The Wikipedia article on coherent sheaves makes the following claim (without any reference), which I had trouble proving or finding a reference for: on an algebraic variety *X* (or I guess possibly even on a locally noetherian scheme), the coherent sheaves can be defined as the smallest class of sheaves of $\mathcal{O}_X$-modules with the following two properties:

i) the sheaf $\mathcal{O}_X$ is itself coherent;

ii) if, in a short exact sequence of sheaves, two of the sheaves are coherent, then so is the third.

I'm skeptical, but I would still like to know if this is true. If so, does anyone know a reference?