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Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.

The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.

Question: Can we write down a complete list of isomorphism classes (up to translation) of indecomposable objects of $\mathcal C$?

It is easy to see that the number of such isomorphism classes is countably infinite.

Here are some indecomposable objects: $$ \cdots\to 0\to\mathbb Z\overset{m}{\to}\mathbb Z\to\mathbb Z/m\to 0\to\cdots, $$

$$ \cdots\to 0\to\mathbb Z_{(m,n)}\to\mathbb Z_n\overset{m}{\to}\mathbb Z_n\to\mathbb Z _{(m,n)}\to 0\to\cdots, $$ where $m$ is a natural number, $n$ is a prime power and $(m,n)$ denotes the greatest common divisor.

But there is more; for instance, if $p$ is prime then the indecomposable object $$ \cdots\to0\to\mathbb Z_p\to\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\cdots\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\to\mathbb Z_p\to 0\to\cdots $$ can have any finite "length."

If this classification problem has been solved, a reference would be great. Otherwise I would very much appreciate any idea/hint towards a general solution.

(I've added the noncommutative-algebra tag because chain complexes can be considered as modules over a certain non-commutative ring. The question I am asking is a sub-problem of classifying all finitely generated indecomposables for this ring.)

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.

The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.

Question: Can we write down a complete list of isomorphism classes (up to translation) of indecomposable objects of $\mathcal C$?

It is easy to see that the number of such isomorphism classes is countably infinite.

Here are some indecomposable objects: $$ \cdots\to 0\to\mathbb Z\overset{m}{\to}\mathbb Z\to\mathbb Z_m\to 0\to\cdots, $$

$$ \cdots\to 0\to\mathbb Z_{(m,n)}\to\mathbb Z_n\overset{m}{\to}\mathbb Z_n\to\mathbb Z _{(m,n)}\to 0\to\cdots, $$ where $m$ is a natural number, $n$ is a prime power and $(m,n)$ denotes the greatest common divisor.

But there is more; for instance, if $p$ is prime then the indecomposable object $$ \cdots\to0\to\mathbb Z_p\to\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\cdots\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\to\mathbb Z_p\to 0\to\cdots $$ can have any finite "length."

If this classification problem has been solved, a reference would be great. Otherwise I would very much appreciate any idea/hint towards a general solution.

(I've added the noncommutative-algebra tag because chain complexes can be considered as modules over a certain non-commutative ring. The question I am asking is a sub-problem of classifying all finitely generated indecomposables for this ring.)

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.

The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.

Question: Can we write down a complete list of isomorphism classes (up to translation) of indecomposable objects of $\mathcal C$?

It is easy to see that the number of such isomorphism classes is countably infinite.

Here are some indecomposable objects: $$ \cdots\to 0\to\mathbb Z\overset{m}{\to}\mathbb Z\to\mathbb Z/m\to 0\to\cdots, $$

$$ \cdots\to 0\to\mathbb Z_{?(m,n)}\to\mathbb Z_n\overset{m}{\to}\mathbb Z_n\to\mathbb Z _{(m,n)}\to 0\to\cdots, $$ where $m$ is a natural number, $n$ is a prime power, $(m,n)$ denotes the greatest common divisor and $?$ is a function I, embarrisingly, lack an explicit describtion of.

But there is more; for instance, if $p$ is prime then the indecomposable object $$ \cdots\to0\to\mathbb Z_p\to\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\cdots\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\to\mathbb Z_p\to 0\to\cdots $$ can have any finite "length."

If this classification problem has been solved, a reference would be great. Otherwise I would very much appreciate any idea/hint towards a general solution.

(I've added the noncommutative-algebra tag because chain complexes can be considered as modules over a certain non-commutative ring. The question I am asking is a sub-problem of classifying all finitely generated indecomposables for this ring.)

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.

The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.

Question: Can we write down a complete list of isomorphism classes (up to translation) of indecomposable objects of $\mathcal C$?

It is easy to see that the number of such isomorphism classes is countably infinite.

Here are some indecomposable objects: $$ \cdots\to 0\to\mathbb Z\overset{m}{\to}\mathbb Z\to\mathbb Z/m\to 0\to\cdots, $$

$$ \cdots\to 0\to\mathbb Z_{(m,n)}\to\mathbb Z_n\overset{m}{\to}\mathbb Z_n\to\mathbb Z _{(m,n)}\to 0\to\cdots, $$ where $m$ is a natural number, $n$ is a prime power and $(m,n)$ denotes the greatest common divisor.

But there is more; for instance, if $p$ is prime then the indecomposable object $$ \cdots\to0\to\mathbb Z_p\to\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\cdots\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\to\mathbb Z_p\to 0\to\cdots $$ can have any finite "length."

If this classification problem has been solved, a reference would be great. Otherwise I would very much appreciate any idea/hint towards a general solution.

(I've added the noncommutative-algebra tag because chain complexes can be considered as modules over a certain non-commutative ring. The question I am asking is a sub-problem of classifying all finitely generated indecomposables for this ring.)

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.

The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.

Question: Can we write down a complete list of isomorphism classes (up to translation) of indecomposable objects of $\mathcal C$?

It is easy to see that the number of such isomorphism classes is countably infinite.

Here are some indecomposable objects: $$ \cdots\to 0\to\mathbb Z\overset{m}{\to}\mathbb Z\to\mathbb Z/m\to 0\to\cdots, $$

$$ \cdots\to 0\to\mathbb Z_{?(m,n)}\to\mathbb Z_n\overset{m}{\to}\mathbb Z_n\to\mathbb Z _{(m,n)}\to 0\to\cdots, $$ where $m$ is a natural number, $n$ is a prime power and $(m,n)$ denotes the greatest common divisor. But there is more; for instance, if $p$ is prime then the indecomposable object $$ \cdots\to0\to\mathbb Z_p\to\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\cdots\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\to\mathbb Z_p\to 0\to\cdots $$ can have any finite "length."

If this classification problem has been solved, a reference would be great. Otherwise I would very much appreciate any idea/hint towards a general solution.

(I've added the noncommutative-algebra tag because chain complexes can be considered as modules over a certain non-commutative ring. The question I am asking is a sub-problem of classifying all finitely generated indecomposables for this ring.)

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.

The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.

Question: Can we write down a complete list of isomorphism classes (up to translation) of indecomposable objects of $\mathcal C$?

It is easy to see that the number of such isomorphism classes is countably infinite.

Here are some indecomposable objects: $$ \cdots\to 0\to\mathbb Z\overset{m}{\to}\mathbb Z\to\mathbb Z/m\to 0\to\cdots, $$

$$ \cdots\to 0\to\mathbb Z_{?(m,n)}\to\mathbb Z_n\overset{m}{\to}\mathbb Z_n\to\mathbb Z _{(m,n)}\to 0\to\cdots, $$ where $m$ is a natural number, $n$ is a prime power, $(m,n)$ denotes the greatest common divisor and $?$ is a function I, embarrisingly, lack an explicit describtion of. 

But there is more; for instance, if $p$ is prime then the indecomposable object $$ \cdots\to0\to\mathbb Z_p\to\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\cdots\overset{p}{\to}\mathbb Z_{p^2}\overset{p}{\to}\mathbb Z_{p^2}\to\mathbb Z_p\to 0\to\cdots $$ can have any finite "length."

If this classification problem has been solved, a reference would be great. Otherwise I would very much appreciate any idea/hint towards a general solution.

(I've added the noncommutative-algebra tag because chain complexes can be considered as modules over a certain non-commutative ring. The question I am asking is a sub-problem of classifying all finitely generated indecomposables for this ring.)