Let $X$ be a Noetherian irreducible scheme of dimension $n$. Let $Y,Z$ be its closed irreducible subschemes of dimensions $k,l$ respectively.
Under what technical conditions the dimension of each irreducible component of $Y\cap Z$ is at least $k+l-n$?
Say is that true for any regular (variety over a field) $X$ ?
Remark. In Propositions I.7.1 and I.7.2 in Hartshorne's book this property is proven in the cases when $X$ is either affine space $\mathbb{A}^n$ or projective space $\mathbb{P}^n$ over a field.