From c3527f025f4f27959cb201ac89424b69a1a1d5aa Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 8 Feb 2026 18:09:30 +0100 Subject: [PATCH] Rings of fractions & localization (#7) * add extended proof of lemma 6.2 * port rest of ch 6 notes (localization, etc) --- commutative-algebra-notes.pdf | Bin 401886 -> 444450 bytes commutative-algebra-notes.tex | 301 ++++++++++++++++++++++++++++++++-- 2 files changed, 288 insertions(+), 13 deletions(-) diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 499f599593cb5ee6135f17f04dfee1c07a530c4c..3a760edf8b066238bdc696d4cca77698f60ab3f3 100644 GIT binary patch delta 292592 zcmZs?Q*fYP^rao!w%xI9+qR94`Np{<)E|sRv|Z4O{XJwZA8bJB7Q_)CRaJvi^`|~3RJKkly<_5ue}@B zg$Bi*z^y=V3&pU%4kC6}W*uq?h5n*ojke(v zC#!mkCY340e?&67996-LEn%)Gh!o!*LDi~pW1f@aqS40}SxofEzU*+fTg$kWNl|EH z`uclmx=c%&ofz6$jzBE~J0kr978?S0jw}?0KaBQD*mN^!cojF1W0=7N?Tut$T8R z;7aASVs}0KK||K0O1ujwqmy!|zkaY)%e}O%EN*b7+HahbXC9VyPZG);s{8k0;GuVt 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Note that in $A_f$ (ring of fractions), $a\f^n = 0$ iff $\exist~ k$ such that $f^k \cdot a = 0$ in $A$. + + So, multiply both sides by $f^k$: + $$f^k \cdot f^n \cdot h(X)=f^k \cdot (Q(X) \cdot (Xf-1) + C)$$ + $$\underbrace{f^k f^n}_{f^{nk}} \cdot h(X)=\underbrace{f^k Q(X)}_{Q'(X)} \cdot (Xf-1) + \underbrace{f^k C}_{0}$$ + + $$\Longrightarrow~~ f^{n+k} \cdot h(X)=Q'(X) \cdot (Xf-1)$$ + $$\Longleftrightarrow~~ f^{n+k} \cdot h(X) \equiv 0 \pmod{Xf-1}$$ + + multiply it by $X^{n+k}$: + $$X^{n+k} \cdot f^{n+k} \cdot h(X) \equiv X^{n+k} \cdot 0 \pmod{Xf-1}$$ + $$(Xf)^{n+k} \cdot h(X) \equiv 0 \pmod{Xf-1}$$ + + Now, since we had $Xf \equiv 1$ in $\frac{A[X]}{(Xf-1)}$, + $$(1)^{n+k} \cdot h(X) \equiv 0 \pmod{Xf-1}$$ + $$\Longrightarrow~~ h(X) \equiv 0 \pmod{Xf-1}$$ + By definition this is saying $h(X) \in (Xf-1) ~\forall~ k \in ker(\psi)$. + + Thus $ker(\psi) \subseteq (Xf-1)$. + + Initially we saw that $(Xf-1) \subseteq ker(\psi)$. Therefore $ker(\psi)=(Xf-1)$. + + Hence, + $$A_f \cong \frac{A[X]}{(Xf-1)}$$ +\end{proof} + + +\vspace{0.4cm} + +Given a ring homomorphism $\psi: A \longrightarrow B$, there is a correspondence + +$$e: \{ \text{ideals of } A \} \longrightarrow \{ \text{ideals of } B \}$$ + +given by $e(I) = \psi(I)B = IB$ (called \emph{extension}), + +and + +$$r: \{ \text{ideals of } B \} \longrightarrow \{ \text{ideals of } A \}$$ + +given by $r(J) = \psi^{-1} J$ (called \emph{restriction}, written $A \cap J$). + +Set $B= S{-1}A$. Then $S^{-1}I = e(I) = \psi(I)B$. + +\begin{prop}{6.3} \label{prop.6.3} + \begin{enumerate}[a.] + \item $\forall$ ideal $J$ of $S^{-1}A$, $e(r(J))=J$ + \item $\forall$ ideal $I$ of $A$, + $$r(e(I)) = \{ a \in A ~|~ as \in I ~\text{for some}~ s \in S \}$$ + \item if $P$ prime and $P \cap S = \emptyset$,\\ + then $e(P)=S^{-1}P$ is a prime ideal of $S^{-1}A$. + \end{enumerate} +\end{prop} +\begin{proof} +\end{proof} + +\begin{cor}{6.3} \label{cor.6.3} + for an ideal $I$ of $A$, the necessary and sufficient condition for + $r(e(I))=I$ is + $$as \in I \Longrightarrow a \in I~~ \forall s \in S$$ +\end{cor} + + +\subsection{Localization} + +If $P$ prime ideal, then $S=A \setminus P$ is a multiplicative set. +Define the set $A_P=S^{-1}A$. + +\begin{prop}{6.4} \label{6.4} + $a/s \in A_P$ is a unit of $A_P ~\Longleftrightarrow~ a \not\in P$. + + Therefore $A_P$ is a \emph{local ring}, with maximal ideal $e(P)=P A_P$. + + The local ring $(A_P, PA_P)$ is called the \emph{localization} of $A$ at $P$. +\end{prop} +\begin{proof} + \begin{enumerate} + \item[$\Longleftarrow$] (if $a \not\in P ~\Longrightarrow~ a/s \in A_P$ is a unit)\\ + + if $a \not\in P$, then by definition of $S,~~ a \in S$. + + In the localization $A_P$, every element of $S$ is invertible. + + The inverse of $\frac{a}{s}$ is $\frac{s}{a}$\\ + \hspace*{2em}$\longrightarrow~~ \frac{a}{s} \cdot \frac{s}{a} = 1$, thus + $a/s \in A_P$ is a unit. + + \item[$\Longrightarrow$] (if $a/s$ unit $~\Longrightarrow~ a \not\in P$)\\ + + Suppose $a/s$ a unit; then $\exist~ \frac{b}{t} \in A_P$ such that + $\frac{a}{s} \cdot \frac{b}{t} = 1$. + + By definition of equality in localization, $\frac{ab}{st}=1$ means $\exist + u \in S$ such that + \begin{equation} + u(ab \cdot 1 - st \cdot 1) = 0~~ \Longrightarrow~ uab=ust + \tag{eq.6.4} + \end{equation} + with $u,s,t \in S$. + + Since $S=A \setminus P$ and $P$ is a prime ideal,\\ + the products of elements outside $P$ must also be outside of $P$. + $$u,s,t \not\in P ~\Longrightarrow~ ust \not\in P$$ + + At \eq{eq.6.4}, we know that $uab=ust \not\in P$, thus $uab \not\in P$. + + If $uab \not\in P$, then $a \not\in P$. + + + Next we will prove that $A_P$ is a local ring.\\ + + A ring is local if it has exactly one maimal ideal (\ref{1.13}); so we + show that the set of non-units forms an ideal. + + $a/s$ is not a unit iff $a \in P$. Let $m= \{ a/s | a \in P,~ s \not\in P \} = PA_P$. + Want to show that $PA_P$ is the unique maximal ideal: + \begin{itemize} + \item it's an ideal: let $\frac{a}{s}, \frac{b}{t} \in PA_P$ with $a,b \in P$. + + Then $\frac{a}{b} + \frac{b}{t} \in PA_P$ with numerator in $P$. + + If multiply a fraction in $PA_P$ by any fraction in $A_P$, the numerator stays in $P$, thus $PA_P$ is an ideal. + + \item every element outside of $PA_P$ is a unit (proven at the beginning of this proposition's proof ($\Longleftarrow$)). + + \item maximality: every ideal strictly larger than $PA_P$ must contain a unit.\\ + If an ideal contains a unit, it must be the entire ring $A_P$.\\ + \hspace*{2em}$\Longrightarrow$ therefore, $PA_P$ is the unique maximal ideal. + \end{itemize} + \end{enumerate} +\end{proof} + + +In other words: + +\emph{Localization}: formal way to 'force' certain elements to have inverses. + +It's the smallest ring that contains $A$ and makes all the elements of $S$ invertible. + +There is a natural map $\psi: A \longrightarrow S^{-1}A$, by $f(a)=\frac{a}{1},~ f(s)=\frac{1}{s}~~ \text{for}~ s \in S$. + + +\subsection{Localization commutes with taking quotients} +Let $A$ ring, $S$ multiplicative set, $I$ ideal. + +Write $T$ for the image of $S$ in $A/I$. + +Then $S^{-1}I=I \cdot S^{-1}A$ is an ideal of $S^{-1}A$, can take the quotient ring $S^{-1}A / S^{-1} I$. + +$\longleftrightarrow$ can take the quotient $A/I$ and then localize to get $T^{-1}(A/I)$. + + +\begin{cor}{6.7} \label{6.7} + $$T^{-1}(A/I) \cong \frac{S^{-1}A}{S^{-1}I}$$ + In particular, for $P$ prime ideal, + $$\frac{A_P}{PA_P} = k(P) = Frac(\underbrace{A/P}_{\substack{\text{integral}\\\text{domain}}})$$ + + From \ref{6.4}, $A_P$: local ring, $PA_P$: unique maximal ideal. + $k(P)$ and $Frac(A/P)$ are field of fractions. +\end{cor} +\begin{proof} + the quotient ring $A/I$ can be viewed as an $A$-module and + $$\underbrace{ T^{-1}(A/I) }_{\text{ring of fractions}} \cong + \underbrace{ \frac{S^{-1}A}{S^{-1}I} }_{\text{module of fractions}}$$ + + The \ref{6.6}.i, gives an isomorphism of modules + $$ T^{-1}(A/I) = S^{-1}(A/I) \cong \frac{S^{-1}A}{S^{-1}I}$$ + it's easy to see that this is a ring homomorphism. +\end{proof} + + + \newpage \section{Exercises}

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