From bd0d2543e7a83c3527ede69e30b839428aa4ad78 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Fri, 2 Jan 2026 19:32:10 +0100 Subject: [PATCH] add Noether normalization proof --- commutative-algebra-notes.pdf | Bin 345113 -> 351227 bytes commutative-algebra-notes.tex | 83 ++++++++++++++++++++++++++++++++-- galois-theory-notes.pdf | Bin 346493 -> 346500 bytes galois-theory-notes.tex | 4 +- 4 files changed, 81 insertions(+), 6 deletions(-) diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 5c31fddce64381fa71bab3ef8c248fcadba743eb..034d87c4ac69ab1f910a95e88277ac440492d145 100644 GIT binary patch delta 116001 zcmZs?Q;;Q0w1ru=ZQHi(F00G7(ba`h>atJSwr$(CZJYDoJM%U-A|rF>Yd-Awa_zO! z!Z1HtFi|L#C8QWxnKS&fbl4_fGur#*9|U|?hmcoMKN|Nae>#~ zxu%0<>#MWIbA{p<`+(q%}h*%GReM!FCwm#tM7;WBIO+`ezzF_#pHbw<)l z{RaHv0ATTzJ$GArcM#))|AP#5crRlEkE$wE-e`fI>-v1_aHZwNg)9x*DqyArH79B$2Db3S=ah8qBr5OO2UaK4Up!STD@TV^2Eh-t4H&~S%zmzAHp0ocr{0lKWqKB(G&^?h8~MVch< zCkeiQ!SOnkJoK_fj8&8hq?Yz!%o{d2P$qU|>*Al; 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zTv$>-3M~XKP5S)3tu_i9LhkQhq;mbdc`#!j9?WU#30U_$w9-sY?H2B0*NQwijv$bT zLJd~wxcev0=71@O?Nh1jRk$`$lp~P8fbiS$u1(ES4>!=8A@C{nQ=3vDiX^^zA(n)$ zvOb(9#BU1cn2<0}H>P!lnMJbHKD79?$76rOB3DTt-@^R9aU%p=*aGaFw^DD8(o$q@ zh$cNR4|!O(*7h!k8rU-44ZHdHE=5h}=rAzRI=A;lXZ9 zo^F9}^Q?dqgg!L5%RDcqfVvNsQ85VbJFIt&+$J$Y2D!Kaq}Af*wgT1zX#Cr(88EWrIk=rg7%{ zRL)-0r}k^v`iP)bGPbVwWwRWiW{yDxX4T6EdJ&*V5ByQLPDn8nP(G=vtXC*ot(!iS z6HNW~4D1_>3Qf0pd9hT6fh+GsBJ&Z7O=*i1*BT!wD11Jet#O zH{ZfdTA;uAQEodPyxy{GZ)OB9|GKM@w>BtpA0$ we have two cases: + \begin{itemize} + \item $y_1, \ldots, y_n$ are algebraically independent over $K$, then $A \cong K[y_1, \ldots, y_n]$, so that $A$ is a finite module over itself. + \item $y_1, \ldots, y_n$ are algebraically dependent over $K$, + $$\exists 0 \neq f \in K[y_1, \ldots, y_n] ~\text{s.th}~ f(y_1, \ldots, y_n)=0$$ + \end{itemize} + + Goal: is to change variables so that $f$ becomes monic in one of the variables; this allows to express one generator as an integral element over the others. + + Following from Lemma \ref{R.4.6.L}, define new variables $y^*_1, \ldots, y^*_{n-1} \in A$ such that $y_n$ is integral over + $$A^* = K[y^*_1, \ldots, y^*_{n-1}] ~\text{and}~ A=A^*[y_n]$$ + + By inductive hypothesis on $A^*,~~ \exists~ z_1, \ldots, z_m \in A^*$ algebraically independent over $K$ and with $A^*$ finite over $B=K[z_1, \ldots, z_m]$. + + Since $y_n$ integral over $A^* ~~\Longrightarrow~ A^*[y_n]$ is finite over $A^*$.\\ + Therefore, each step of $B \subset A^* \subset A^*[y_n]=A$ is finite, and $A$ is finite over $B$ as required. +\end{proof} + + +\begin{eg}{ } + $A = K[X,Y]/(XY-1)$. $Y$ is algebraic over $K[X]$, but not integral over $K[Y]$. + + This corresponds to the fact that the hyperbola $XY=1$ has the line $X=0$ as an asymptotic line (so that its projection to the $X$-axis misses a root over $X=0$). + + Take $X' = X- \epsilon Y$ as the element of $A$ instead of $X$; then the relation becomes $(X' + \epsilon Y) Y=1$, monic in $Y$ if $\epsilon \neq 0$. + + This corresponds geometrically to tilting the hyperbola a little before projecting, so that no longer has a vertical asymtotic line. +\end{eg} + + \newpage \section{Exercises} diff --git a/galois-theory-notes.pdf b/galois-theory-notes.pdf index ebc857ad2a0126eedb5bec0490751a370b2a4f7f..1b2bc5b439fad1704104a952565a6e3e1254ef82 100644 GIT binary patch delta 3789 zcmV;;4l?omk`;uL6|mz`0yr>}acwAn)mm9^+eQ$6_pc!OL;>Mu?;}MYS|Ev=7VZ%N zeQ@(2+q9)thbt=0-|y@$Npb0=NGgieAh4uRtJ&E(zWHWIoIE5rd3%9d?>|g0UcR9^ zF~~?Nlj(gzm_ZmmN}&*CBAMPL*Xd$`CLgBn>YpXonZP(L=9>xC>8DA?7*BV9^V!a> zdijQthCpGE5CVSYB>jmg@r#08HIlSmvnZ@Umd?D~`DO+GwvY2gv3g~Hn`D9`ZDf)W zfwUE1fAz~vk#GOC`sA-7Y7mW#Qb|S$V%!{5Mj(b83JV+&$B=U>GZpp-4r51 z_qfkmj1RDw=(89ZVlmY`-Z0yr!%`eaO7QJ49Dtk`)o%>0$dZN6B|9`Y9!bBy;Jn2n_C*nt=X8|{N1g$$Ps z=eZT2e=~I$tEChlw}0gmLDJjZ;@>)RN6Z36h}O_mo>U?Mln&ku(cRf>lTYx3 zB3T9BnR<*oDp0Z-`CgPoD-Mzv0#<*`mJB$%otDX@r+F}ch~wHie}i>snr~wSl7r%8 zOe7ABXi~ZapzSt~LA%t*P#(4@9ln8+Sm%&l=cwbI!<=)}Am@l7qn-n$!NNJG!&&=J zHIpNThJ(eE4g@jG 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