Say, we have a topological space $X$ and a sheaf $\mathcal{F}$ over a sheaf of rings $\mathcal{A}$ on $X$. The sheaf $\mathcal{F}$ is said to be coherent, if the following two conditions are satisfied:
i) $\mathcal{F}$ is of finite type
ii) If $s_1,...s_p$ are sections of $\mathcal{F}$ over an open $U \subset X$, then the sheaf of relations between the $s_i$ is of finite type.
Now I want to prove that any coherent sheaf is isomorphic to the cokernel of a homomorphism $\phi: \mathcal{A}^q \to \mathcal{A}^p$.
My ideas:
For a point $x \in X$, we find an open neighbourhood $x \in U \subset X$, where $\mathcal{F}$ is generated by sections $s_1, ..., s_p$.
So on $U$, every $s \in \Gamma(U,\mathcal{F})$ is of the form $\sum \limits_{i = 1}^{n} a_i s_i$ for $a_i \in \Gamma(U, \mathcal{A})$.
By identifying the tuple $(a_1,...,a_p) \in \mathcal{A}^p$ with $\sum \limits_{i = 1}^{n} a_i s_i$, it seems that we have to quotient by some relations. But I am not sure how exactly to define the $\phi:\mathcal{A}^q \to \mathcal{A}^p$ here.