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In studying to write an expository paper in representation theory,
I am reading Abelian Categories with Applications to Rings and Modules
by Popescu and I have not been able to figure out something which
appears to go without saying in the book.

In Section 5.1, Theorem 1.3 (Azumaya) says that, in an AB5 category, decomposition
of an object into indecomposable objects with local endomorphism rings
is unique in the Krull-Remak-Schmidt-Azumaya sense (i.e. up to reordering
and isomorphism).

The following characterization of the condition AB5 is used in the
proof (at the moment, I do not have the original book, but only the
notes I have made while reading, so I might be using a different notation).
Let $mathscrA$ be an abelian category which has arbitrary coproducts.
Let $left A_iright _iin I$ be a set of objects of $mathscrA$
and let
$$
iota_i^I:A_itocoprod_iin IA_i
$$
denote the coprojection for $iin I$. For
any finite subset $F$ of the set $I$, let $iota_f^F$ denote
the coprojection of $A_f$ into $coprod_fin FA_f$, and let
$A_F$ denote the source of the image (that is the image itself
if an image is thought of as an object and not an arrow) of the canonical
arrow
$$
u_F:coprod_fin FA_ftocoprod_iin IA_i
$$
defined such that
$$
u_Fiota_f^F=iota_f^I
$$
for all $fin F$. If for every subobject $A$ of the object $coprod_iin IA_i$
holds the equality
$$
A=sum_Fin Tleft(Acap A_Fright),
$$
with $T$ denoting the set of all finite subsets of $I$, it is said
that $mathscrA$ verifies the condition AB5 or that it is an AB5
category.

In the same section, Theorem 1.4 says that, in an Abelian category
in which every double chain is stationary, every object has a unique
(in the Krull-Remak-Schmidt-Azumaya sense) decomposition into finitely many
indecomposable objects with local endomorphism rings.

By double chain, one means
$$
left(A_nrightleftarrowsA_n+1right)_ninmathbbN,
$$
with $A_n$ being objects, $i_n:A_n+1to A_n$ being monomophisms and $p_n:A_nto A_n+1$
being epimorphisms. The chain is stationary if there is $n_0inmathbbN$
such that $i_n$ and $p_n$ are isomorphisms for all $ngeq n_0$.

For the proof of the uniqueness part of Theorem 1.4, it is said in
the book that Theorem 1.3 is used. No proof is given, though.

My question: How does one prove that the condition for using Theorem
1.3 is satisfied in Theorem 1.4, namely that a category described
in Theorem 1.4 indeed verifies the aforementioned condition AB5 needed
for Theorem 1.3?

Thank you for your time and attention!

The point seems to be that the double chain condition is strictly stronger than AB5. More precisely, I claim that

An abelian category satisfying the double chain stabilizing condition admits only finite coproducts.

For such categories, note that AB5 is trivially satisfied since then, with the above notation, $I in T$ and thus, $A=sum_F in T(A cap A_F)$ trivially holds.

To show the claim, suppose there is an infinite collection of nonzero objects $A_i_i in I$ such that $coprod_i in IA_i$ exists. Fix a countable subset $i_n ;$ of $I$.

Set $B_0=coprod_i in IA_i$, and let $B_1=mathrmcoker(iota_i_1:A_i_1 rightarrowcoprod_i in IA_i)$. This is just $coprod_i in I setminusi_1 A_i$
(when someone brings morphisms $A_i rightarrow X, ; i neq i_1,$ add the zero morphism $A_i_1 rightarrow X$ to get a morphism $coprod_i in IA_i rightarrow X$ from the universal property, and this in turn factorizes through a map $B_1 rightarrow X$; the point of this awkwad description is to make sure that we do not use any coproducts that might not exist). Now it's easy to see that $B_0= A_0 oplus B_1,$ so one has the canonical maps $B_0 rightleftarrows B_1$ from the biproduct description. Now repeat the process with $B_1$ to get $B_1 rightleftarrows B_2=coprod_i in I setminusi_1, i_2A_i$, and so on.

In the end, one has a double chain $(B_n)_n$ that clearly does not stabilize (kernel of each $B_n rightarrow B_n+1$ is $A_i_n+1 neq 0$). So this contradicts the double chain stabilizing assumption.