I'm studying blow-ups in connection with an introduction course in algebraic geometry. I've some problems with the details in the below set-up, which my textbook introduces in order to define the blow up of an affine variety $X$ in $(f_{0},\dots, f_{r})$.
The set-up:
Let $X\subseteq \mathbb{A}^{n}$ be an affine variety and let $f_{0}, \dots,f_{r}\in k[x_{1},\dots, x_{n}]$ be polynomial functions that do not vanish identically on $X$. Then $U = X\setminus Z(f)$ is a non empty open subset of $X$, and there is a well-defined morphism:
$f: U \rightarrow \mathbb{P}^{r}, P \mapsto (f_{0}(P):...:f_{r}(P))$
Now consider the graph $\Gamma = \lbrace (p,f(p)) \vert p \in U \rbrace \subseteq X \times \mathbb{P}^{r}$ which is isomorphic to $U$.
Problem:
From the book morphisms are defined to be a map between ringed spaces. I know that $U$ is a ringed space, because it is an open subset of an affine variety. But I can't really see why $\Gamma$ is a ringed space and thus why we (in the first place) can define a morphism between $U$ and $\Gamma$. Someone who can clarify this?
I don't think it's really necessary to talk about sheaves here, since $\Gamma$ is a subset of a variety ($X\times\mathbb{P}^r$) cut out by some equations ($f_i y_j = f_j y_i$), so it belongs to the category of varieties rather naturally.
But in general, any subset of a locally ringed space acquires the structure of a locally ringed space via pullback along the inclusion.