Improvements to section on local rings

Thanks to Daniel Apsley
https://stacks.math.columbia.edu/tag/07BH#comment-8512

CONTRIBUTORS

@@ -19,6 +19,7 @@ Konstantin Ardakov
 Dima Arinkin
 Kenneth Asher
 Aravind Asok
+Daniel Apsley
 Clément Baillet
 Dario Balboni
 Badam Baplan

algebra.tex

@@ -3168,7 +3168,7 @@ \section{The spectrum of a ring}
 $$
 where the topology on the right hand side is that induced from the
 Zariski topology on $\Spec(R)$. The inverse map is given
-by $\mathfrak p \mapsto S^{-1}\mathfrak p$.
+by $\mathfrak p \mapsto S^{-1}\mathfrak p = \mathfrak p(S^{-1}R)$.
 \end{lemma}
 
 \begin{proof}
@@ -3245,77 +3245,6 @@ \section{The spectrum of a ring}
 
 
 
-\begin{remark}
-\label{remark-fundamental-diagram}
-A fundamental commutative diagram associated to a ring map
-$\varphi : R \to S$, a prime $\mathfrak q \subset S$ and
-the corresponding prime $\mathfrak p = \varphi^{-1}(\mathfrak q)$
-of $R$ is the following
-$$
-\xymatrix{
-\kappa(\mathfrak q) = S_{\mathfrak q}/{\mathfrak q}S_{\mathfrak q}
-&
-S_{\mathfrak q} \ar[l]
-&
-S \ar[r] \ar[l]
-&
-S/\mathfrak q \ar[r]
-&
-\kappa(\mathfrak q)
-\\
-\kappa(\mathfrak p) \otimes_R S =
-S_{\mathfrak p}/{\mathfrak p}S_{\mathfrak p} \ar[u]
-&
-S_{\mathfrak p} \ar[u] \ar[l]
-&
-S \ar[u] \ar[r] \ar[l]
-&
-S/\mathfrak pS \ar[u] \ar[r]
-&
-(R \setminus \mathfrak p)^{-1}S/\mathfrak pS \ar[u]
-\\
-\kappa(\mathfrak p) =
-R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p} \ar[u]
-&
-R_{\mathfrak p} \ar[u] \ar[l]
-&
-R \ar[u] \ar[r] \ar[l]
-&
-R/\mathfrak p \ar[u] \ar[r]
-&
-\kappa(\mathfrak p) \ar[u]
-}
-$$
-In this diagram the arrows in the outer left and outer right columns
-are identical. The horizontal maps induce on the associated spectra
-always a homeomorphism onto the image. The lower two rows
-of the diagram make sense without assuming $\mathfrak q$ exists.
-The lower squares induce fibre squares of topological spaces.
-This diagram shows that $\mathfrak p$ is in the image
-of the map on Spec if and only if $S \otimes_R \kappa(\mathfrak p)$
-is not the zero ring.
-\end{remark}
-
-\begin{lemma}
-\label{lemma-in-image}
-Let $\varphi : R \to S$ be a ring map. Let $\mathfrak p$
-be a prime of $R$. The following are equivalent
-\begin{enumerate}
-\item $\mathfrak p$ is in the image of
-$\Spec(S) \to \Spec(R)$,
-\item $S \otimes_R \kappa(\mathfrak p) \not = 0$,
-\item $S_{\mathfrak p}/\mathfrak p S_{\mathfrak p} \not = 0$,
-\item $(S/\mathfrak pS)_{\mathfrak p} \not = 0$, and
-\item $\mathfrak p = \varphi^{-1}(\mathfrak pS)$.
-\end{enumerate}
-\end{lemma}
-
-\begin{proof}
-We have already seen the equivalence of the first two
-in Remark \ref{remark-fundamental-diagram}. The others
-are just reformulations of this.
-\end{proof}
-
 \begin{lemma}
 \label{lemma-quasi-compact}
 \begin{slogan}
@@ -3379,10 +3308,13 @@ \section{Local rings}
 \begin{definition}
 \label{definition-local-ring}
 A {\it local ring} is a ring with exactly one maximal ideal.
-The maximal ideal is often denoted $\mathfrak m_R$ in this case.
-We often say ``let $(R, \mathfrak m, \kappa)$ be a local ring''
-to indicate that $R$ is local, $\mathfrak m$ is its unique maximal
-ideal and $\kappa = R/\mathfrak m$ is its residue field.
+If $R$ is a local ring, then the maximal ideal is often denoted
+$\mathfrak m_R$ and the field $R/\mathfrak m_R$ is called the
+{\it residue field} of the local ring $R$.
+We often say ``let $(R, \mathfrak m)$ be a local ring''
+or ``let $(R, \mathfrak m, \kappa)$ be a local ring''
+to indicate that $R$ is local, $\mathfrak m$ is its unique
+maximal ideal and $\kappa = R/\mathfrak m$ is its residue field.
 A {\it local homomorphism of local rings} is a ring map
 $\varphi : R \to S$ such that $R$ and $S$ are local rings and such
 that $\varphi(\mathfrak m_R) \subset \mathfrak m_S$.
@@ -3392,8 +3324,41 @@ \section{Local rings}
 \end{definition}
 
 \noindent
-A field is a local ring. Any ring map between fields is a local homomorphism
-of local rings.
+A field is a local ring. Any ring map between fields is a local
+homomorphism of local rings.
+
+\medskip\noindent
+The localization $R_\mathfrak p$ of a ring $R$ at a prime $\mathfrak p$
+is a local ring with maximal ideal $\mathfrak p R_\mathfrak p$. Namely, by
+Lemma \ref{lemma-spec-localization} every prime ideal of $R_\mathfrak p$
+is contained in the prime ideal $\mathfrak p R_\mathfrak p$
+(hence this is a maximal ideal and the only maximal ideal of $R_\mathfrak p$).
+The residue field of $R_\mathfrak p$ is denoted $\kappa(\mathfrak p)$;
+we call it the {\it residue field of $\mathfrak p$}; by
+Proposition \ref{proposition-localize-quotient} we may identify
+$\kappa(\mathfrak p)$ with the field of fractions of the domain $R/\mathfrak p$.
+Via the composition
+$$
+\Spec(\kappa(\mathfrak p)) \to \Spec(R_\mathfrak p) \to \Spec(R)
+$$
+the unique point of the source maps to the point $\mathfrak p$ of the target.
+
+\medskip\noindent
+Let $\varphi : R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime
+and consider the prime $\mathfrak p = \varphi^{-1}(\mathfrak q)$ of $R$.
+Since $\varphi(\mathfrak p) \subset \mathfrak q$ the induced ring map
+$$
+R_\mathfrak p \to S_\mathfrak q,\quad r/g \mapsto \varphi(r)/\varphi(g)
+$$
+is a local ring map and we obtain an induced map of residue fields
+$\kappa(\mathfrak p) \to \kappa(\mathfrak q)$.
+
+
+\begin{example}
+\label{example-not-local}
+If $R$ is a local ring and $\mathfrak p \subset R$ is a non-maximal
+prime ideal, then $R \to R_\mathfrak p$ is not a local homomorphism.
+\end{example}
 
 \begin{lemma}
 \label{lemma-characterize-local-ring}
@@ -3428,13 +3393,6 @@ \section{Local rings}
 a contradiction. Thus (4) and (1) are equivalent.
 \end{proof}
 
-\noindent
-The localization $R_\mathfrak p$ of a ring $R$ at a prime $\mathfrak p$
-is a local ring with maximal ideal $\mathfrak p R_\mathfrak p$. Namely,
-the quotient $R_\mathfrak p/\mathfrak pR_\mathfrak p$ is the fraction
-field of the domain $R/\mathfrak p$ and every element of $R_\mathfrak p$
-which is not contained in $\mathfrak pR_\mathfrak p$ is invertible.
-
 \begin{lemma}
 \label{lemma-characterize-local-ring-map}
 Let $\varphi : R \to S$ be a ring map. Assume $R$ and $S$ are local rings.
@@ -3457,10 +3415,110 @@ \section{Local rings}
 contrapositive of (2) by Lemma \ref{lemma-characterize-local-ring}.
 \end{proof}
 
-\noindent
-Let $\varphi : R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime
-and set $\mathfrak p = \varphi^{-1}(\mathfrak q)$. Then the induced ring
-map $R_\mathfrak p \to S_\mathfrak q$ is a local ring map.
+\begin{remark}
+\label{remark-fundamental-diagram}
+A fundamental commutative diagram associated to a ring map
+$\varphi : R \to S$ and a prime $\mathfrak p \subset R$ is the following
+$$
+\xymatrix{
+\kappa(\mathfrak p) \otimes_R S =
+S_{\mathfrak p}/{\mathfrak p}S_{\mathfrak p}
+&
+S_{\mathfrak p} \ar[l]
+&
+S \ar[r] \ar[l]
+&
+S/\mathfrak pS \ar[r]
+&
+(R \setminus \mathfrak p)^{-1}S/\mathfrak pS
+\\
+\kappa(\mathfrak p) =
+R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p} \ar[u]
+&
+R_{\mathfrak p} \ar[u] \ar[l]
+&
+R \ar[u] \ar[r] \ar[l]
+&
+R/\mathfrak p \ar[u] \ar[r]
+&
+\kappa(\mathfrak p) \ar[u]
+}
+$$
+In this diagram the outer left and outer right columns are identical.
+On spectra the horizontal maps induce homeomorphisms
+onto their images and the squares induce fibre squares of topological spaces
+(see Lemmas \ref{lemma-spec-localization} and \ref{lemma-spec-closed}).
+This shows that $\mathfrak p$ is in the image of the map on Spec if and
+only if $S \otimes_R \kappa(\mathfrak p)$ is not the zero ring.
+If there does exist a prime $\mathfrak q \subset S$ lying over
+$\mathfrak p$, i.e., with $\mathfrak p = \varphi^{-1}(\mathfrak q)$
+then we can extend the diagram to the following diagram
+$$
+\xymatrix{
+\kappa(\mathfrak q) = S_{\mathfrak q}/{\mathfrak q}S_{\mathfrak q}
+&
+S_{\mathfrak q} \ar[l]
+&
+S \ar[r] \ar[l]
+&
+S/\mathfrak q \ar[r]
+&
+\kappa(\mathfrak q)
+\\
+\kappa(\mathfrak p) \otimes_R S =
+S_{\mathfrak p}/{\mathfrak p}S_{\mathfrak p} \ar[u]
+&
+S_{\mathfrak p} \ar[u] \ar[l]
+&
+S \ar[u] \ar[r] \ar[l]
+&
+S/\mathfrak pS \ar[u] \ar[r]
+&
+(R \setminus \mathfrak p)^{-1}S/\mathfrak pS \ar[u]
+\\
+\kappa(\mathfrak p) =
+R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p} \ar[u]
+&
+R_{\mathfrak p} \ar[u] \ar[l]
+&
+R \ar[u] \ar[r] \ar[l]
+&
+R/\mathfrak p \ar[u] \ar[r]
+&
+\kappa(\mathfrak p) \ar[u]
+}
+$$
+In this diagram it is still the case that the outer left and outer right
+columns are identical and that on spectra the horizontal maps induce
+homeomorphisms onto their image.
+\end{remark}
+
+\begin{lemma}
+\label{lemma-in-image}
+Let $\varphi : R \to S$ be a ring map. Let $\mathfrak p$
+be a prime of $R$. The following are equivalent
+\begin{enumerate}
+\item $\mathfrak p$ is in the image of
+$\Spec(S) \to \Spec(R)$,
+\item $S \otimes_R \kappa(\mathfrak p) \not = 0$,
+\item $S_{\mathfrak p}/\mathfrak p S_{\mathfrak p} \not = 0$,
+\item $(S/\mathfrak pS)_{\mathfrak p} \not = 0$, and
+\item $\mathfrak p = \varphi^{-1}(\mathfrak pS)$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+We have already seen the equivalence of the first two
+in Remark \ref{remark-fundamental-diagram}. The others
+are just reformulations of this.
+\end{proof}
+
+
+
+
+
+
+