# Homological dimension in algebraic geometry

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!

• What does $A$ stand for? – T. Amdeberhan Dec 19 '16 at 5:34
• The field $k$(this is the affine space). – Paulo Rossi Dec 19 '16 at 16:00

## 1 Answer

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?

• Thanks abx. For 2) I forgot to write that $Y$ is a subvariety of $X$. – Paulo Rossi Dec 19 '16 at 16:04