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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).

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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 '11 at 23:39
@Donu, why don't you make that an answer? :) –  Gjergji Zaimi Aug 22 '11 at 0:28
Thank you, Donu. Is there a simpler example? –  Victor Aug 22 '11 at 0:51
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 '11 at 1:50
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 '11 at 1:53
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1 Answer

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 $mod (R_3)$.

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Welcome to MO! :) –  Mariano Suárez-Alvarez Oct 24 '11 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 '11 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 '11 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 '11 at 8:25
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