From 6c35d6cbcce437f960a587074f6f4dde1027694f Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 11 Jan 2026 22:28:29 +0100 Subject: [PATCH] add weak nullstellensatz theorem and proof --- .github/workflows/typos.toml | 2 + commutative-algebra-notes.pdf | Bin 367209 -> 370173 bytes commutative-algebra-notes.tex | 135 ++++++++++++++++++++++++++-------- 3 files changed, 105 insertions(+), 32 deletions(-) diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml index acadd66..8effe42 100644 --- a/.github/workflows/typos.toml +++ b/.github/workflows/typos.toml @@ -16,4 +16,6 @@ nd = "nd" # names Strang = "Strang" +Pinter = "Pinter" Bootle = "Bootle" +Groth = "Groth" diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index a45cec5a6ff92796ba4f9e9db9fd3da99d242a88..8f8cf2c0284763c5e14e1c7177c1ba3ae3f97aef 100644 GIT binary patch delta 71508 zcmZs?Q*hu<^z9qloY=N)PwY%=+y2I$m=oJJCbn&JVq5q3Kj+k~I(6?$*Y1a|e(GJH 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--git a/commutative-algebra-notes.tex b/commutative-algebra-notes.tex index 84686ae..108b445 100644 --- a/commutative-algebra-notes.tex +++ b/commutative-algebra-notes.tex @@ -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. @@ -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, with $m=n$. + \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} 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 $K[y_1, \ldots, y_{n-1}]$. + $$y_n^d+ a_{d-1} y_n^{d-1} + \ldots + a_1 y_n + a_0 = 0~~~~~~~a_i \in K[y_1^*, + \ldots, y_{n-1}^*]$$ + \hspace*{2em} ie. monic, so that by definition (\ref{R.4.1}), $y_n$ is + integral over $K[y_1^*, \ldots, y_{n-1}^*]$. $~\longrightarrow~$ 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 - $$A^* = K[y^*_1, \ldots, y^*_{n-1}] ~\text{and}~ A=A^*[y_n]$$ + $$A^* = K[y^*_1, \ldots, y^*_{n-1}], ~~\text{and}~ A=A^*[y_n]$$ - Setting $y_i^* = y_i - y_n^{r_i}$, so that $y_i = y_i^* + y_i^{r_i}$ $\forall i \in [n-1],~~ r_1, \ldots, r_{n-1} \geq 1 \in \mathbb{Z}$. + 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 $f(y_1, \ldots, y_n)=0$: - $$f(y_1^* + y_n^{r_1}, y_2^* + y_n^{r_2}, \ldots, y_n^* + y_n^{r_n}, y_n) = 0$$ + $$f(y_1^* + y_n^{r_1}, y_2^* + y_n^{r_2}, \ldots, y_{n-1}^* + y_n^{r_{n-1}}, y_n) = 0$$ 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} - Now, $A$ is a finite module over $A^*=K[y_1^*, \ldots, y_{n-1}^*]$, so that $A^*$ is generated by $n-1$ elements. + 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 $A^*$ (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} @@ -1292,6 +1300,69 @@ Recall: a $K$-algebra $A$ is fingen over $K$ if $A=K[y_1, \ldots, y_n]$ for some 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