From c0484e630494765ceeb0a38e55df96af77403615 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 1 Feb 2026 17:12:22 +0100 Subject: [PATCH] add extended proof of lemma 6.2 --- commutative-algebra-notes.pdf | Bin 401886 -> 419998 bytes commutative-algebra-notes.tex | 120 +++++++++++++++++++++++++++++++++- 2 files changed, 118 insertions(+), 2 deletions(-) diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 499f599593cb5ee6135f17f04dfee1c07a530c4c..7565d94baaa88c3e2c1743fd35f418697068ccd2 100644 GIT binary patch delta 248795 zcmV)QK(xQ!!x)~?84)E=L`E$!E;R}z(A34)3NbU2Q4bRYF*G+blW|ojkx&eOg=S0! zH-hA&o79cd+S5tYHdDtBkOavb^Wu_J66fo?%LQpkaV$HTPMp4oxGccM*>e^v@?aJ4 z;Fl+S&-170lcz^22}BY}5$j;O3?xq@lLbb^k;-H+T?A+BSzVSBY1q26UH%SRZW1ak z*PJt)0D62KE2@brHDfPiy$V&~86$`%V)on;KM%vcS-DXTBZ z7X!&VDwymI;C$M%jD@;sV^S}S%mNe!zT8tyI}t#Kb*GtXPs9^gdAj%j#V7e$Tk zzo$OhqyK!{bKAE0swx40MtE@tZYX$Bk++>g_q1fTqp>puZuG3U<@|rNYg-_EN{*qx zX5;K)B3K^`>~Ej1r9nah8dvtJpK*NwZIe6`vskPIqc(qO4_G!;f

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zfBu9W(RWv_d)vLEzSii`J!8TBdj&rJ_7<#vd35uhr>Fl1wph+Ilbmu&Bg4(nmW~rxTGjG jF&D(jPs-vdE=epZsVGWK<1)6iv@|p0QdM>JcjE#8KN%#V diff --git a/commutative-algebra-notes.tex b/commutative-algebra-notes.tex index 60eb50b..ffdcfeb 100644 --- a/commutative-algebra-notes.tex +++ b/commutative-algebra-notes.tex @@ -117,8 +117,12 @@ $x \in A$ such that $xy=1$ for some $y \in A$. ie. element \emph{which divides 1}. \end{defn} +\begin{cor}{1.8} \label{1.8} + $A = A^{\times} \sqcup \bigcup m$ (where $\sqcup$ denotes ``disjoint union"), ie. $f \in A$ is either a unit or it is contained in a maximal ideal, not both. +\end{cor} + \begin{defn}{}[zerodivisor] - $x \in A$ such that $\exists 0 \neq y \in A$ such that $xy=0 \in A$. ie. $x$ \emph{divides 0}.. + $x \in A$ such that $\exists~ 0 \neq y \in A$ such that $xy=0 \in A$. ie. $x$ \emph{divides 0}.. If a ring does not have zerodivisors is an integral domain. @@ -172,10 +176,15 @@ \end{defn} \begin{defn}{}[local ring] - A \emph{local ring} has a unique maximal ideal. + A ring is \emph{local} if it has a unique maximal ideal. Notation: local ring $A$, its maximal ideal $\mM$, residue field $K=A/\mM$: $$A \supset \mM ~\text{or}~ (A, \mM) ~\text{or}~ (A, \mM, K)$$ + + \vspace{0.3cm} + By Corollary \ref{1.8}, $A$ is local\\ + \hspace*{4em}$\Longleftrightarrow$ $A$ has only one maximal ideal.\\ + \hspace*{4em}$\Longleftrightarrow$ all the nonunits of $A$ form an ideal. \end{defn} \subsection{Z and K[X], two Principal Ideal Domains} @@ -1693,6 +1702,113 @@ Note: for $k$ a field, $k[X_1, \ldots, X_n]$, $m$ maximal ideal; the residue fie where $X = V(\mathfrak{P})$. \end{proof} + +\section{Rings of fractions $S^{-1}A$ and localization} + +\begin{defn}{6.1}[ring of fractions] + let $A$ a ring, $S \subset A$ a multiplicative set ($1 \in S$, and $st \in S + ~\forall s, t \in S$). + + Introduce th following relation $\sim$ on $A \times S$: + $$(a, s) \sim (b, t) \Longleftrightarrow \exist y \in S ~\text{such that}~ u(at - bs) = 0$$ + + (write $a/s$ for the equivalence class of $(a, s)$. + + Then, the \emph{ring of fractions of $A$ with respect to $S$} is + $$S^{-1}A = (A \times S)/\sim$$ + + with ring op'ns defined by the usua arithmetic op'ns on fractions: + $$\frac{a}{s} \pm \frac{b}{t} = \frac{(at \pm bs)}{st} ~~\text{and}~~ \frac{a}{s} \cdot \frac{b}{t} = \frac{ab}{st}$$ +\end{defn} + +\begin{eg}{ } + TODO +\end{eg} + +\begin{lemma}{6.2} \label{6.2} + For $f \in A$, write $S = \{ 1, f, f^2, \ldots \}$, and $A_f \cong A[X]/(Xf-1)$. + + Then + $$A_f \cong \frac{A[X]}{(Xf-1)}$$ +\end{lemma} +\begin{proof} + Define the homomorphism + \begin{align*} + \psi: A[X] &\longrightarrow A_f\\ + a &\longmapsto a/1\\ + X &\longmapsto 1/f + \end{align*} + +By the 1st isomorphism theorem: + + \begin{tikzpicture}[node distance=1.5cm, auto] + \node (G) {$A[X]$}; + \node (H) [right of=G] {$A_f$}; + \node (GmodK) [below of=G, xshift=0.75cm] {$A[X]/\ker(\psi)$}; + + \draw[->] (G) to node {$\psi$} (H); + \draw[->] (G) to node [swap] {$\phi$} (GmodK); + \draw[->] (GmodK) to node [swap] {$\eta$} (H); + \end{tikzpicture} + + Thus we want to prove that $ker(\psi) = (Xf-1)$, so that $\frac{A[X]}{ker(\psi)} = \frac{A[X]}{(Xf-1)}$, and the lemma is proven. + + \vspace{0.3cm} + First, observe that $\psi(Xf-1) = \psi(X)\psi(f)-\psi(1) = \frac{1}{f} \frac{f}{1} - 1 = 1-1=0$, + so $Xf-1 \in ker(\psi)$, ie. $(Xf-1) \subseteq ker(\psi)$. + + Now, we want to prove that $ker(\psi) \subseteq (Xf-1)$. + + Take $h \in ker(\psi)$, will prove that $h \in (Xf-1)~ \foracll~ h \in ker(\psi)$, and thus $ker(\psi) \subseteq(Xf-1)$. + + \vspace{0.3cm} + Want to prove that $h(X)$ is a multiple of $(Xf-1)$. + + Let + $$h(X)=a_n X^n + a_{n-1} X^{n-1} + \ldots + a_1 X + a_0$$ + + % Since $h \in ker(\psi) ~~\Longrightarrow~~ \psi(h)=0$, so + % $$\psi(h) = \frac{a_n}{f^n} + \frac{a_{n-1}}{f^{n-1}} + \ldots \frac{a_1}{f} + \frac{a_0}{1} = 0 \in A_f$$ + + multiply $h(X)$ by $f^n$: + $$f^n \cdot h(X)=a_n (f^n X^n) + a_{n-1} f (f^{n-1} X^{n-1}) + a_{n-2} f^2 (f^{n-2} X^{n-2}) \ldots$$ + + Note that since $\forall~ i \geq 1,~~ f^i X^i = (Xf -1)\cdot (f^{i-1} X^{i-1} + f^{i-2} X^{i-2} + \ldots + 1)$, then $f^i X^i \equiv 1 \pmod{Xf-1}$. + + So, + $$f^n \cdot h(X)=\underbrace{a_n (1) + a_{n-1} f (1) + a_{n-2} f^2 (1) + \ldots + a_0 f^n}_{C~~\text{(constant)}} \pmod{Xf-1}$$ + + $$\Longrightarrow~~ f^n \cdot h(X)=C \pmod{Xf-1}$$ + $$\Longleftrightarrow~~ f^n \cdot h(X)=Q(X) \cdot (Xf-1) + C$$ + + Want to remove $C$, but it is non-zero. 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} + + + \newpage \section{Exercises}