From 15f00f14ffc7c5adbfa56ffe00f1ede2fafee2bd Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 4 Jan 2026 19:20:43 +0100 Subject: [PATCH] add lemma 4.6 proof, polish Noether normalization lemma's proof, add aux lemma on integrality implies finiteness --- .github/workflows/typos.toml | 4 + commutative-algebra-notes.pdf | Bin 351227 -> 367209 bytes commutative-algebra-notes.tex | 143 ++++++++++++++++++++++++++-------- typos.toml | 7 -- 4 files changed, 114 insertions(+), 40 deletions(-) delete mode 100644 typos.toml diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml index de9c696..acadd66 100644 --- a/.github/workflows/typos.toml +++ b/.github/workflows/typos.toml @@ -1,3 +1,7 @@ +# usage: +# install `typos`: https://github.com/crate-ci/typos +# run: typos -c .github/workflows/typos.toml + [default.extend-words] iddeal = "ideal" iddeals = "ideals" diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 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\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. @@ -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 $K=A/\mM$: @@ -262,7 +262,7 @@ A subset $S \subset \Sigma$ is \emph{totally ordered} if for every pair $s_1,s_2 Every non-unit of $A$ is contained in a maximal ideal. \end{cor} -\begin{defn}{}{Jacobson radical} +\begin{defn}{}[Jacobson radical] The \emph{Jacobson radical} of a ring $A$ is the intersection of all the maximal ideals of $A$. Denoted $Jac(A)$. @@ -698,7 +698,7 @@ $A[\psi] \subset End M$ is the subring geneerated by $A$ and the action of $\psi \subsection{Sequences} -\begin{defn}{R.2.9.a}{Exact Sequence} +\begin{defn}{R.2.9.a}[Exact Sequence] Let a sequence of homomorphisms $$L \stackrel{\alpha}{\longrightarrow} M \stackrel{\beta}{\longrightarrow} N$$ It is \emph{exact} at $M$ if $im(\alpha)=ker(\beta)$. @@ -707,7 +707,7 @@ ie. $\beta \circ \alpha = 0$ and $\alpha$ maps surjectively to $ker(\beta)$. \end{defn} -\begin{defn}{R.2.9.b}{Short Exact Sequence (s.e.s.)} \label{2.9} +\begin{defn}{R.2.9.b}[Short Exact Sequence (s.e.s.)] \label{2.9} $$0 \longrightarrow L \stackrel{\alpha}{\longrightarrow} M \stackrel{\beta}{\longrightarrow} N \longrightarrow 0$$ is exact $\Longleftrightarrow~ L \subset M$ and $N=M / L$. @@ -815,7 +815,7 @@ $$ \section{Noetherian rings (and modules)} -\begin{defn}{}{Ascending Chain Condition} +\begin{defn}{}[Ascending Chain Condition] A partially orddered set $\Sigma$ has the \emph{ascending chain condition} (a.c.c.) if every chain $$s_1 \leq s_2 \leq \ldots \leq s_k \leq \ldots$$ eventually breaks off, that is, $s_k = s_{k+1} = \ldots$ for some $k$. @@ -824,7 +824,8 @@ $$ $\Longrightarrow~ \Sigma$ has the a.c.c. iff every non-empty subset $S \subset \Sigma$ has a maximal element.\\ \hspace*{2em} if $\empty \neq S \subset \Sigma$ does not have a maximal element, choose $s_1 \in S$, and for each $s_k$, an element $s_{k+1}$ with $s_k < s_{k+1}$, thus contradicting the a.c.c. -\begin{defn}{R.3.2}{Noetherian ring} +\subsection{Noetherian rings and modules} +\begin{defn}{R.3.2}[Noetherian ring]\\ Let $A$ a ring; 3 equivalent conditions: \begin{enumerate}[i.] \item the set $\Sigma$ of ideals of $A$ has the a.c.c.; in other words, every increasing chain of ideals @@ -839,7 +840,7 @@ $\Longrightarrow~ \Sigma$ has the a.c.c. iff every non-empty subset $S \subset \ TODO \end{proof} -\begin{defn}{R.3.4.D}{Noetherian modules} +\begin{defn}{R.3.4.D}[Noetherian modules]\\ An $A$-module $M$ is Noetherian if the submoles of $M$ have the a.c.c.,\\ that is, ay increasing chain $$M_1 \subset M_2 \subset \ldots \subset M_k \subset \ldots$$ @@ -924,8 +925,9 @@ As in with rings, it is equivalent to say that \vspace{0.5cm} -\begin{thm}{R.3.6}{Hilbert basis theorem} \label{hilbert-basis} - if $A$ a Noetherian ring, then so is the polynomial ring $A[x]$. +\subsection{Hilbert basis} +\begin{thm}{R.3.6}{Hilbert basis theorem.} \label{hilbert-basis} + If $A$ a Noetherian ring, then so is the polynomial ring $A[x]$. \end{thm} \begin{proof} Prove that any ideal $I \subset A[x]$ is fingen. @@ -973,6 +975,7 @@ As in with rings, it is equivalent to say that Thus, any ideal of $A[x]$ is finitely generated. \end{proof} +\vspace{0.5cm} \begin{cor}{R.3.6.C} if $A$ a Noetherian ring, and $\psi: A \longrightarrow B$ a ring homomorphism such that $B$ is a fingen extension ring of $\psi(A)$, then $B$ is Noetherian. @@ -993,11 +996,11 @@ As in with rings, it is equivalent to say that \vspace{1cm} -\section{Finite ring extensions and Noether normalisation} +\section{Finite ring extensions and Noether normalization} \subsection{A-algebras and integral domains} -\begin{defn}{}{A-algebra.} +\begin{defn}{}[A-algebra] An $A$-algebra is a ring $B$ with a ring homomorphism $\psi: A \longrightarrow B$. $B$ is an $A$-module with multiplication defined by $\psi(a) \cdot b~~~ (a \in A, b \in B)$. @@ -1132,8 +1135,40 @@ As in with rings, it is equivalent to say that \end{enumerate} \end{proof} +\begin{lemma}{4.3.Aux}[Integrality implies finiteness] \label{integral-implies-finite} + If $y_n$ integral over $A$ then $A[y_n]$ is finite over $A$ -\begin{defn}{4.4}{Integral closure.} + This extends on point (b) from the previous proposition \ref{R.4.3}: +\end{lemma} +\begin{proof} + Suppose $y_n$ is integral over $A$. By definition $\exists~~ f \in A[T]$, with $f$ monic, such that $f(y_n)=0$. + + Let $deg(f)=d$, so that for $f(y_n)=0$ we have + $$y_n^d + a_{d-1} y_n^{d-1} + \ldots + a_1 y_n + a_0 = 0 ~~~~ a_i \in A$$ + + Since it is monic (leading coefficient is $1$), we can rearrange it to isolate the highest power: + \begin{equation} + y_n^d = -(a_{d-1} y_n^{d-1} + \ldots + a_1 y_n + a_0) + \label{eq:yn} + \end{equation} + Thus $y_n^d$ can be written using lower powers of $y_n$ with coefficients in $A$. + + \vspace{0.5cm} + Consider any element $p \in A[y_n]$, $p = c_m y_n^m + c_{m-1} y_n^{m-1} + \ldots + c_0$. + + if $m0$ 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 $K$, - $$\exists 0 \neq f \in K[y_1, \ldots, y_n] ~\text{s.th}~ f(y_1, \ldots, y_n)=0$$ + \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, with $m=n$. + \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. + Want $f$ to be \emph{monic}, so that $y_n$ is integral over new defined variables $y_1^*, \ldots, y_{n-1}^*$. In other words, want some polynomial like + $$y_n^d+ a_{d-1} y_n^{d-1} + \ldots + a_1 y_n + a_0 = 0~~~~~~~a_i \in K[y_1, \ldots, y_{n-1}]$$ + \hspace*{2em} ie. monic, so that by definition (\ref{R.4.1}), $y_n$ is integral over $K[y_1, \ldots, y_{n-1}]$. - 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 + $~\longrightarrow~$ Change variables so that $f$ becomes monic in one of the variables ($y_n$); this allows to express one generator ($y_n$) as an integral element over the others. + + + \vspace{0.3cm} + Following from Lemma \ref{R.4.6.L}, define the new variables $y^*_1, \ldots, y^*_{n-1} \in A$ such that $y_n$ is integral over $$A^* = K[y^*_1, \ldots, y^*_{n-1}] ~\text{and}~ A=A^*[y_n]$$ + Setting $y_i^* = y_i - y_n^{r_i}$, so that $y_i = y_i^* + y_i^{r_i}$ $\forall i \in [n-1],~~ r_1, \ldots, r_{n-1} \geq 1 \in \mathbb{Z}$. + + Use those new variables at $f(y_1, \ldots, y_n)=0$: + $$f(y_1^* + y_n^{r_1}, y_2^* + y_n^{r_2}, \ldots, y_n^* + y_n^{r_n}, y_n) = 0$$ + + Then the highest power of $y_n$ in each term of $f$ will look like $y_n^{(\sum a_i r_i)}$, and with $r_i$ growing fast enough we ensure that each monomial in $f$ produces a unique power of $y_n$. + + Then we have $c \cdot y_n^D + \text{(terms with lower powers of $y_n$)} = 0$ with $c \in K \setminus \{0\}$. So that dividing by $c$ we get the shape $y_n^D + \ldots =0$, thus $y_n$ is integral over $A^* = K[y_1^*, \ldots, y_{n-1}^*]$. + + \vspace{0.3cm} + Now, $A$ is a finite module over $A^*=K[y_1^*, \ldots, y_{n-1}^*]$, so that $A^*$ is generated by $n-1$ elements. + 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 $A^*$.\\ + Since $y_n$ integral over $A^* ~~\Longrightarrow~ A^*[y_n]$ is finite over $A^*$ (by \ref{integral-implies-finite}).\\ Therefore, each step of $B \subset A^* \subset A^*[y_n]=A$ is finite, and $A$ is finite over $B$ as required. \end{proof} +\vspace{0.5cm} \begin{eg}{ } $A = K[X,Y]/(XY-1)$. $Y$ is algebraic over $K[X]$, but not integral over $K[Y]$. @@ -1212,7 +1289,7 @@ Recall: a $K$-algebra $A$ is fingen over $K$ if $A=K[y_1, \ldots, y_n]$ for some 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. + This corresponds geometrically to tilting the hyperbola a little before projecting, so that no longer has a vertical asymptotic line. \end{eg} diff --git a/typos.toml b/typos.toml deleted file mode 100644 index 0788b15..0000000 --- a/typos.toml +++ /dev/null @@ -1,7 +0,0 @@ -# usage: -# install `typos`: https://github.com/crate-ci/typos -# run: typos --config typos.toml - -[default.extend-words] -groth = "groth" -pinter = "pinter"