions(+), 29 deletions(-) diff --git a/.github/workflows/typos.yml b/.github/workflows/typos.yml index de7bc65..1c0695a 100644 --- a/.github/workflows/typos.yml +++ b/.github/workflows/typos.yml @@ -1,7 +1,7 @@ name: typos on: pull_request: - branches: [ main ] + branches: [ master ] types: [ready_for_review, opened, synchronize, reopened] push: branches: diff --git a/commutative-algebra-notes.pdf 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R$ with $ab \in P$ and $P \neq R$ ($P$ a prime ideal), implies $a in P$ or $b \in P$. \end{defn} -\begin{defn}{principal ideal} +\begin{defn}{}{principal ideal} generated by a single element, $(a)$. $(a)$: principal ideal, the set of all multiples $xa$ with $x \in R$. \end{defn} -\begin{defn}{maximal ideal} +\begin{defn}{}{maximal ideal} $\mM \subset A$ ($A$ ring) with $m \neq A$ and there is no ideal $I$ strictly between $\mM$ and $A$. ie. if $\mM$ maximal and $\mM \subseteq I \subseteq A$, either $\mM=I$ or $I=A$. \end{defn} -\begin{defn}{unit} +\begin{defn}{}{unit} $x \in A$ such that $xy=1$ for some $y \in A$. ie. element \emph{which divides 1}. \end{defn} -\begin{defn}{zerodivisor} +\begin{defn}{}{zerodivisor} $x \in A$ such that $\exists 0 \neq y \in A$ such that $xy=0 \in A$. ie. $x$ \emph{divides 0}.. If a ring does not have zerodivisors is an integral domain. \end{defn} -\begin{defn}{prime spectrum - $Spec(A)$} +\begin{defn}{}{prime spectrum - $Spec(A)$} set of prime ideals of $A$. ie. $$Spec(A) = \{ P ~|~ P \subset A~ \text{is a prime ideal} \}$$ \end{defn} -\begin{defn}{integral domain} +\begin{defn}{}{integral domain} Ring in which the product of any two nonzero elements is nonzero. ie. no zerodivisors. @@ -140,15 +140,15 @@ Every field is an integral domain, not the converse. \end{defn} -\begin{defn}{principal ideal domain - PID} +\begin{defn}{}{principal ideal domain - PID} integral domain in which every ideal is principal. ie. ie. $\forall I \subset R,~ \exists~ a \in I$ such that $I = (a) = \{ ra ~|~ r \in R \}$. \end{defn} -\begin{defn}{nilpotent} +\begin{defn}{}{nilpotent} $a \in A$ such that $a^n=0$ for some $n>0$. \end{defn} -\begin{defn}{nilrad A} +\begin{defn}{}{nilrad A} set of all nilpotent elements of $A$; is an ideal of $A$. if $nilrad A = 0 ~\Longrightarrow$ $A$ has no nonzero nilpotents. @@ -157,11 +157,11 @@ $$nilrad A = \bigcap_{P \in Spec(A)} P$$ \end{defn} -\begin{defn}{idempotent} +\begin{defn}{}{idempotent} $e \in A$ such that $e^2=e$. \end{defn} -\begin{defn}{radical of an ideal} +\begin{defn}{}{radical of an ideal} $$rad I = \{ f \in A | f^n \in I~ \text{for some} n \}$$ $rad I$ is an ideal. @@ -171,7 +171,7 @@ $rad I = \bigcap_{\substack{P \in \operatorname{Spec}(A)\\ P \supset I}} P$ \end{defn} -\begin{defn}{local ring} +\begin{defn}{}{local ring} A \emph{local ring} has a unique maximal ideal. Notation: local ring $A$, its maximal ideal $\mM$, residue field $K=A/\mM$: @@ -233,7 +233,7 @@ -\subsection{Lemmas, propositions and corollaries} +\subsection{Zorn's lemma and Jacobson radicals} Let $\Sigma$ be a partially orddered set. Given subset $S \subset \Sigma$, an \emph{upper bound} of $S$ is an element $u \in \Sigma$ such that $s0$ we have two cases: + \begin{itemize} + \item $y_1, \ldots, y_n$ are algebraically independent over $K$, then $A \cong K[y_1, \ldots, y_n]$, so that $A$ is a finite module over itself. + \item $y_1, \ldots, y_n$ are algebraically dependent over $K$, + $$\exists 0 \neq f \in K[y_1, \ldots, y_n] ~\text{s.th}~ f(y_1, \ldots, y_n)=0$$ + \end{itemize} + + Goal: is to change variables so that $f$ becomes monic in one of the variables; this allows to express one generator as an integral element over the others. + + Following from Lemma \ref{R.4.6.L}, define new variables $y^*_1, \ldots, y^*_{n-1} \in A$ such that $y_n$ is integral over + $$A^* = K[y^*_1, \ldots, y^*_{n-1}] ~\text{and}~ A=A^*[y_n]$$ + + By inductive hypothesis on $A^*,~~ \exists~ z_1, \ldots, z_m \in A^*$ algebraically independent over $K$ and with $A^*$ finite over $B=K[z_1, \ldots, z_m]$. + + Since $y_n$ integral over $A^* ~~\Longrightarrow~ A^*[y_n]$ is finite over $A^*$.\\ + Therefore, each step of $B \subset A^* \subset A^*[y_n]=A$ is finite, and $A$ is finite over $B$ as required. +\end{proof} + + +\begin{eg}{ } + $A = K[X,Y]/(XY-1)$. $Y$ is algebraic over $K[X]$, but not integral over $K[Y]$. + + This corresponds to the fact that the hyperbola $XY=1$ has the line $X=0$ as an asymptotic line (so that its projection to the $X$-axis misses a root over $X=0$). + + Take $X' = X- \epsilon Y$ as the element of $A$ instead of $X$; then the relation becomes $(X' + \epsilon Y) Y=1$, monic in $Y$ if $\epsilon \neq 0$. + + This corresponds geometrically to tilting the hyperbola a little before projecting, so that no longer has a vertical asymtotic line. +\end{eg} + \newpage @@ -1353,7 +1575,7 @@ Prove that $\bigoplus A/I_i$ is a Noetherian $A$-module, and deduce that if $\bi \begin{enumerate}[i.] \item by Corollary \ref{R.3.5} (i), if $M_i$ Noetherian modules, then $\bigoplus M_i$ is Noetherian. 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