My question comes from an example in Hartshorne (Example V.2.11.4) which I'm having trouble following. It is claimed that the Blow up of a point $p \in \mathbb{P}^n$ is isomorphic to $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$.
The specific part which I'm having trouble justifying is:
On the other hand, if $\mathcal{E} = (\mathcal{O} \oplus \mathcal{O}(1))$ on $\mathbb{P}^n$, then $\mathbb{P}(\mathcal{E})$ is defined as $\operatorname{Proj} S(\mathcal{E})$, where $S(\mathcal{E})$ is the symmetric algebra of $\mathcal{E}$. Now $\mathcal{E}$ is generated by the global sections $1$ of $\mathcal{O}$ and $y_1, ... ,y_{n+ 1}$ of $\mathcal{O}(1)$. Therefore $S(\mathcal{E})$ is a quotient of the polynomial algebra $\mathcal{O}[x_0,...,x_{n+ 1}]$ by the mapping $x_0 \to 1$, $x_i \to y_i$ for $i = 1,... ,n + 1.$ The kernel of this map is the ideal generated by all $x_iy_j - x_jy_i$, $i =1, ... ,n + 1$. Therefore $\mathbb{P}(\mathcal{E})$ is isomorphic to the subscheme of $\mathbb{P}^n \times \mathbb{P}^{n+ 1}$ defined by these equations, which is the same as the variety $V \subseteq \mathbb{P}^n \times \mathbb{P}^{n+ 1}$ defined above.
Specifically, how does Hartshorne conclude that the kernel of this map is generated by all $x_iy_j - x_jy_i$, $i =1, ... ,n + 1$? This seems to be the crux of my issue, as I am interpreting the $x_i$'s and $y_j$'s as living in two different rings. Thanks in advance.
I think you can understand this locally. $y_i$ are global sections of $\mathcal{O}(1)$, and we have that $\mathcal{O}(D(y_k)) = k\left[\frac{y_1}{y_i},\ldots,\frac{y_{n+1}}{y_k}\right]$ where $y_j|_{D(y_k)}$ is identified with $\frac{y_j}{y_k}$.
We have a surjection of graded $\mathcal{O}$-algebras $\mathcal{O}[x_0,\ldots,x_{n+1}] \rightarrow S(\mathcal{E})$ given by mapping $x_0\mapsto 1$, $x_i \mapsto y_i$.
The kernel of this map is going to be a subsheaf of $\mathcal{O}[x_0,\ldots,x_{n+1}]$, where on the open $D(y_k)$ is given by the ideal generated by $\left(x_i\frac{y_j}{y_k} - x_j \frac{y_i}{y_k}\right) \subset \mathcal{O}(D(y_k))[x_0,\ldots,x_{n+1}]$.