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zqjB%)*WO~k{_nk)<=-79WMGO2(50*5%Z6~QK_A>=FcqDYfpNR>JQ!z`WQ8tDg9Zy8 z9>$LN7ezuT{AF{7r}}GZ4b(8xsQs1plyPwM~VO1?Dw)bT=XmE7oRa) z)i+YIm`PQ~7Cm1`$(%h24qH`yIDHN{h8*hs&!qtEe)qwDk-uwJ+%k}v=^}E6$uWqC JAk6d`{sY)p^zHxv diff --git a/commutative-algebra-notes.tex b/commutative-algebra-notes.tex index 1b35444..afa4e2f 100644 --- a/commutative-algebra-notes.tex +++ b/commutative-algebra-notes.tex @@ -2531,6 +2531,148 @@ $$\{ 1, X \} \times \{ 1,Y, Y^2 \} = \{ 1, Y, Y^2, X, XY, XY^2 \}$$ \end{proof} + +\subsection{Exercises Chapter 6} + +\begin{ex}{R.6.3.a} + Let $A = A' \times A''$; prove that $A'$ and $A''$ are rings of fractions of $A$. +\end{ex} +\begin{proof} + ring of fractions of $A$ = $S^{-1}A$, so want to prove that $S^{-1}A \cong A'$. + + Localization map + \begin{align*} + \psi: A &\longrightarrow A'\\ + (a', a'') &\longmapsto a' + \end{align*} + + note that $\psi$ is surjective, since $\forall~ a' \in A',~ \psi(a', 0)=a'$. + + Let the multiplicative set $S$ be $S=\{ e_1 \}$ with $e_1 = (1,0)$. + + Want $\underbrace{S^{-1}A}_{\text{localization}} \cong A'$. + + In $S^{-1}A$, $x \in A$ maps to $0$ iff $s x = 0$ for some $s \in S$. + + Let $x=(a', a'')\in A$; then $sx=0 ~\Longrightarrow~ s \cdot (a', a'') =0$, + + with $s \in S$, so $s=(1,0)$, hence + $$(1,0)\cdot(a',a'')=(0,0)$$ + $$\Longrightarrow~ (a', 0) = (0, 0)$$ + \hspace*{4em} which implies $a' = 0$. + + $\Longrightarrow$ the elements that become zero in the localization are of the form $(0, a'')$ + + $$ker(\psi) = \{ (a',a'')\in A ~|~ \psi(a', a'')=0 \} = \{(0, a'') ~|~ a'' \in A'' \}$$ + + By the 1st isomorphism theorem, + + \begin{tikzpicture}[node distance=1.5cm, auto] + \node (G) {$A$}; + \node (H) [right of=G] {$A'$}; + \node (GmodK) [below of=G, xshift=0.75cm] {$\frac{A}{ker(\psi)}$}; + + \draw[->] (G) to node {$\psi$} (H); + \draw[->] (G) to node [swap] {$\phi$} (GmodK); + \draw[<->] (GmodK) to node [swap] {$\eta$} (H); + \end{tikzpicture} + + $\Longrightarrow~ \frac{A}{ker(\psi)} \cong A'$ + + + \vspace{0.5cm} + Take the localization map + \begin{align*} + \phi: A &\longrightarrow S^{-1}A\\ + a &\longmapsto a/1 + \end{align*} + + $$ker(\phi) = \{ x \in A ~|~ sx=0 ~\text{for some}~ s \in S \}$$ + + and we've seen that $a'=0$, so $x \in A ~\Rightarrow~ x=(a', a'') = (0, a'')$ + + $$ker(\phi) = \{ (0, a'') ~|~ a'' \in A'' \}$$ + + which is the same as $ker(\psi)$; $ker(\psi) = ker(\phi)$. + + \vspace{0.2cm} + $\Longrightarrow~~ A'$ and $S^{-1}A$ are both surjective images of $A$ with exact same kernel, + thus $A' \cong S^{-1}A$. +\end{proof} + +\begin{ex}{R.6.4} + \begin{enumerate}[a.] + \item Give an example of a ring $A$ and distinct multiplicative sets $S,~ T$ such that $S^{-1}A = T^{-1}A$. + \item Prove that for fixed $S$, there is a maximal multiplicative set $T$ with this property defined by + $$T = \{ t \in A ~|~ at \in S ~\text{for some}~ a \in A \}$$ + \end{enumerate} +\end{ex} +\begin{proof} + \begin{enumerate}[a.] + \item + (\emph{saturated sets})\\ + + General rule of saturation:\\ + two multiplicative sets $S,~T$ yield the same localization, $S^{-1}A = T^{-1}A$, iff the have the same saturation. + + Saturation of $S$: $\hat{S}$, set of all elements in $A$ that divide some element of $S$: + $$\hat{S} = \{ a \in A ~|~ \exist~ b \in A ~\text{s.th.}~ ab \in S \}$$ + + \vspace{0.4cm} + Example 1: $A=\mathbb{Z}$,\\ + + $$S = \{ 2^n ~|~ n \geq 0 \} = \{ 1, 2, 4, 8, 16, \ldots \}$$ + localization: + $$S^{-1} \mathbb{Z} = \{ \frac{a}{2^n} ~|~ a \in \mathbb{Z},~ n \in \mathbb{N} \}$$ + + $$T = \{ \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \ldots \}$$ + + Notice that despite $S \neq T$, we have $S^{-1}\mathbb{Z} = T^{-1} \mathbb{Z}$,\\ + since once $n \in \mathbb{Z}$ is invertible, $-n$ is automatically invertible: $(-n)^{-1} = - (n^{-1})$. + + \vspace{0.4cm} + Example 2: $A= k[x]$, + $$S = \{ x^n ~|~ n \geq 0 \}$$ + $$T = \{ (2x)^n ~|~ n \geq 0 \}$$ + $$\Longrightarrow~ S^{-1}A = T^{-1} A = k[x, x^{-1}]$$ + + \item + Show that $T$ is a multiplicative set: + + (must contain $1$ and be closed under multiplication)\\ + Since $1 \in A$ and $1 \cdot 1 = 1 \in S$, then $1 \in T$. + + Let $t_1,~ t_2 \in T$; by definition of $T$, + $$\exists~ a_1, a_2 \in A ~\text{such that}~ a_1 t_1,~ a_2 t_2 \in S$$ + + Since $S$ a multiplicative set, $(a_1 t_1)(a_2 t_2) \in S$\\ + rearrange it: $(a_1 a_2)(t_1 t_2) \in S$.\\ + Since $a_1 a_2 \in A,~~ t_1 t_2 \in T$. + + Thus $T$ is a multiplicative set. + + Next, we show that $T$ is maximal: + + Suppose $U$ is the maximal instead of $T$, such that $U^{-1}A \cong S^{-1}A$. + + Let $u \in U$; the image of $u$ in $S^{-1}A$ must be a unit. + + Units in $S^{-1}A$ are the fractions $\frac{s}{a}$ such that their inverse exists.\\ + \hspace*{2em} $\exist~ x=\frac{a}{s}$ such that $u \frac{a}{s} = \frac{1}{1}$\\ + \hspace*{2em} $\Longrightarrow~ \frac{ua}{s} = \frac{1}{1} ~\Longrightarrow~ \exist s' \in S$ such that $s'(ua-s)=0$\\ + $\Longrightarrow~ s'ua=s's \in S$ + + since $s,s' \in S$, their product is also in $S$. + + Let $b=s'a$, then $ub \in S$. + + By definition of $T$, $u \in T$. + + Therefore, $U \subseteq T$; thus $T$ is maximal. +\end{enumerate} +\end{proof} + + \bibliographystyle{unsrt} \bibliography{commutative-algebra-notes.bib}