From 5e08a390b92df422074b7350459ed0d03a713578 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 25 Jan 2026 22:04:47 +0100 Subject: [PATCH] port Nullstellensatz & Varieties notes (ch5) (#6) * 5.1, Corollary 5.2 * variety def, Nullstellensatz proof, correspondences V - I * port notes on propositions 5.3, 5.7, 5.8 * add missing conclusion at proposition 5.3 --- .github/workflows/typos.toml | 2 + commutative-algebra-notes.pdf | Bin 381425 -> 401886 bytes commutative-algebra-notes.tex | 323 +++++++++++++++++++++++++++++++++- 3 files changed, 323 insertions(+), 2 deletions(-) diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml index 8effe42..0455301 100644 --- a/.github/workflows/typos.toml +++ b/.github/workflows/typos.toml @@ -3,9 +3,11 @@ # run: typos -c .github/workflows/typos.toml [default.extend-words] +ieal = "ideal" iddeal = "ideal" iddeals = "ideals" allpha = "alpha" +fieldd = "field" # strings that are not a typo: thm = "thm" diff --git 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zs|fLGLlyn0kPB87z|N%8-7f8F!lKZ1#BPY*!nmrDrX~sp>9IYsaLi#srdwgZ+A8! zK4nI_?0ko)XtmSxZ33r(2b>ipwkwWJ($(K{zM`oswefiaREypJxu=c-QP7{(#e+&6 zy3UQXA3pZxQ2lZuRpYa_o~IcR-ZBLi32KaRu=&jKJOQH5s};6&%oCbf6FYR*tJh3! z7Gx+_PxneYLkgO(uuMd$;HwV*`xD1|4X&4X2{OW}c)M1?DJF^I5h(V)nZ0wL`OTx^ zar>wq_Qo?Iw$0!6Vj3lITM#_RFda*oq`U#-CkN!}C>KD!*0e&f_D zPfOsZS|8;F!Etd5PvS`Xqlr9n1W!EaeOpv=$koGjDRt}}yfiDeeXg-~DtaZpUNq52 zkvK2W#?JaZPCsK$N1YhToX_Z|y}$}KtJmhYBjsCznJz;T->C0SCZFqn_&svy+kwo# zlKWGi|740y*mRm}AHq&$+s*M`u9R^SkH+7-wl-LfeC;E=aaUbqhluN62iq(VDNH+r z4OU!H5`wwv7lk@ptq$n8)vr~*70 \}$. + + Want to show that if a polynomial vanishes at all points where $g_1, \ldots, g_m$ vanish, then $f \in rad(g_1, \ldots, g_m)$. + + Consider the ring $k[X_1, \ldots, X_n, Y]$ and the ideal $J'$ generated by $\{ g_1, \ldots, g_m, 1-Y f \}$ + + Suppose there is a point $(a_1, \ldots, a_n, a_{n+1})$ that is a zero of $J'$. ie. + $$\exists~ (a_1, \ldots, a_n, a_{n+1}) \in V(J')$$ + + Since $g_i(a)=0$, our hypothesis says $f(a)=0$. However, the last generator $(1-Yf)$ requires + $$1 - a_{n+1} f(a) = 0 ~~\Longrightarrow~ \text{implies}~ 1 - a_{n+1} \cdot 0 = 0 ~\Longrightarrow~ 1-0=0$$ + a contradiction. + + Therefore, $V(J') = \emptyset$. + + \vspace{0.4cm} + + Since $V(J')=\emptyset$, by the Weak Nullstellensatz/Zariski (\ref{zariski}),\\ + \hspace*{2em} if $V(J')=\emptyset$ then $J'=(1)$, so $1 \in J'=(1)$. + + Every element in an ideal is a linear combination of its generators: $J'$ is generated by $\{ g_1, \ldots, g_m, 1-Yf \}$ + + $$\Longrightarrow~ \forall j \in J',~~ j=(\sum \text{(polynomial)} g_i) + \text{(polynomial)} \cdot (1 - Yf)$$ + + which, since $1 \in J'$, + $$1= \left( \sum_{i=1}^m p_i(X,Y) g_i(X) \right) + q(X,Y) \cdot (1 - Y f(X))$$ + + substitute $Y=1/f$, + $$1= \left( \sum_{i=1}^m p_i(X,\frac{1}{f}) g_i(X) \right) + q(X,\frac{1}{f}) \cdot \underbrace{(1 - \frac{1}{f} f(X))}_{0}$$ + thus + $$1= \sum_{i=1}^m p_i(X,\frac{1}{f}) g_i(X)$$ + multiply by $f^n$, + $$f^n= \sum_{i=1}^m A_i(X) g_i(X)$$ + + thus $f^n$ is a linear combination of $g_i$.\\ + Thus $f^n \in J$, so $f \in rad~J$. +\end{enumerate} +\end{proof} + + +\vspace{0.4cm} +\subsection{Irreducible varieties} + +\begin{defn}{5.7}[Irreducible variety] + a variety $X \subset k^n$ is \emph{irreducible} if it is nonempty and not the union of two proper subvarieties; that is, if + $$X = X_1 \cup X_2 ~~\text{for varieties}~ X_1, X_2 ~\Longrightarrow~ X= X_1 ~\text{or}~ X_2$$ +\end{defn} + +\begin{prop}{5.7} \label{5.7} + a variety $X$ is irreducible iff $I(X)$ is prime. +\end{prop} +\begin{proof} + set $I=I(X)$. + + if $I$ not prime, then $f,g \in A \setminus I$ be such that $fg \in I$. + + Define new ideals + $$J_1=(I, f) ~~\text{and}~~ J_2=(I, g)$$ + + Then, since $f \not\in I(X)$, it follows that $V(J_1) \subsetneq X$ + + \hspace*{2em}$\Longrightarrow$ so $X=V(J_1) \cup V(J_2)$ is reducible. + + The converse is similar. +\end{proof} + +\begin{cor}{5.8} \label{cor.5.8} + let $k$ algebraically closed field. Then $V$ and $I$ induce one-to-one correspondences + + \begin{align*} + \{ \text{radical ideals}~J~\text{of}~k[X_1, \ldots, X_n] \} &\longleftrightarrow \{ \text{varieties}~ X \subset k^n \}\\ + \{ \text{prime ideals}~P~\text{of}~k[X_1, \ldots, X_n] \} &\longleftrightarrow \{ \text{irreducible varieties}~ X \subset k^n \} + \end{align*} + + Therefore, + $$Spec k[X_1, \ldots, X_n] = \{ \text{irreducible varieties}~ X \subset k^n \}$$ +\end{corollary} +\end{cor} + +\begin{prop}{5.8} \label{5.8} + let $A= k[x_1, \ldots, x_n]$ a fingen $k$-algebra ($k$ an algebraically closed field). + + Write $J$ for the ideal of relations holding between $x_1, \ldots, x_n$, so that $A=k[X_1, \ldots, X_n]/J$. + + Then there is a one-to-one correspondence + $$Spec A \longleftrightarrow \{ \text{irreducible subvarieties}~ X \subset V(J) \}$$ +\end{prop} +\begin{proof} + By definition, $Spec A = \{ P ~|~ P \subset A ~\text{is prime ideal} \}$. + + By Corollary \ref{cor.5.8}: + $$\{ \text{prime ideals}~P~\text{of}~k[X_1, \ldots, X_n] \} &\longleftrightarrow \{ \text{irreducible varieties}~ X \subset k^n \}$$ + + About varieties: + \begin{itemize} + \item $I(X)$ in $R=k[X_1, \ldots, X_n]$ is the ideal of the variety $X$\\ + \hspace*{2em}ie. the set of all polynomials that vanish on every point of $X$. + \item $I(X)$ in $A=k[X_1, \ldots, X_n]/J$, we're not looking at all possible polynomials but at the residue classes. + \end{itemize} + + If $P$ is prime in $A~~\Longrightarrow~$ it must correspond to some prime ideal $\mathfrak{P}$ in $R$ (also $J \subset R$). + + Then from Nullstellensatz (\ref{nullstellensatz}), every prime ideal $\mathfrak{P}$ in $R$ is the ideal of some irreducible variety $X$.\\ + \hspace*{2em}$\Longrightarrow~~ \mathfrak{P}=I(X)$. + + + Since we're restricted to the ring $A=k[X_1, \ldots, X_n]/J$, the ideal $P$ are the elements of $I(X)$ viewed through the lens of the quotient\\ + $$\Longrightarrow~~ P = \{ f+J ~|~ f \in I(X) \}~~ = I(X) ~\text{mod}~J$$ + + Now, $J$ is the set of equations defining our "universe" $V(J)$. + + Since $A=R/J ~~ \Longrightarrow~$ we thus have $J \subseteq R$. + + Also we have a correspondence between $A=R/J$ and $R$. + + Let $\mathfrak{P}$ be the preimage of $P$ in $R=k[X_1, \ldots, X_n]$, ie. + \begin{align*} + R &\longrightarrow A=R/J\\ + \mathfrak{P} &\longmapsto P\\ + I(X) &\longmapsto I(X) ~\text{mod}~J + \end{align*} + + $\Longrightarrow~ \text{thus}~ J \subseteq \mathfrak{P}$. + + + Now, $J \susbset \mathfrak{P} = I(X)$;\\ + by \ref{5.5}, the set of points where $\mathfrak{P}$ vanishes must be inside the set of points where $J$ vanishes, so + $$V(J) \supseteq V(\mathfrak{P}) = V(I(X)) = X$$ + $\Longrightarrow~~ X \subseteq V(J)$, ie. the irreducible variety $X$ must be a subvariety of $V(J)$. + + Therefore, + $$\mathfrak{P} \in Spec A \longleftrightarrow X \subseteq V(J)$$ + where $X = V(\mathfrak{P})$. +\end{proof} \newpage