Skip to content
Permalink
Browse files

Jordan-Holder

  • Loading branch information...
aisejohan committed May 30, 2019
1 parent 299d9d1 commit 94cbee6173564a2568416d7a569710edd6aecfc4
Showing with 132 additions and 5 deletions.
  1. +132 −5 homology.tex
@@ -1767,6 +1767,135 @@ \section{Localization}




\section{Jordan-H\"older}
\label{section-jordan-holder}

\noindent
The Jordan-H\"older lemma is Lemma \ref{lemma-jordan-holder}.
First we state some definitions.

\begin{definition}
\label{definition-simple}
Let $\mathcal{A}$ be an abelian category. An object $A$ of $\mathcal{A}$
is said to be {\it simple} if it is nonzero and the only subobjects
of $A$ are $0$ and $A$.
\end{definition}

\begin{definition}
\label{definition-Artinian}
Let $\mathcal{A}$ be an abelian category.
\begin{enumerate}
\item We say an object $A$ of $\mathcal{A}$ is {\it Artinian} if and only if
it satisfies the descending chain condition for subobjects.
\item We say $\mathcal{A}$ is {\it Artinian} if every object of
$\mathcal{A}$ is Artinian.
\end{enumerate}
\end{definition}

\begin{definition}
\label{definition-Noetherian}
Let $\mathcal{A}$ be an abelian category.
\begin{enumerate}
\item We say an object $A$ of $\mathcal{A}$ is {\it Noetherian} if and only if
it satisfies the ascending chain condition for subobjects.
\item We say $\mathcal{A}$ is {\it Noetherian} if every object of
$\mathcal{A}$ is Noetherian.
\end{enumerate}
\end{definition}

\begin{lemma}
\label{lemma-ses-artinian}
Let $\mathcal{A}$ be an abelian category. Let $0 \to A_1 \to A_2 \to A_3 \to 0$
be a short exact sequence of $\mathcal{A}$. Then $A_2$ is Artinian
if and only if $A_1$ and $A_3$ are Artinian.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{lemma}
\label{lemma-ses-noetherian}
Let $\mathcal{A}$ be an abelian category. Let $0 \to A_1 \to A_2 \to A_3 \to 0$
be a short exact sequence of $\mathcal{A}$. Then $A_2$ is Noetherian
if and only if $A_1$ and $A_3$ are Noetherian.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{lemma}
\label{lemma-finite-length}
Let $\mathcal{A}$ be an abelian category. Let $A$ be an object
of $\mathcal{A}$. The following are equivalent
\begin{enumerate}
\item $A$ is Artinian and Noetherian, and
\item there exists a filtration
$0 \subset A_1 \subset A_2 \subset \ldots \subset A_n = A$
by subobjects such that $A_i/A_{i - 1}$ is simple for $i = 1, \ldots, n$.
\end{enumerate}
\end{lemma}

\begin{proof}
Assume (1). If $A$ is zero, then (2) holds. If $A$ is not zero, then
there exists a smallest nonzero object $A_1 \subset A$ by the Artinian
property. Of course $A_1$ is simple. If $A_1 = A$, then we are done.
If not, then we can find $A_1 \subset A_2 \subset A$ minimal
with $A_2 \not = A_1$. Then $A_2/A_1$ is simple. Continuing in this way, we
can find a sequence $0 \subset A_1 \subset A_2 \subset \ldots $
of subobjects of $A$ such that $A_i/A_{i - 1}$ is simple. Since $A$
is Noetherian, we conclude that the process stops. Hence (2) follows.

\medskip\noindent
Assume (2). We will prove (1) by induction on $n$. If $n = 1$, then
$A$ is simple and clearly Noetherian and Artinian. If the result holds
for $n - 1$, then we use the short exact sequence
$0 \to A_{n - 1} \to A_n \to A_n/A_{n - 1} \to 0$
and Lemmas \ref{lemma-ses-artinian} and \ref{lemma-ses-noetherian}
to conclude for $n$.
\end{proof}

\begin{lemma}[Jordan-H\"older]
\label{lemma-jordan-holder}
Let $\mathcal{A}$ be an abelian category. Let $A$ be an object
of $\mathcal{A}$ satisfying the equivalent conditions of
Lemma \ref{lemma-finite-length}. Given two filtrations
$$
0 \subset A_1 \subset A_2 \subset \ldots \subset A_n = A
\quad\text{and}\quad
0 \subset B_1 \subset B_2 \subset \ldots \subset B_m = A
$$
with $S_i = A_i/A_{i - 1}$ and $T_j = B_j/B_{j - 1}$ simple objects we have
$n = m$ and there exists a permutation $\sigma$ of $\{1, \ldots, n\}$
such that $S_i \cong T_{\sigma(i)}$ for all $i \in \{1, \ldots, n\}$.
\end{lemma}

\begin{proof}
Let $j$ be the smallest index such that $A_1 \subset B_j$.
Then the map $S_1 = A_1 \to B_j/B_{j - 1} = T_j$ is an isomorphism.
Moreover, the object $A/A_1 = A_n/A_1 = B_m/A_1$
has the two filtrations
$$
0 \subset A_2/A_1 \subset A_3/A_1 \subset \ldots \subset A_n/A_1
$$
and
$$
0 \subset (B_1 + A_1)/A_1 \subset \ldots \subset
(B_{j - 1} + A_1)/A_1 = B_j/A_1 \subset B_{j + 1}/A_1
\subset \ldots \subset B_m/A_1
$$
We conclude by induction.
\end{proof}








\section{Serre subcategories}
\label{section-serre-subcategories}

@@ -2115,11 +2244,8 @@ \section{K-groups}
This means that there exist
\begin{enumerate}
\item a finite set $I = I^{+} \amalg I^{-}$,
\item for each $i \in I$ a short exact sequence
$$
0 \to A_i \to B_i \to C_i \to 0
$$
in the abelian category $\mathcal{A}$
\item for $i \in I$ a short exact sequence $0 \to A_i \to B_i \to C_i \to 0$
in $\mathcal{A}$
\end{enumerate}
such that
$$
@@ -2222,6 +2348,7 @@ \section{K-groups}




\section{Cohomological delta-functors}
\label{section-cohomological-delta-functor}

0 comments on commit 94cbee6

Please sign in to comment.
You can’t perform that action at this time.