From 6df4b3793920f2bd24826c245e1f1c9a606e3c9a Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 25 Jan 2026 12:59:52 +0100 Subject: [PATCH] add missing conclusion at proposition 5.3 --- .github/workflows/typos.toml | 1 + commutative-algebra-notes.pdf | Bin 401173 -> 401886 bytes commutative-algebra-notes.tex | 11 +++++++++++ 3 files changed, 12 insertions(+) diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml index bee3b93..0455301 100644 --- a/.github/workflows/typos.toml +++ b/.github/workflows/typos.toml @@ -3,6 +3,7 @@ # run: typos -c .github/workflows/typos.toml [default.extend-words] +ieal = "ideal" iddeal = "ideal" iddeals = "ideals" allpha = "alpha" diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 24fa6afd2c15e1309582fa3e3ba623b5b8e802d6..499f599593cb5ee6135f17f04dfee1c07a530c4c 100644 GIT binary patch delta 36054 zcmV)NK)1h@z8K!a7_ip`0XLITK`4?2d;wXLeFjbyG2;ogz!U7d@PwT41Yh6@{#|%N 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zYlmVb-m9_XChN%^>D9m1p2gBCs7pvSRRpZ-ft~$s+(W@WXBVEeI*t38^of$%#@PaX giz=jv`-h&1{EOmGuhE3=7B|6|i0kSSo$bZ{1MOs}YXATM diff --git a/commutative-algebra-notes.tex b/commutative-algebra-notes.tex index f83804c..60eb50b 100644 --- a/commutative-algebra-notes.tex +++ b/commutative-algebra-notes.tex @@ -1486,6 +1486,17 @@ Note: for $k$ a field, $k[X_1, \ldots, X_n]$, $m$ maximal ideal; the residue fie If $J \subseteq m'$, then $\forall~ f \in J$ must vanish at $P$. By definition, the set of points where all polynomials in $J$ vanish is the \emph{variety}, $V(J)$. + + \vspace{0.4cm} + Thus,\\ + every maximal ideal in $A$ corresponds to a point $(a_1, \ldots, a_n) \in k^n$, ie. + $$m-Spec A \longleftrightarrow k^n$$ + + The condition that the ideal belongs to the quotient ring $A=k[X_1, \ldots, X_n]/J$ forces that point to lie in $V(J)$, so + \begin{align*} + m-Spec A &\longleftrightarrow V(J)\\ + \text{maximal spectrum} &\longleftrightarrow \text{variety} + \end{align*} \end{proof} \begin{prop}{5.5}[Correspondeces $V$ and $I$] \label{5.5}