I am self studying some introductory algebraic geometry and the author of lecture notes make the following claim without explaining the reason.
Let $F\subset k[x_1,...,x_n]$, $Z(F)\subset \mathbb{A}^n$ be the zero set of $F$, $\pi=(x_1,...,x_{n-1}): Z(F)\rightarrow \mathbb{A}^{n-1}$.Then
$k[x_1,...,x_n]/I(Z(F)) = \pi^{*}(k[x_1,...,x_{n-1}])\cdot [w_n]$
where $[w_n]$ is the class of $x_n$ in $k[x_1,...,x_n]/I(Z(F))$
I have no idea how he get this. It looks right because $k[x_1,...,x_{n-1}]$ is regular on $\mathbb{A}^{n-1}$ and if I pull it back it is regular on $Z(F)$ but without containing $x_n$. But I can not quite see why RHS and LHS are equal.