I am learning about schemes and my level is still very basic. Currently, I would, as an exercise, like to solve the following problem:
Let $X = Spec(\mathbb{Z})$: Describe the structure sheaf $\mathcal{O}_X$ of $X$ by exhibiting all restriction morphisms. Do the same for $X = Spec(\mathbb{C}[t]/(t^2-t))$ and $X = Spec(\mathbb{C}[t]/(t^3-t^2))$.
I guess these are very basic and rather simple examples but I have a lot of questions, especially about how to tackle such problems.
Here is what I am aware of: Let $A$ be a ring. For $U\subset X$ open $\mathcal{O}_X(U)$ is the projective limit of all $A_f$ s.t. $D(f)\subset U$, where $D(f)$ is the set of all $x\in X$ s.t. $f\notin p_x$ (or $f(x)\neq 0$). The restriction morphisms $\rho^V_U:\mathcal{O}_X(V)\rightarrow \mathcal{O}_X(U)$ are induced by the universal property of the projective limit, since the limits come with maps $\mathcal{O}_X(U)\rightarrow A_f$ and $\mathcal{O}_X(V)\rightarrow A_f$ for $f\in A$ with $D(f)\subset U\subset V$.(right?)
I have read several times that one might identify $\mathcal{O}_X(U)$ with regular "functions" on $U$ that locally around $x\in U$ look like $f/g$ with $g(x)\neq0$ but I don't see why.
Now back to the case where $A=\mathbb{Z}$: How do the rings $\mathcal{O}_X(U)$ look like? Or can I ignore the detailed structure of these rings and somehow find the restriction morphisms right away?
I'd be happy about any hint, comment or remark! :)