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Can somebody give me an example of a subcategory of an abelian category which is also an abelian category, but not an abelian subcategory (which means some kernels or cokernels are different from the initial ones).

flag	asked Aug 21 2011 at 23:32 Victor 428●2●8

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The category of presheaves of abelian groups on a space $X$ is abelian, as is the subcategory of sheaves. But the inclusion is not generally right exact, so the cokernels can differ. – Donu Arapura Aug 21 2011 at 23:39

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@Donu, why don't you make that an answer? :) – Gjergji Zaimi Aug 22 2011 at 0:28

Thank you, Donu. Is there a simpler example? – Victor Aug 22 2011 at 0:51

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Take any nontrivial abelian category A and any object a not isomorphic to the zero object. Then the subcategory consisting of the object a together with its identity map is abelian, but not an abelian subcategory. – Tom Leinster Aug 22 2011 at 1:50

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Also, the question states that the only way in which a subcategory can fail to be a sub-Abelian-category is if it has different (co)kernels. But this isn't so: another way it can fail is if it has different (co)products or a different zero object (as in my example). – Tom Leinster Aug 22 2011 at 1:53

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Dag Oskar Madsen · Answer 1 · 2011-10-24 22:24:39Z UTC

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Here is a small example of a full subcategory that is abelian, but not an abelian subcategory:

Let $k$ be a field and let $R_n$ denote the ring of upper triangular $n \times n$-matrices over $k$. Then $mod (R_3)$ is an abelian category with $6$ indecomposable objects (up to isomorphism). There is a unique indecomposable object $P$ of length $3$. It has a simple socle $s(P)$ and a simple top $t(P)$. Consider the full additive subcategory with indecomposable objects $P$, $s(P)$, and $t(P)$. This category is equivalent to $mod (R_2)$ and therefore abelian. It is not an abelian subcategory since the lengths do not add up in $mod (R_3)$.

edited Oct 24 2011 at 22:24

answered Oct 24 2011 at 21:06

Dag Oskar Madsen
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	Welcome to MO! :) – Mariano Suárez-Alvarez Oct 24 2011 at 22:28
	Thanks, I guess I am not supposed to ask how you are doing in the comments, so I won't do that :) – Dag Oskar Madsen Oct 24 2011 at 22:37
	@Dag. Do you consider the category of finitely generated R_n modules? Could you give a more explicit description of the indecomposable representations of R_n? It must be classical, do you know a good reference? – Victor Oct 26 2011 at 1:18
	@Victor. Yes, it is the category of finitely generated $R_n$-modules, but I don't think it matters in this example. Honestly, I was considering the equivalent category of repreresentations of the quiver $A_n$ with "linear orientation". Here is an online reference: arXiv:math/0505082v1. Hope this helps. – Dag Oskar Madsen Oct 26 2011 at 8:25

Abelian subcategory

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