Definition of the structure sheaf on $\text{Spec} A$

Question

In his book Algebraic Geometry, Hartshorne defines the structure sheaf of $\text{Spec} A$ to be the set of functions $s:U\to\coprod_{p\in U}A_p$ such that $s(p)\in A_p$ and $s$ is locally a quotient of elements of $A$, i.e. for every $p\in U$, there exists an open neighborhood of $p$ (say $V$) such that for all $q\in V$, we have $s(q)=a/f$ where $f\notin q$. I am not sure I understand this last part. If $s(p)\in A_p$ for all $p\in U$, then doesn't that mean that $s(q)\in A_q$ and thus automatically looks like $a/f$, with $f\notin q$? Perhaps I am not understanding what is meant by $s(q)$?

Answer 1

The question was answered by Arthur in the comments. Here is a more formal definition of the structure sheaf which should make it more clear:

$$\mathcal{O}(U) = \left\{s \in \prod_{\mathfrak{p} \in U} A_{\mathfrak{p}} : s \text{ is locally consistent}\right\},$$

where "$s$ is locally consistent" is defined by

$$\forall \mathfrak{p} \in U ~ \exists a,f \in A ~ \forall \mathfrak{q} \in U \cap D(f).~ s(\mathfrak{q})=a/f$$ An alternative definition is $$\mathcal{O}(U) = \varprojlim_{D(f) \subseteq U} \, A[f^{-1}].$$