It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (say, over $\mathbb C$), the category of coherent sheaves of $\mathcal O\_X$-modules is abelian.


Now, a rigid analytic space $X$ inherits a Grothendieck topology and we can similarly define coherent sheaves of $\mathcal O\_X$-modules with respect to this. So I would like to ask



> 
> **Question:** Do coherent sheaves on rigid analytic spaces form an abelian category?
> 
> 
> 


In the first place, we know from [BGR's book](https://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf) that taking kernels and images will preserve coherence in this setting. So the answer might be positive. But the axioms of abelian categories are too abstract for me, and I have no idea about how to prove or disprove this.



Do coherent sheaves on rigid analytic spaces form an abelian category?
The answer is yes.


An abelian category is a category $\mathcal C$ with the following properties:


1. $\mathcal C$ is additive.
2. $\mathcal C$ has kernels and cokernels.
3. Images and coimages coincide. That is, for every morphism $f$ in $\mathcal C$, the canonical map $coker(ker(f)) \to ker(coker(f))$ is an isomorphism.


(1) is obvious in our case. (2) follows because $Coh(X) \subseteq Sh(X)$ is a full subcategory closed under kernels and cokernels, and $Sh(X)$ has all kernels and cokernels. (3) is then automatic: because this is an isomorphism in $Sh(X)$ and $Coh(X)$ is a full subcategory, it is also an isomorphim in $Coh(X)$.


In conclusion: if $\mathcal C \subseteq \mathcal D$ is a full subcategory containing 0, closed under finite direct sums, and closed under kernels and cokernels, and if $\mathcal D$ is abelian, then so is $\mathcal C$.


