Let $X$ be a Noetherian irreducible scheme of dimension $n$. Let $Y,Z$ be its closed irreducible subschemes of dimensions $k,l$ respectively. 



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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.



Basic question on dimension of intersection of subschemes
This is true for $X$ a non-singular variety, see Fulton, *Intersection Theory*, section 8.2, p. 137.


