My question concerns an argument in following former thread: How to show for $Y\subset \mathbb{P}^n$ we have $\dim C(Y)=\dim Y+1$
One considers here a problem in Hartshorne's "Algebraic Geometry" . (Chapter $1$, Section $2$, $10$th problem):
Let $Y\subset \mathbb{P}^n$ be a nonempty algebraic set, and let $\theta :\mathbb{A}^{n+1}\setminus\{(0,\ldots,0)\} \to \mathbb{P}^n$ be the map which sends the point with affine coordinates $(a_0,\ldots,a_n)$ to the point with homogeneous coordinates $(a_0,\ldots,a_n).$ We define the affine cone over $Y$ to be $C(Y)=\theta^{-1}(Y)\cup \{(0,\ldots,0)\}.$
Two qestions:
- In the thread the autor states that the statement
c) $\dim C(Y)=\dim Y+1.$
should be clear by the argument that
$$\dim C(Y)=\dim K[x_0,\ldots,x_n]/I(Y)=\dim Y+1.$$
I don't see why $\dim K[x_0,\ldots,x_n]/I(Y)=\dim Y+1.$ holds.
- What means the notation $Z_{\mathbb{P}^n}(\mathfrak{a})$ for an ideal $\mathfrak{a}$?