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Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<\cdots<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

  1. Is the endomorphism ring of a uniserial object local?

  2. Assume $A$ has the property that every indecomposable object is uniserial. Does there exist a projective-injective object $M$ such that for every projective object $P$ there is a monomorphism $P \rightarrow M$?

  3. Is there a name for such $A$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

  4. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

  5. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

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  • $\begingroup$ Are you missing a hypothesis for Q2? $\endgroup$ – Jeremy Rickard Apr 13 at 20:35
  • $\begingroup$ @JeremyRickard Im not sure. The question is motivated by the fact that it is true for Nakayama algebras. Maybe one should say that $A$ has enough projetive and injectives but maybe this follows automatically. $\endgroup$ – Mare Apr 13 at 21:07
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    $\begingroup$ I meant that it doesn’t mention anything about uniserial objects. I’m guessing that, as in questions 3-5, you want every object of the category to be uniserial? But even then, there are trivial examples, such as the category of $\mathbb{Z}$-graded vector spaces with finite total dimension, where there are infinitely many isomorphism classes of projective-injective objects. $\endgroup$ – Jeremy Rickard Apr 14 at 2:09
  • $\begingroup$ @JeremyRickard Thanks, I added the assumption on A. For me any example that is not the module category of a finite dimensional (connected!) algebra is non-trivial. So your example gives a counterexample to 2, since you would need an infinite direct sum (with infinite dimension) to embed all projectives. $\endgroup$ – Mare Apr 14 at 8:40
  • $\begingroup$ @JeremyRickard In question 2 I wanted to ask more or less whether A has dominant dimension at least one always (since Nakayama algebras have that property). But I probably choose the wrong generalisation of dominant dimension at least one. Maybe a better question 2 is: Does every projective embed into a projective-injective? $\endgroup$ – Mare Apr 14 at 8:44
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Every finite length indecomposable object in an abelian category has local endomorphism ring (the proof is exactly the same as for modules), and a uniserial object must be indecomposable, so the answer to question 1 is “yes”.

Here’s an example that addresses some of the other questions.

Let $A$ be the category of representations, with finite total dimension, of the infinite quiver $$\dots\to\bullet\to\bullet\to\bullet$$

Then every object is a representation of some finite subquiver, and so every indecomposable object is uniserial.

There are enough projective objects, but no nonzero injectives, so not every projective embeds in a projective-injective, answering question 2 (even the alternative version in comments).

Also, $A$ has global dimension equal to one, answering the last part of question 4.

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