If I take $\mathcal{A}$ = coherent sheaves on $X$\* up to isomorphism, then there are two things I could do which come to mind.


The first is noticing that $(\mathcal{A},\oplus)$ is a monoid and subsequently applying Grothendieck's $K$-functor to it obtaining a group $K(\mathcal{A})$.


The second is to take the free abelian group generated by $\mathcal{A}$ quotiented out by relations


$B = A + C$ for any exact sequence
$$0 \to A \to B \to C \to 0$$
obtaining a second group $L(\mathcal{A})$.


Are these two groups $K(\mathcal{A})$ and $L(\mathcal{A})$ isomorphic?
If not, which one is the 'correct' one?


\*I'm not sure what $X$ should stand for. I'd be happy to assume smooth and projective variety over $\mathbb{C}$, or maybe just locally noetherian scheme, or maybe just locally ringed space or etc.



How is K-theory defined for coherent sheaves?
The Grothendieck group on $X$ is defined (in your notation) as $K(\mathcal{B})$, where $\mathcal{B}$ is the monoid of f.g. locally free sheaves on $X$, (i.e. vector bundles on $X$.) The group $L(\mathcal{A})$ is usually denoted as $G(X)$, or $K'(X)$. If $X$ is smooth, the embedding $\mathcal{B} \subset \mathcal{A}$ induces an isomorphism from $K(\mathcal{B})$ to $L(\mathcal{A})$ (i.e. from $K\_0(X)$ to $G(X)$.)


Chuck Weibel's notes for his K-theory book (which can be found on his website) is a good place to find some of this information.


