Chapter 1, Exercise 2.7: Basic properties of projective dimension

(a) $\dim\mathbb P^n=n$.

(b) If $Y\subseteq\mathbb P^n$ is a quasi-projective variety, then $\dim Y=\dim\bar Y$.
[Hint: Use (Ex. 2.6) to reduce to (1.10).]

(a) Note $I(\mathbb P^n)=(0)$, so $S(\mathbb P^n)=S$. But $\dim S=n+1$ and $\dim\mathbb P^n=\dim S(\mathbb P^n)-1=n$.

(b) Consider any nonempty component $Y_i=Y\cap U_i$: by 2.6 $\dim Y_i=\dim Y$. Further, $U_i$ is open, so $\bar Y_i=\bar Y\cap U_i$. By Proposition 1.10 and the fact that the $Y_i$ are Zariski homeomorphic to quasi-affine varieties, we find $\dim Y_i=\dim\bar Y_i$. But $\dim\bar Y_i=\dim\bar Y$ again by 2.6, so $\dim Y = \dim Y_i=\dim\bar Y_i=\dim\bar Y$.