M .github/workflows/typos.toml → .github/workflows/typos.toml

@@ -3,6 +3,7 @@ # install `typos`: https://github.com/crate-ci/typos
 # run: typos -c .github/workflows/typos.toml
 
 [default.extend-words]
+ieal = "ideal"
 iddeal = "ideal"
 iddeals = "ideals"
 allpha = "alpha"

M commutative-algebra-notes.pdf → commutative-algebra-notes.pdf

M commutative-algebra-notes.tex → commutative-algebra-notes.tex

@@ -1486,6 +1486,17 @@
   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}