From 5b80edc1c9c919bfb4c0296128a539c04443080c Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sat, 24 Jan 2026 23:38:50 +0100 Subject: [PATCH] port notes on propositions 5.3, 5.7, 5.8 --- .github/workflows/typos.toml | 1 + commutative-algebra-notes.pdf | Bin 392661 -> 401173 bytes commutative-algebra-notes.tex | 138 +++++++++++++++++++++++++++++++++- 3 files changed, 138 insertions(+), 1 deletion(-) diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml index 8effe42..bee3b93 100644 --- a/.github/workflows/typos.toml +++ b/.github/workflows/typos.toml @@ -6,6 +6,7 @@ iddeal = "ideal" iddeals = "ideals" allpha = "alpha" +fieldd = "field" # strings that are not a typo: thm = "thm" diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index bd8f076ff8a4ea7ea0eb72ff8450a2f84a3ff387..24fa6afd2c15e1309582fa3e3ba623b5b8e802d6 100644 GIT binary patch delta 76548 zcmY(KQ;;UG)~?&OZQHgrZQFKF8(-VDjcMDqZQGc(ac1woj&4#bS(Q}fCVBF%H_IQG 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k^n | f(P)=0 ~\forall~ f \in J \}$$ @@ -1445,9 +1447,45 @@ Note: for $k$ a field, $k[X_1, \ldots, X_n]$, $m$ maximal ideal; the residue fie \end{defn} \begin{prop}{5.3} \label{5.3} - TODO + $k$ an algebraically closed field, and $A=k[X_1, \ldots, X_n]$ a fingen $k$-algebra of the form $A=k[X_1, \ldots, X_n]/J$, where $J$ is an ideal of $k[X_1, \ldots, X_n]$. + (notation: $x_i = X_i \pmod J$) + + Then every maximal ideal of $A$ is of the form + $$(x_1 -a_1, \ldots, x_n - a_n)$$ + for some point $(a_1, \ldots, a_n) \in V(J)$. + + Therefore, $\exists$ a one-to-one correspondence + \begin{align*} + V(X) &\longleftrightarrow m-Spec A\\ + \text{given by}~~~(a_1, \ldots, a_n) &\longleftrightarrow (x_1 -a_1, \ldots, x_n - a_n) + \end{align*} \end{prop} \begin{proof} + the ideals of $A$ are given by ideals of $k[X_1, \ldots, X_n]$ containing $J$, since for $Q=R/I$, $\exists$ one-to-one correspondence between ideals of $Q$ and ideals of $R$ that contain $I$, ie. + \begin{align*} + m \subset A \longleftrightarrow &m' \subseteq k[X_1, \ldots, X_n]\\ + &\text{s.th.}~ J \subseteq m' + \end{align*} + + Thus, every maximal ideal of $A$ is of the form + $$\underbrace{(x_1 -a_1, \ldots, x_n - a_n)}_{i.} ~\text{such that}~ \underbrace{J \subset (X_1 - a_1, \ldots, X_n - a_n)}_{ii.}$$ + + \begin{enumerate}[i.] + \item Since $k$ is algebraically closed, the maximal ideals of $k[X_1, \ldots, X_n]$ look like $m=(X_1 - a_1, \ldots, X_n - a_n)$ for some point $(a_1, \ldots, a_n) \in k^n$. + + Which when projected to the quotient ring $A$, $X_i \longmapsto x_i$ (residue class), giving $(x_1 - a_1, \ldots, x_n - a_n)$. + + \item for $m$ to exist in $A$, the corresponding $m'$ must contain $J$; since if it didn't contain $J$ it wouldn't "survive" the quotient process. + \end{enumerate} + + \vspace{0.3cm} + However, since $(X_1 - a_1, \ldots, X_n - a_n)$ is the kernel of the evaluation map $f \longmapsto f(a_1, \ldots, a_n)$ + + $\Longrightarrow~$ means that $m'=(X_1 - a_1, \ldots, X_n - a_n)$ consists of all polynomials that vanish at point $P=(a_1, \ldots, a_n)$. + + If $J \subseteq m'$, then $\forall~ f \in J$ must vanish at $P$. + + By definition, the set of points where all polynomials in $J$ vanish is the \emph{variety}, $V(J)$. \end{proof} \begin{prop}{5.5}[Correspondeces $V$ and $I$] \label{5.5} @@ -1468,6 +1506,10 @@ Note: for $k$ a field, $k[X_1, \ldots, X_n]$, $m$ maximal ideal; the residue fie $$J \subset J' ~\Longrightarrow~ V(J) \supset V(J') ~~~~\text{and}~~~~ X \subset Y ~\Longrightarrow~ I(X) \supset I(Y)$$ \end{prop} + +\vspace{0.4cm} +\subsection{Nullstellensatz} + \vspace{0.5cm} \begin{thm}{5.6}[Nullstellensatz] \label{nullstellensatz} Let $k$ algebraically closed field. @@ -1545,6 +1587,100 @@ Note: for $k$ a field, $k[X_1, \ldots, X_n]$, $m$ maximal ideal; the residue fie \end{proof} +\vspace{0.4cm} +\subsection{Irreducible varieties} + +\begin{defn}{5.7}[Irreducible variety] + a variety $X \subset k^n$ is \emph{irreducible} if it is nonempty and not the union of two proper subvarieties; that is, if + $$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