Gluing schemes to get $x_0^2+x_1^2-x_2^2=0$ Exercise 4.5.A from Vakil asks to think through how to define a scheme that should be
interpreted as $x_0^2+x_1^2-x_2^2=0$ "in $\mathbb P^2_k$". According to the hint, I need to consider the following three affine schemes (here $x_{0/2}=x_0/x_2$ and other indices are defined similarly), assuming I didn't mess up anything:

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*$X_0=\operatorname{Spec} k [x_{1/0}, x_{2/0}]/(x_{2/0}^2-x_{1/0}^2-1)$


*$X_1=\operatorname{Spec} k [x_{0/1},x_{2/1}]/(x_{2/1}^2-x_{0/1}^2-1)$


*$X_2=\operatorname{Spec} k [x_{0/2},x_{1/2}]/(x_{0/2}^2+x_{1/2}^2-1)$
From what I understand, to say how we glue them, we need to define the following:

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*open subschemes $X_{ij}\subset X_i$ such that $X_{ii}=X_i$


*isomorphisms $f_{ij}:X_{ij}\to X_{ji}$ such that $f_{ii}$ is the identity
and then prove that

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*$C.\text{   }f_{ik}|_{X_{ij}\cap X_{ik}}=f_{jk}|_{X_{ji}\cap X_{jk}}\circ f_{ij}|_{X_{ij}\cap X_{ik}}$
The first question is whether what I wrote above is correct ("ideologically" at least), and the second question is what should $X_{ij}$ and $f_{ij}$ be?
 A: Question: @hm2020 But to check the cocycle condition, we need to have an explicit definition of the map fij, not just the fact that it exists, right? Thats' why I was trying to get a simple explicit definition of fij. (I guess I could try to define fij by using the universal property of quotient rings and also the universal property of localization, but it seems so bulky and complicated...) And when we consider intersections like Xij∩Xik should I just consider it as the intersection of two quotient rings, or is such an intersection isomorphic to something more simple?
Answer: If you look at the intersection $D(x)\cap D(y) \cap D(z)$ you get the ring $C:=k[ \frac{ x}{y },\frac{z }{y },\frac{y }{x },\frac{z }{x },\frac{x }{z },\frac{y }{z }]$ and the ideals $(f^x), (f^y)$ and $(f^z)$ are equal in this ring. Let $f^x:=1+(\frac{y}{x})^2-(\frac{z}{x})^2, f^y:=(\frac{x}{y})^2+1-(\frac{z}{y})^2, f^z:=(\frac{x}{z})^2+(\frac{y}{z})^2-1$. It follows $(\frac{x}{y})^2f^x=f^y, (\frac{x}{z})^2f^x=f^z$ and $(\frac{y}{z})^2f^y=f^z$ and since the elements $\frac{z}{y}$ etc are units you get an equality of ideals. Hence when passing to the quotient rings you get canonical isomorphisms
$$C/(f^x) \cong C/(f^y) \cong C/(f^z)$$
of $k$-algebras. This proves that the cocycle condition $C$ is fulfilled.
Example: If $S:=k[x,y,z]/(x^2+y^2-z^2)$, it follows $S$ is a graded ring and if $S_{(x)}$ is the sub ring of degree zero elements in $S_x$  you get
$$S_{(x)} \cong k[\frac{y}{x}, \frac{z}{x}]/(f^x),$$
$$S_{(y)} \cong k[\frac{x}{y}, \frac{z}{y}]/(f^y)$$
and
$$A/(f^x) \cong (S_{(x)})_y  \cong S_{(xy)} \cong (S_{(y)})_x \cong A/(f^y)$$
You may consider the rings $(S_{(x)})_z \cong (S_{(z)})_x \cong S_{(xz)}$ and so on. All these localizations will be canonically isomorphic since they are all constructed using $S$. It may seem like a "mystery" that you get such isomorphisms, but when you realize you are localizing one fixed graded ring this is no longer such a "mystery".
Note: The Proj-construction constructs a locally ringed space $C:=Proj(S)$, and $C$ has an open cover $D(x),D(y),D(z)$ where $D(x) \cong Spec(k[\frac{y}{x},\frac{z}{x}]/(f^x)$ etc, and "by construction" $C$ is isomorphic to the scheme you construct in your post. The Vakil exercise constructs $C$ via glueing. There is a general construction of $Proj(S)$ for any $\mathbb{N}$-graded ring $S$. When you glue the schemes $X_i$ in the exercise it is not immediate that you end up with a projective curve $C \subseteq \mathbb{P}^2_k$.
Question: "Exercise 4.5.A from Vakil asks to think through how to define a scheme that should be interpreted as $F:=x^2_0+x^2_1−x^2_2=0$ in $P^2_k$"."
Note: A subscheme $X \subseteq \mathbb{P}^n_k$ is a closed subscheme of projective n-space iff $X$ has a linebundle with certain properties.
One exercise could be to give a direct construction of invertible sheaves $L_x,L_y,L_z$ on the various rings corresponding to the open sets $D(x),D(y),D(z)$, and to glue to an invertible sheaf $L$ giving a closed immersion into $\mathbb{P}^n_k$. Then you must look up this property in Vakils book (This is HH.Prop.II.7.2). Hence the exercise wants you to study the relation between maps to projective space and invertible sheaves: There is a "1-1" correspondence between maps $\phi: X \rightarrow \mathbb{P}^n_k$ of $k$-schemes and (equivalence classes of) surjections $\phi': \mathcal{O}_X^{n+1} \rightarrow \mathcal{L} \rightarrow 0$ where $\mathcal{L}\in Pic(X)$.
"Solve little problems on a napkin while sipping coffee with a friend and draw doodles on that napkin."
