# stacks/stacks-project

Jordan-Holder

 @@ -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}