Given an algebraic variety $W\subset \mathbb{k}^n$, with $\mathbb{k}$ any field, one can consider the coordinate ring $K[W]=\mathbb{k}[x_1,...,x_n]/I(W)$. I am wondering whether there is a geometric interpretation for the Krull dimension of this ring.
Can we say something about $W$ (geometrically) knowing the Krull dimension of $K[W]$?
Thank you for your insights.