1-Do we have some relation between the projective dimension of $k[X]$ as $K[x\_{1},..x\_{n}]$-module and the Krull dimension of the affine variety $X$ of $A^{n}$.


2- If we have to affine subvarieties $X$,$Y$ of $A^{n}$. Do we have some relation between 
$pdk[X]$ and $pdk[Y]$?
Thanks!



Homological dimension in algebraic geometry
1) Assuming for simplicity that $X$ is integral, what you can say is $\ \mathrm{pd}(K[X])\geq n-\dim(X)$. After localizing at the (prime) ideal of $X$, this is an easy case of the Auslander-Buchsbaum theorem. And, of course, $\ \mathrm{pd}(K[X])\leq n$.


2) I just don't understand the question. If $X$ and $Y$ are arbitrary subvarieties, how can you expect any relations between the projective dimensions?


