From bafd55f27f153ba5535bcefe0c8f70cbc02c2ec8 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 11 Jan 2026 22:35:04 +0100 Subject: [PATCH] Noether normalization & weak nullstellensatz thm (#4) * add lemma 4.6 proof, polish Noether normalization lemma's proof, add aux lemma on integrality implies finiteness * add weak nullstellensatz theorem and proof --- .github/workflows/typos.toml | 6 + commutative-algebra-notes.pdf | Bin 351227 -> 370173 bytes commutative-algebra-notes.tex | 232 ++++++++++++++++++++++++++++------ typos.toml | 7 - 4 files changed, 196 insertions(+), 49 deletions(-) delete mode 100644 typos.toml diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml index de9c696..8effe42 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" @@ -12,4 +16,6 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zg|;@Dq1ay94}>4oCI~}$@b57m1Q(uQ5)v(u890&+jU%HbP%>ijLK;SFdlh(mYe}>H zMf+V)0@~dXZwMu+Rzbo5w4gA)>xvBiY*IIkY2p|}*bJxv)zA=X0Kfz+VhmmZM6&)` zEjr3KEo$vYQUr!E&o?#PB#0KqF!H$X^D7$sPwWf}SB|tK2#K?UU~Yuo?i^l%kT9EO zV>u!AyT;-9wBW*FD`qD#Pqe;7om^>EU2>N!m%BS9^c$?l?WXN-5?v@$?oqi}tUX@!~7V+lyP+8m$J-GvX0^F)uTMxW<{Q zm{mm%OG57w(2Fl3CLi9*Ux|;X*^FSBV@#ov&A`9wc{m?KZceyjowKpdsN4}`?ue&w z{&DEeb7ebICeB}lH_D-7O!n;UFM3rPwDB>rk0$. \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 $K=A/\mM$: @@ -241,7 +241,7 @@ A \emph{maximal element} of $\Sigma$, is $m \in \Sigma$ such that $m0$ 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 by + definition \ref{R.4.6.D} $A \cong K[y_1, \ldots, y_n]$, 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 - $$A^* = K[y^*_1, \ldots, y^*_{n-1}] ~\text{and}~ A=A^*[y_n]$$ + $~\longrightarrow~$ 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 $y_i = y_i^* + y_n^{r_i}$ $\forall i \in [n-1],~~ r_1, \ldots, r_{n-1} \geq 1 \in \mathbb{Z}$. + + Use those new variables at $f(y_1, \ldots, y_n)=0$: + $$f(y_1^* + y_n^{r_1}, y_2^* + y_n^{r_2}, \ldots, y_{n-1}^* + y_n^{r_{n-1}}, y_n) = 0$$ + + 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} + Induction:\\ + Since $y_n$ integral over $A^* ~~\Longrightarrow~ A=A^*[y_n]$ is finite over $A^*=K[y_1^*, \ldots, y_{n-1}^*]$ (by \ref{integral-implies-finite}). 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} +\vspace{0.5cm} \begin{eg}{ } $A = K[X,Y]/(XY-1)$. $Y$ is algebraic over $K[X]$, but not integral over $K[Y]$. @@ -1212,9 +1297,72 @@ 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} +\vspace{1cm} + +\subsection{Weak Nullstellensatz} + +\begin{prop}{R.4.9}\label{R.4.9} + let $A \subset B$ be an integral extension of integral domain,\\ + then $A$ is a field $\Longleftrightarrow~ B$ is a field. +\end{prop} +\begin{proof} + $\Longrightarrow$:\\ + let $0 \neq x \in B$, then $\exists~~ x^n + a_{n-1} x^{n-1} + \ldots + a_0 = 0 ~~~~ a_i \in A$, monic. + + Since $A$ is a field, $\exists$ inverse, observe that: + \begin{align*} + x^n &+ a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0\\ + x(x^{n-1} &+ a_{n-1} x^{n-2} + \ldots + a_1) = - a_0\\ + -a_0^{-1}(x^{n-1} &+ a_{n-1} x^{n-2} + \ldots + a_1) = x^{-1} \in B + \end{align*} + thus there exists inverse in $B$, so $B$ is a field too. + + $\Longleftarrow$:\\ + if $B$ is a field and $0 \neq x \in A$, then $x^{-1} \in B$, so $x^{-1}$ is integral over $A$. + + So there is a relation of the form + $$(x^{-1})^n + a_{n-1} (x^{-1})^{n-1} + \ldots + a_0 =0$$ + + Therefore + \begin{align*} + (x^{-1})^n &+ a_{n-1} (x^{-1})^{n-1} + \ldots + a_0 = 0\\ + (x^{-1})^n &= -a_{n-1} (x^{-1})^{n-1} - \ldots - a_0\\ + x^{-n} &= -a_{n-1} x^{-n+1} - \ldots - a_0 ~~\text{(mult by $x^{n-1}$)}\\ + x^{-n+(n-1)} &= -a_{n-1} x^{-n+1+(n-1)} - \ldots - a_0 x^{n-1}\\ + x^{-1} &= -a_{n-1} - \ldots - a_0 x^{n-1} \in A + \end{align*} + + thus there exists inverse in $A$, so $A$ is a field too. +\end{proof} + +\begin{thm}{R.4.10}[Weak Nullstellensatz] + let $k$ a field, $K$ a $k$-algebra which + \begin{enumerate} + \item is finitely generated as a $k$-algebra + \item is a field + \end{enumerate} + Then $K$ is algebraic over $k$, so that $k \subset K$ is a finite field + extension. That is, $[K:k] < \infty$. +\end{thm} +\begin{proof} + by Noether normalization \ref{noether-normalization}, $\exists~ z_1, \ldots, + z_m \in K$ which are algebraically independent, and such that $K$ is finite + over $A=k[z_1, \ldots, z_m]$. +\\ + Now we're at the situation of \ref{R.4.9}: + + $A \subset K$ is integral, $K$ is a field $~~\Longrightarrow~$ therefore $A$ is a field. + + Since $z_1, \ldots, z_m \in K$ are algebraically independent,\\ + \hspace*{2em}$\Longrightarrow~ A=k[z_1, \ldots, z_m]$ is a polynomial ring in $m$ indeterminates, and this is a field only if $m=0$, and $K$ is finite over $k$. +\end{proof} + + + + \newpage 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"