Timeline for Categories with every indecomposable object being uniserial
Current License: CC BY-SA 4.0
13 events
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| Apr 14, 2019 at 10:15 | comment | added | Mare | @JeremyRickard Yes, probably I should add some assumption. Maybe a better definition of uniserial would also have been that the subobjects form a chain (to allow $n= \infty$) for question 1. | |
| Apr 14, 2019 at 10:08 | comment | added | Jeremy Rickard | By the way, there are abelian categories with no indecomposable objects, for which it is vacuously true that every indecomposable is uniserial. I imagine that these aren’t the kind of examples you’re interested in? | |
| Apr 14, 2019 at 9:48 | answer | added | Jeremy Rickard | timeline score: 3 | |
| Apr 14, 2019 at 8:44 | comment | added | Mare | @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? | |
| Apr 14, 2019 at 8:40 | comment | added | Mare | @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. | |
| Apr 14, 2019 at 8:37 | history | edited | Mare | CC BY-SA 4.0 |
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| Apr 14, 2019 at 5:17 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Apr 14, 2019 at 2:09 | comment | added | Jeremy Rickard | 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. | |
| Apr 13, 2019 at 21:07 | comment | added | Mare | @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. | |
| Apr 13, 2019 at 20:35 | comment | added | Jeremy Rickard | Are you missing a hypothesis for Q2? | |
| Apr 13, 2019 at 19:27 | history | edited | Mare | CC BY-SA 4.0 |
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| Apr 13, 2019 at 18:28 | history | edited | Mare | CC BY-SA 4.0 |
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| Apr 13, 2019 at 14:04 | history | asked | Mare | CC BY-SA 4.0 |