I was doing Professor vakil's FOAG, in exercise 7.6B needs to interpret the rational functions on the integral scheme as a rational map to $\Bbb{A}_\Bbb{Z}^1$
There may exist some different definitions of the rational function for the integral scheme $X$, I mean here is the stalk at the (unique) generic point of $X$, say $\mathcal{O}_{X,\eta}$. Given any affine open subset $\text{Spec } A$ we can also compute it as the fractional field of $A$.
while the rational map to $\Bbb{A}_\Bbb{Z}^1$ means an equivalent class of morphism from some dense open subset of $X$ to $\Bbb{A}_\Bbb{Z}^1$. where two $U\to \Bbb{A}_{\Bbb{Z}}^1$ and $ V\to \Bbb{A}_{\Bbb{Z}}^1$ are equivalent if exist some dense open $W \subset U\cap V$, that two representative agree on $W$.
Given a representative of the rational map say $\text{Spec }A \to \Bbb{A}_{\Bbb{Z}}^1$ then it one to one corresponds to $\Bbb{Z}$- algebra homomorphism $\Bbb{Z}[t]\to A$ the set of this homomorphism is, therefore, bijective to $A$ itself. I have no idea how to proceed then, I want to get $K(A)$?
Note that we don't only consider morphisms from $X$ resp. $\operatorname{Spec}(A)$ to $\mathbb{A}^1_{\mathbb{Z}}$, but rational maps. That is how we get to the fraction field.
Let $\varphi\in\mathcal{O}_{X,\eta}$ be a rational map defined on some open set $U$. Then there is a unique $\mathbb{Z}$-algebra morphism $\mathbb{Z}[t]\to \Gamma(\mathcal{O}_U,U)$ mapping $t$ to $\varphi$. By Hartshorne Exercise II.2.4 this corresponds to a morphism $U\to\mathbb{A}^1_{\mathbb{Z}}$ via contraction.
On the other hand, given any $U\to {A}^1_{\mathbb{Z}}$ defines an element of $\Gamma(\mathcal{O}_X,U)$ by taking global sections and evaluating at $t$.
You can then check that these constructions are inverse to each other modulo equivalence of rational maps and equivalence of germs at $\eta$.