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).
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $mod (R_3)$. 

