From d2fad9d1776dd73e185421a211c643c76d43dd60 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sat, 17 Jan 2026 19:14:27 +0100 Subject: [PATCH] 5.1, Corollary 5.2 --- commutative-algebra-notes.pdf | Bin 381425 -> 383723 bytes commutative-algebra-notes.tex | 38 +++++++++++++++++++++++++++++++++- 2 files changed, 37 insertions(+), 1 deletion(-) diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 34913138ff08f5778c751e2dfc099ef3bacb56d4..a82e2ebd97a93a25885590a7fe556bbf0bc86b1f 100644 GIT binary patch delta 54470 zcmZs?Lv)}`+qE6rwr$&X(y^U%Y@@H(wylnBqhs4nI!?#VcfXVWpFC^Tpa#2YSZmj@ z_xZ0F`=SNA-th{XmxP7H(bNV`Kmd+e!NTFEn-vKg2QN=DDKagftK+yOh4yQ;Zciu_ zj3c)<-s*9WAqf(#f2yN8pIddDyx(`Qkl9RvzInQZvk{FzKS}$X%ClwnLG4@;(#= z?c3D)SFiCI(<@u|6Bahh_S}&+6=hMmmDW4tc zW+L4_>yEXB1;$k$&ynZNy2O)S!h>s3X#GYrVWsZ9bj?#?fXi)lF)BUgfOrpwAy zhH>GwpvysMkD4*U%ja77ly+qHc=%N6`w`bA;KiF43};s-{kkSs3$$$)0W?{h*tI`$ zrWLmK_hEEzb*}J+sXgM+84t&YHX53*HmQyi)o&reTm~<`-!mSPueC2_55-k}Ju}8o zC+V=19QDBZ&ThB=CSt54*^~!b%^lr|3tIE;^b=Dnv#I;3OJ;hTb4&DCNDw-Cmn0C! z%StWx_t&3KUKQBk2=f6PXF0+&MN~;0{zWo9!6Qe{OymIM$pfq*Ph1+9EfIV~biE>q z=bHwV@bQb#qGouet@ARFuFvb@OVZt=uIh1yYuY@PLZ=>NFqB{vm6vliNVGgvW*dp_@aGq0Tv2J zV^pGqvww*L3P-HuTy=SEn$A=@CHYdte2Q3{-}(b7ofpO&teK6+yN%3$^WD5Jbt8M@EM(WWfTJVA=>sfexdRlfg@JnTw%mlfWo`$ z8FcnC1YtFlvqL~SLb{&xdNi0azJMFLc;+OOJL^tUmVgH0>;_lJPaW{V@HBr(@(gFF zmE0ksvn3)(JY!@9;c&qq8SA)aihqf;G~E!~*aU0IE?+xUrmoqeq!>$gxSN07I|1*A zpe6}JvY!&Wvyq4we$jN+XzTK`DEBt7e3ds2tzGV|bV3%G#wo;=nKJY#MKZrQ(2}EN z+L>`eFXilE44E2X158R!!9)qrVNRm|4B$*d>}~J;)_!u5Q1$HIpB&ZW7q98?@e$>y>!gC#;h1O?(~nF#po*a}+7ut4X z#D&ZX62w6sllOa|{ln%igpW$O;%79nyF#9dL7s>niGvrL;X5x<3romq5)5#6@UIwJ zr{bW1fR!{TCeup~XW%f2hwlJ_Io!HTN&1yLceIVydL1D}@{vcR>#FBlGQ)Luq>7T- zf@UY0NAp3*CE_QPgNt9B6*y?_n~9dMdo@yWoQj}uPmV1W3tZ}mhIxEcF!s4<4?ejU z0=rlnqyp!w;L?KgokOXY%{@>oyF#$qk`YN_;NmXV_)C?~N63)sOZe4_inp1u$vMud z68i!sx%hzcgs;HK_F0e`0lA*UmHn$=ivUe;C@*68iT*xj*8EcLHQ`4i67EZra?+-I z;BQoD&kSl(&Nu8LL}j?~UiK-zYt?Q?Uy|~QyY0?3a}7f>)MX605PqOfq6OkN-X(W} zj{(8E8RN{=A73hw1OJ#KH5nv3RV5NU|56}S$X@NdbiUUbA0{}nx>^)MbJ_U{rW?bE z?^W7>N!wy7*0Fm^@nq2&X^JR#wWNVhL2YoDUB)EBB9Pk0*M^3(QYq5ExQA}1_6{B> zu9n5WYt!qt8IB`^Oa(%oGXK(1)q=AmN=IIp9$0?;)N&2wQ{St5ADzA``e#rB8Y4Xn zlHX%RUlr!fF+Kj{FDb;^6ec-NU8^(I;Hg#R#r>H^_x#9f!{3O<9`>_cE0YV)toDcv&5hr%$Bat|D2XtWG7ssc!<_z6gY%rgMi_;wYV*YMd#wJB{J zcY2}spO9XOkr${SmCdFtX04u&{`Zg9Fj9TYE5|cH&%tlrriu4FYRs}=@GRRQ{4IMh z^ivU^gX=1X3o|6Cx!lcDQTJ#KJCzG5df;rL&@41hd33FdDrSS4Ljau133s0pq;qTP zkzcaUepfrmO&Bpoo=N!d}*Z{Jsn5cIwf{mC?o1*rQt1yFuRI?%m zG48PhHF?x03Wq)~TJ9T9c0SFCGS1p21Kk&|Pb#BX}4jTOc zLo3atY-Wig9<-4lLp7L^=??}m3$yY*d(X{fx=m1x1@)4s+i`Xg@VYx(1@+c%7ShWA z8YzFu#=$ffgWMxe5p4pLZQGBk_dKgZ;+bp&y*^;_tzgvGeeV5V#X&vtgyqJZ>5Kh9 zexSd;@w-h^6FkF{ER zS4FPaUWEH2;wQihllRsX{8QMO->R=Zz9rIb28D6zrvsC=BjCDtO-9Y19JKazkF!=v z=m&hU>!+G1ljyhd%ps&ww>X=TcJGO#auOk|7}|6(`_)fdg81)@PSG0MKk83cE_S~F z=bDQiy^*;NDBJ=ApnXFl-)PJeJe&ZXbKL!?Y%L{unWE`HLwK4_FvRzkTMzc3cKq6*v%J+1?w zFrtgO$*9Zd$(Iip;H+%C$t3?-ykl*7*F!FZ-gm9~@`=TWeQ=7A>!W)9MK?>|br1hF z>!e^)`ber@=Jw@^U;d)WniSB#O1pXEY>H%a3mQ&mIy-2c7$k2CotMyqY0&TqitRSd ze%GgEV?>qv?=u5 zR+84JH0G;f&qpWtgxDB#?M<=eMiC)c`nWGdu%BWS>hK6rd_sjS?{D|R72 z*|MK4S}za#9wsiSY8hQorWutgy1gd{I9|VF*UX&|@csyFW`8{YL>G$&b5JiXHFPs| zrKM0m`~epMHZ9tP0ib1Tx91_z!*OHbOAJwANJ9VdG?>)pwa^bbKC|a6EeNS8r$GvF zGmLYry6`SuhEdm@p0wbMYO6JitJsr0V(UVF;|oGX3K076{p(qC74{{+m`rU_$ECNE z2r9{h{^v3zYw)#b4ZKCHvPY$9rwy%FEOjL=nkA(f9w34n{#yA2qMPaQqjr?7Dh z2CdN#4^o2Uufux$+fZ~`0cPM(@SY;!Rz9txkR~;&nhQf#8g!gSzpo|&Gs}n_DvHEq zNUVFJd?fHw(5*mi(d3-uwA>#ZC)GShrtl2g_MqP@aET-ROK0X{#Q*bvSO{Iv z54?)XNi?hTIzz3QZ{IYV&v{F`B_)r7M%m|fw|hH}zkAFRNT4NH!e}eAq7oB*2`&k4 zs&amR4%xfsp9Cw{Cgm2PZ#n)m@F#(VzYt1WAUgu;0aEg6Oy)#!_=&E6&XP-#4Zpj@ zS~;-n8u-DCZoMzznO!#FgQrzJ@N#)^Xs1*D!+F}a>bH%&8L0cxFjr*x=W%CXDJaJW zZ8ANki_Ih`OcPD2d!XVDk?fC%>wfH9W;n-MX#v~6{)(6z#8jftp zl#xBt$(Ih7nYnAp%z?8l!=ecl>HYpA1CL_9i-1!E8QJ3Rz@BnZpO1&wa4l5w&;j1g zzeK5hv=Xf-gK;GxJmc4a!1r_6O+4tl;LY2ni?s~@>(AaoGK3NkBzlP})j)4IQrt}qlGkAGY`6!c)AeHR zM%+*F3aRTxo;8d$$}7G{$WPI!MYy9uhk;W8!DPB^+r!e&U)4Nq4 zyEAB^E=IpSxnWcXV1Bk#G5Nh1d>q88tXK9Uf<9@)h6vW%_kWw=fXGRMtKXjh#hP04?9v`>RhjX5Y%Bq2>n_N_AAMvjX%^S|a(An}q7Bq7;lFk97cv z$|&v`Ro`F4FE|TrYBKjmEPK-~X#C#F{XYpLiCQcMl!=NoKApAkVI4%1BAjk?6aJqn zr3y&xrvjf$ei8L>l_Lc#LwH458pQxI-IZ3bS%RQ@`onuFa2t?FUt7@qI^_b^R#v3w z-+d*2erOLgZP`6jP7S?rRcW3cJaO$FXFnqQQVD{CeX0ohu|%FKyO-1{fHtJ!qv{

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zsk#v|u`iMx@7B8fsC?da%cHtx<`Isnyx~7_+wgUGoO}TECL?wwqSt>7k$Y0IyK+Ii z?`$Ymg+VoNet63-sdu14obl^m?5SUG$h#N92(J^R_KP({<4W*JTFt0%8>Js;rlA~JX2^d+bXYNwVEtQLz-D`_E4h=z|8d|; zJ^qnQTkZS0(q+Uk0{_8)>aDNcv>Tcl?eEm!cF9C15HW2A-@#tBrQ%4Z2!G! f$^X|uk3jT|B*sPt;z0;B2Bob7Qcy6n$AbO^m%b0e diff --git a/commutative-algebra-notes.tex b/commutative-algebra-notes.tex index 503301d..29771d4 100644 --- a/commutative-algebra-notes.tex +++ b/commutative-algebra-notes.tex @@ -1338,7 +1338,7 @@ Recall: a $K$-algebra $A$ is fingen over $K$ if $A=K[y_1, \ldots, y_n]$ for some thus there exists inverse in $A$, so $A$ is a field too. \end{proof} -\begin{thm}{R.4.10}[Weak Nullstellensatz - Zariski's lemma] +\begin{thm}{R.4.10}[Weak Nullstellensatz - Zariski's lemma] \label{zariski} let $k$ a field, $K$ a $k$-algebra which \begin{enumerate} \item is finitely generated as a $k$-algebra @@ -1372,6 +1372,42 @@ Recall: a $K$-algebra $A$ is fingen over $K$ if $A=K[y_1, \ldots, y_n]$ for some \end{proof} +\vspace{1cm} + +\section{Nullstellensatz} + +Note: for $k$ a field, $k[X_1, \ldots, X_n]$, $m$ maximal ideal; the residue field $K=k[X_1, \ldots, X_n]/m$ satisfies the Zariski's lemma (\ref{zariski}), thus $K$ is a finite algebraic extension of $k$. + +\vspace{0.3cm} +\begin{cor}{5.2} \label{5.2} + $k$ algebraically closed. Then every maximal ideal of $A = k[X_1, \ldots, + X_n]$ is of the form + $$m = (X_1 - a_1, \ldots, X_n -a_n),~~ a_i \in k$$ + + The map $k[X_1, \ldots, X_n] \longrightarrow k[X_1, \ldots, X_n]/m=k$ is the natural evaluation map $f(X_1, \ldots, X_n) \longmapsto f(a_1, \ldots, a_n)$. + + Thus + \begin{align*} + k^n &\longleftrightarrow m-Spec A\\ + (a_1, \ldots, a_n) &\longleftrightarrow f(a_1, \ldots, a_n) + \end{align*} +\end{cor} +\begin{proof} + let $m \subset k[X_1, \ldots, X_n]$ be a maximal ideal. + + By fundamental property of maximal ideals, $K=A/m$ is a field. + + Since $A$ is a fingen $k$-algebra (generated by $X_1, \ldots, X_n$), then $K=A/m$ is also a fingen $k$-algebra, generated by residues $x_i' = x_i +m$. + + By Zariski's lemma (\ref{zariski}), $K=A/m$ is algeraic over $k$. + + Since by hypothesis $k$ is algebraically closed, it has no proper algebraic extensions\\ + \hspace*{2em} $\Longrightarrow~~ K=k~~ \Longrightarrow~~ k \cong A/m$. + + So, $\forall x_i \in k$, its image in the quotient field $A/m$ must be an element of $k$. + $$\Longrightarrow x_i'=a_i \in k, ~\forall i \in [n]$$ + $$\Longrightarrow x_i - a_i \in m$$ +\end{proof} \newpage